Chapter 1

FIXED POINTS OF HOLOMORPHIC MAPPINGS: A METRIC APPROACH

Tadeusz Kuczumow, Simeon Reich, David Shoikhet

Contents

1. Introduction

2. Preliminaries

3. The Kobayashi distance on bounded convex domains

4. The Kobayashi distance on the Hilbert ball

5. Fixed points in Banach spaces

6. Fixed points in the Hilbert ball

7. Fixed points in finite powers of the Hilbert ball

8. Isometries on the Hilbert ball and its finite powers

9. The extension problem

10. Approximating sequences in the Hilbert ball

11. Fixed points in infinite powers of the Hilbert ball

12. The Denjoy-Wolff theorem in the Hilbert ball and its powers

13. The Denjoy-Wolff theorem in Banach spaces

14. Retractions onto fixed point sets

15. Fixed points of continuous semigroups

16. Final notes and remarks References

1. Introduction

Let X_{1} and X_{2} be two complex normed linear spaces and let D_{1} be a domain (that is, a nonempty open connected subset) in X_{1}. A mapping f : D_{1} → X_{2} is said to be holomorphic in D_{1} if it is Fréchet differentiable at each point of D_{1}. If D_{1} and D_{2} are domains in X_{1} and X_{2}, respectively, then H(D_{1}, D_{2}) will denote the family of all holomorphic mappings from D_{1} into D_{2}.

A system of pseudodistances which assigns a pseudodistance to each domain in every complex normed linear space is called a Schwarz-Pick system [68] if the following two conditions hold:

(i) The pseudodistance assigned to the open unit disc Δ of the complex plane ℂ is the Poincaré distance, that is,

ρ_{Δ} (z, w) = arg tanh |\frac{z − w}{1 − z\overline{w}} | ,  z, w ∈ Δ.  (1.1)

(ii) If ρ_{D_{1}} and ρ_{D_{2}} are two pseudodistances assigned to the domains D_{1} and D_{2}, respectively, then

ρ_{D_{2}} (f(x), f(y)) ≤ ρ_{D_{1}} (x, y)  (1.2)

for all f ∈ H(D_{1}, D_{2}) and x, y ∈ D_{1}.

In particular, taking D = D_{1} = D_{2} and f ∈ H(D, D) = H(D) in (1.2) we obtain the inequality

ρ_{D} (f(x), f(y)) ≤ ρ_{D} (x, y) ,  x, y ∈ D.  (1.3)

In other words, each holomorphic self-mapping f of D is nonexpansive with respect to any pseudodistance ρ_{D} assigned to D by a Schwarz-Pick system. This fact is the basis of our metric approach to holomorphic fixed point theory.

The present exposition is by no means a complete survey of all aspects of this theory. Rather, our primary focus will only be on the following three issues: existence and uniqueness of fixed points, the behavior of iteration and approximation processes, and the structure of fixed point sets.

2. Preliminaries

We begin this section with a few notions which are of a general metric character. Let (M, d) be a metric space.

Definition 2.1 [106] .An admissible subset of M is any intersection of closed balls.

Definition 2.2 A mapping f : M → M is nonexpansive if

d (f(x), f(y)) ≤ d (x, y)

for all x, y ∈ M.

Remark 2.3 The above definition can, of course, be extended to the case of two metric spaces and a mapping from one of them to the other.

The identity operator on M will always be denoted by I. An isometry from M onto M will be denoted by I. The fixed point set of a mapping f : M → M will be denoted by Fix(f). Now we recall the following theorem.

Theorem 2.4 [106]. Let (M, d) be a metric space and let f : M → M be nonexpansive. Then any nonempty admissible subset A of M which is f-invariant contains a nonempty admissible subset A_{0} of M which is also f-invariant and in addition satisfies

diam (A_{0}) ≤ \frac{1}{2} (diam (A) + r (A)) ,

where

r (A) = inf {r (x, A) = sup {d (x, y) : y ∈ A} : x ∈ A} .

We will also need a result due to A. Całka [32] concerning the behavior of the iterates of a nonexpansive mapping on a finitely compact metric space.

Theorem 2.5 [32]. Let f be a nonexpansive mapping of a finitely compact metric space (M, d) into itself. If for some x_{0} ∈ M the sequence of iterates {f^{k}(x_{0})} contains a bounded subsequence {f^{k_{m}}(x_{0})}, then for every x ∈ M the whole sequence {f^{k}(x)} is bounded.

Remark 2.6 In his 1971 paper [104] about local isometries, W. A. Kirk asked when an isometry I of a metric space with some bounded subsequence of iterates {(I)^{k_{m}}(x_{0})} had to have bounded orbits. Całka’s theorem solves Kirk’s problem. In view of the example due to M. Edelstein ([47]); see also Section 12), the assumption that M is finitely compact seems to be quite natural.

Now we turn our attention to Banach spaces. Let (X, ∥ · ∥) be a Banach space and let ∅ ≠ D ⊂ X. We say that a mapping f : D → D is α_{∥·∥}-condensing with respect to Kuratowski’s measure of noncompactness α_{∥·∥} [137] if

α_{∥·∥} (f (C)) < α_{∥·∥} (C)

for each bounded C ⊂ D with α_{∥·∥}(C) > 0.

Theorem 2.7 [176] (see also [13] , [17] , [19] , [38] and [148]). Let D be a nonempty bounded closed convex subset of a Banach space (X, ∥ · ∥) and let α_{∥·∥} denote the Kuratowski measure of noncompactness in X. If f : D → D is a continuous and α_{∥·∥}- condensing mapping, then f has a fixed point.

Remark 2.8 [176]. Let D be a nonempty closed convex subset of a Banach space (X, ∥·∥) and let α_{∥·∥} denote the Kuratowski measure of noncompactness in X. Suppose f : D → D is a continuous and α_{∥·∥}-condensing mapping. If there exists a point x ∈ D such that the sequence of iterates {f^{k}(x)} is bounded, then α_{∥·∥}({f^{k}(x)}) = 0.

In the sequel, all Banach spaces will be complex, unless stated otherwise.

In order to formulate a characterization of holomorphic mappings, we need the concept of a norming set.

Definition 2.9 [45]Let (X, ∥ · ∥) be a Banach space and let N be a nonempty subset of its dual X^{∗}. If there exist positive constants c and C such that

sup {|l (x)| : l ∈ N , ∥l∥ ≤ C} ≥ c ∥x∥

for each x ∈ X, then we say that N is a norming set for X.

It is obvious that a norming set generates a Hausdorff linear topology σ(X, N) on X which is weaker than the weak topology σ(X, X^{∗}).

Theorem 2.10 [15] , [16] , [20] , [34] , [37] , [45] , [53] , [75] . Let (X_{1}, ∥·∥_{1}) and (X_{2}, ∥· ∥_{2}) be Banach spaces, D ⊂ X_{1} a domain in X_{1}, and let N be a norming set for (X_{2}, ∥ · ∥_{2}). For a ∈ D and x ∈ X\{0}, let D(a, x) denote the set

D (a, x) = {z ∈ ℂ : a + zx ∈ D} .

Then the mapping f : D → X_{2} is holomorphic in D if and only if f is locally bounded on D and for each a ∈ D, x ∈ X\{0} and l ∈ N, the function

l ∘ f_{|D(a,x)} : D (a, x) → ℂ

is holomorphic in D(a, x) in the classical one-variable sense.

Corollary 2.11 Let D_{1} and D_{2} be bounded convex domains in the Banach spaces (X_{1}, ∥ · ∥_{1}) and (X_{2}, ∥ · ∥_{2}), respectively. Let N be a norming set for (X_{2}, ∥ · ∥_{2}). If {f_{λ}}_{λ∈J} is a net of holomorphic mappings f_{λ} : D_{1} → D_{2} which is pointwise convergent in σ(X_{2}, N) to a mapping f : D_{1} → D_{2}, then f is also holomorphic.

Finally, we recall the Maximum Modulus Theorem for unit balls. First we need two definitions.

Definition 2.12 [192] A point x in a convex subset C of a Banach space (X, ∥ · ∥) is said to be a complex extreme point of C if there is no non-zero vector y ∈ X such that

{x + αy ∈ X : α ∈ ℂ, |α| = 1} ⊂ C.

Definition 2.13 [39] , [41] A Banach space (X, ∥ · ∥) is said to be strictly convex if the following implication holds for all x, y ∈ X :

\begin{matrix}∥x∥ ≤ 1 \\ ∥y∥ ≤ 1 \\ ∥x − y∥ > 0 \end{matrix} \} ⟹ ∥\frac{1}{2}x + \frac{1}{2}y∥ < 1.

Directly from these definitions we get the following fact.

Corollary 2.14 [192]. In a strictly convex Banach space (X, ∥ · ∥) each point on the boundary of its closed unit ball \overline{B} is a complex extreme point.

Theorem 2.15 [192]. Let (X_{1}, ∥ · ∥_{1}) and (X_{2}, ∥ · ∥_{2}) be Banach spaces, D a domain in X_{1} and f : D → X_{2} a holomorphic mapping. If ∥f(·)∥_{2} has a maximum at a point in D, then ∥f(·)∥_{2} is a constant function on D.

Corollary 2.16 [192]. Let f satisfy the hypotheses of Theorem 2.15. If every point on the unit sphere of (X_{2}, ∥ · ∥_{2}) is a complex extreme point of the closed unit ball \overline{B}, then f is a constant mapping on D.

Corollary 2.17 [192]. Let f satisfy the hypotheses of Theorem 2.15 and let (X_{2}, ∥·∥_{2}) be a strictly convex Banach space. Then f is a constant mapping on D.

3. The Kobayashi distance on bounded convex domains

Let Δ be the open unit disc in the complex plane ℂ. Recall that the Poincaré distance on Δ is given by (see (1.1))

k_{Δ} (z, w) = ρ_{Δ} (z, w) = arg tanh |\frac{z − w}{1 − z\overline{w}} | = arg tanh (1 − σ (z, w))^{\frac{1}{2}} ,

where

σ (z, w) = \begin{matrix}(1 − |z|^{2}) & (1 − |w|^{2}) \\ |1 − z\overline{w}| & 2 \end{matrix}, z, w ∈ Δ

(see [60] and [61]).

Note that following the usual practice in complex analysis we call the metric k_{Δ} a distance (although it is, in fact, a metric in the topological sense).

Next, for any a ∈ Δ, consider the Möbius transformation m_{a} : Δ → Δ defined by

m_{a} (z) = \frac{z + a}{1 + z\overline{a}}, z ∈ Δ.

This mapping is not only a biholomorphic automorphism of Δ, but also a k_{Δ}-isometry. Using these Möbius transformations and the Schwarz Lemma ([30], [42]), we obtain the following simple but fundamental fact.

Theorem 3.1 Each holomorphic function f : Δ → Δ is nonexpansive in the metric space (Δ, k_{Δ}), i.e.,

k_{Δ} (f (z) , f (w)) ≤ k_{Δ} (z, w)  (3.1)

for all z, w ∈ Δ.

From now on we let D be a bounded convex domain in a complex Banach space (X, ∥ · ∥). We will use the following two equivalent definitions ([43] , [138] , [172]) of the Kobayashi distance k_{D} on D ([109] , [110]).

The first definition is, in fact, that of the Lempert function δ ([138]; see also [82]).

k_{D} (x, y) = δ_{D} (x, y)

= inf {k_{Δ} (0, γ) : there exists f ∈ H(Δ, D) s.t. f (0) = x and f (γ) = y} .

To arrive at the second definition we first define the function α_{D} : D × X → ℝ^{+} by

α_{D} (x, v) = inf {η > 0 : ∃ g ∈ H(Δ, D) with g (0) = x, g′ (0) η = v}

for all x ∈ D and v ∈ X. The function α_{D} is called the infinitesimal Kobayashi pseudometric for D. Directly from its definition we get the equality

α_{D} (x, tv) = tα_{D} (x, v) ,  t > 0,  (3.2)

and the following lemma.

Lemma 3.2 If f : D_{1} → D_{2} is a holomorphic mapping between the bounded convex domains D_{1} ⊂ X_{1} and D_{2} ⊂ X_{2}, then

α_{D_{2}} (f (x) , f′ (x) v) ≤ α_{D_{1}} (x, v)

for all x ∈ D_{1} and v ∈ X_{1}.

Given x, y ∈ D, we now consider the family of all curves γ̃ : [0, 1] → D that join x and y and have piecewise continuous derivatives. We call such curves admissible and define

L \binom{}{γ̃} = ∫_{0}^{1} α_{D} (γ̃ (t) , γ̃′ (t)) dt.

Then we have

k_{D} (x, y) = inf \{L \binom{}{γ̃} : γ̃ is admissible with γ̃ (0) = x and γ̃ (1) = y\} .

This second definition means that the Kobayashi distance k_{D} for a bounded convex domain D is the integrated form of α_{D} [172].

Our next lemma follows directly from the definition of the Lempert function.

Lemma 3.3 [85], [135]. Let D be a bounded convex domain in a Banach space (X, ∥·∥).

(i) If x, y, w, z ∈ D and s ∈ [0, 1] , then

k_{D} (sx + (1 − s) y, sw + (1 − s) z) ≤ max [k_{D} (x, w) , k_{D} (y, z)] ;

(ii) if x, y ∈ D and s, t ∈ [0, 1] , then

k_{D} (sx + (1 − s) y, tx + (1 − t) y) ≤ k_{D} (x, y) .

Another immediate consequence of the first definition is the following important extension of (3.1): If D_{1} and D_{2} are bounded domains in the Banach spaces (X_{1}, ∥ · ∥_{1}) and (X_{2}, ∥ · ∥_{2}), respectively, and k_{D_{1}} and k_{D_{2}} are the Kobayashi distances on D_{1} and D_{2}, respectively, then each holomorphic f : D_{1} → D_{2} is nonexpansive, i.e.,

k_{D_{1}}(f(x), f(y)) ≤ k_{D_{2}}(x, y)  (3.3)

for all x, y ∈ D_{1}.

As a matter of fact, Kobayashi’s original definition [109] applies to all domains, not just to the convex ones, and leads to a Schwarz-Pick system which, of all Schwarz-Pick systems, assigns the largest possible pseudodistance to each domain [68].

We also recall at this point that all the pseudodistances assigned to a convex domain D by Schwarz-Pick systems coincide ([43], [138], [144]; see also [82]). In the special case of a bounded convex domain D they are all equal, of course, to the Kobayashi distance k_{D}. This unique distance is sometimes called the hyperbolic metric on D.

The following theorem shows that the Kobayashi distance k_{D} is locally equivalent to the norm ∥ · ∥. We denote by dist_{∥·∥}(x, ∂D) the distance in (X, ∥ · ∥) between the point x and the boundary ∂D of the domain D and by diam_{∥·∥} D the diameter of D in (X, ∥ · ∥).

Theorem 3.4 [68]. If D is a bounded convex domain in a Banach space (X, ∥ · ∥), then

arg tanh (\frac{∥x − y∥}{diam_{∥·∥} D} ) ≤ k_{D}(x, y)

for all x, y ∈ D and

k_{D}(x, y) ≤ arg tanh (\frac{∥x − y∥}{dist_{∥·∥}(x, ∂D)})

whenever ∥x − y∥ < dist_{∥·∥}(x, ∂D).

A subset C of D is said to lie strictly inside D if dist_{∥·∥}(C, ∂D) > 0. Thus any closed subset C lying strictly inside a bounded convex domain D is complete with respect to k_{D}.

Theorem 3.5 [68]. Let D be a bounded convex domain in a Banach space (X, ∥ · ∥). A subset C of D is k_{D}-bounded if and only if C lies strictly inside D.

We continue with the following simple lemma on sequences of holomorphic mappings. We denote the norm closure of a subset C by \overline{C}.

Lemma 3.6 [83], [123]. Let D_{1} and D_{2} be bounded convex domains in the Banach spaces (X_{1}, ∥ · ∥_{1}) and (X_{2}, ∥ · ∥_{2}), respectively, and let N be a norming set for (X_{2}, ∥ · ∥_{2}). If {f_{λ}}_{λ∈J} is a net of holomorphic mappings f_{λ} : D_{1} → D_{2} which is pointwise convergent in the topology σ(X_{2}, N) to a mapping f : D_{1} → \overline{D_{2}} and there exists a point z_{0} ∈ D_{1} such that w_{0} = f(z_{0}) ∈ D_{2}, then f maps D_{1} holomorphically into D_{2}.

Proof. By Corollary 2.11, f maps D_{1} holomorphically into \overline{D_{2}}. Next, we observe that the mapping

g_{k} = \frac{1}{k} w_{0} + (1 − \frac{1}{k} ) f

(k ∈ ℕ) transforms D_{1} into A_{k} = \frac{1}{k}w_{0} + (1 − \frac{1}{k})\overline{D_{2}}, and that the set A_{k} lies strictly inside D_{2}. Finally, let z be any point in D_{1}. Since

k_{D_{2}} (w_{0}, g_{k} (z)) = k_{D_{2}} (g_{k} (z_{0}) , g_{k} (z)) ≤ k_{D_{1}} (z_{0}, z) ,

the sequence {g_{k} (z)} lies strictly inside D_{2} by Theorem 3.5. Therefore

f (z) = lim_{k} g_{k} (z)

is an element of D_{2} and this completes the proof of Lemma 3.6. ■

We end this section by studying properties of certain sequences in (D, k_{D}). First we observe that in analogy with the norm, the Kobayashi distance is lower semicontinuous with respect to a suitably chosen topology.

Theorem 3.7 [83] , [117] , [123]. Let X be a Banach space, N a norming set for X and let D ⊂ X be a bounded convex domain such that its norm closure \overline{D} is compact in σ(X, N). If {x_{β}}_{β∈J} and {y_{β}}_{β∈J} are nets in D which are convergent in σ(X, N) to x and y, respectively, and x, y ∈ D, then

k_{D} (x, y) ≤ lim inf_{β} k_{D} (x_{β}, y_{β}) .

If in place of the nets {x_{β}}_{β∈J} and {y_{β}}_{β∈J} we have sequences {x_{k}}_{k∈ℕ} and {y_{k}}_{k∈ℕ}, then the compactness of \overline{D} in σ(X, N) can be replaced by its sequential compactness in σ(X, N).

Proof. Without loss of generality we may assume that

sup_{β} k_{D} (x_{β}, y_{β}) < ∞.

Let {β_{λ}}_{λ∈J′} be an ultranet in J ([14], [50], [87]) such that

lim_{λ} k_{D} (x_{β_{λ}}, y_{β_{λ}}) = lim inf_{β} k_{D} (x_{β}, y_{β}) .

Let ϵ > 0. Then there exist holomorphic functions f_{β_{λ}} : Δ → D and points γ_{β_{λ}} in the open unit disc Δ such that

x_{β_{λ}} = f_{β_{λ}} (0) , y_{β_{λ}} = f_{β_{λ}} (γ_{β_{λ}}) and k_{Δ} (0, γ_{β_{λ}}) ≤ k_{D} (x_{β_{λ}}, y_{β_{λ}}) + ϵ

for each λ ∈ J′. Hence

sup_{λ} k_{Δ} (0, γ_{β_{λ}}) ≤ [sup_{λ} k_{D} (x_{β_{λ}}, y_{β_{λ}})] + ϵ < ∞.

Since \{β_{λ}\}_{λ∈J′} is an ultranet in J, \overline{D} is compact in the topology σ (X, N) , and

x = σ (X, N) - lim_{λ} x_{β_{λ}} = σ (X, N) - lim_{λ} f_{β_{λ}} (0) ,

Lemma 3.6 implies the existence of a holomorphic mapping f : Δ → D and a point γ ∈ Δ such that

σ (X, N) - lim_{λ} f_{β_{λ}} (z) = f (z)

for each z ∈ Δ, and γ_{β_{λ}} →_{λ} γ. These observations and Theorem 3.4 yield

f (0) = x,

k_{Δ} (0, γ) = lim_{λ} k_{Δ} (0, γ_{β_{λ}}) ≤ lim_{λ} k_{D} (x_{β_{λ}}, y_{β_{λ}}) + ϵ = lim inf_{β} k_{D} (x_{β}, y_{β}) + ϵ < ∞,

lim_{λ} k_{Δ} (γ, γ_{β_{λ}}) = 0,

lim sup_{λ} [arg tanh (\frac{∥f_{β_{λ}} (γ) − f_{β_{λ}} (γ_{β_{λ}})∥}{diam_{∥·∥} D})]

≤ lim sup_{λ} k_{D} (f_{β_{λ}} (γ) , f_{β_{λ}} (γ_{β_{λ}})) ≤ lim_{λ} k_{Δ} (γ, γ_{β_{λ}}) = 0,

and finally,

f (γ) − y = σ (X, N) - lim_{λ} [f_{β_{λ}} (γ) − y_{β_{λ}}] = σ (X, N) - lim_{λ} [f_{β_{λ}} (γ) − f_{β_{λ}} (γ_{β_{λ}})] = 0.

This means that

k_{D} (x, y) = k_{D} (f (0) , f (γ)) ≤ k_{Δ} (0, γ)

and

k_{D} (x, y) ≤ lim inf_{β} k_{D} (x_{β}, y_{β}) + ϵ

for each ϵ > 0. Therefore k_{D}(x, y) ≤ lim inf_{β} k_{D}(x_{β}, y_{β}), which completes the first part of the proof. If in place of the nets {x_{β}}_{β∈J} and {y_{β}}_{β∈J} we are given sequences {x_{k}}_{k∈ℕ} and \{y_{k}\}_{k∈ℕ}, and in place of the compactness of \overline{D} in σ(X, N) we have its sequential compactness in σ(X, N), then we can repeat our earlier arguments taking into account the following facts: (Δ, k_{Δ}) is a separable metric space, the distance k_{Δ} is locally equivalent to the Euclidean metric and holomorphic mappings are nonexpansive with respect to Kobayashi distances. We also replace the ultranet technique by the usual diagonal procedure. ■

Remark 3.8 Theorem 3.7 can be used to show that Lemma 3.6 has a nonexpansive analog.

We now replace the general bounded convex domain D with the open unit ball in X. We begin with the definition of the Kadec-Klee property with respect to σ(X, N).

Definition 3.9 [17] , [58] .Let N be a norming set for a Banach space (X, ∥ · ∥). The space (X, ∥ · ∥) is said to have the Kadec-Klee property with respect to σ(X, N)if for every sequence {x_{k}} in X the following implication holds:

\begin{matrix}∥x_{k}∥ ≤ 1 \\ sep \{x_{k}\} = inf \{∥x_{k} − x_{j}∥ : j ≠ k\} > 0 \\ σ (X, N) - lim_{k→∞} x_{k} = x \end{matrix} \} ⇒ ∥x∥ < 1.

Lemma 3.10 [83], [123]. Let N be a norming set for a strictly convex Banach space (X, ∥ · ∥) and let B denote the open unit ball in (X, ∥ · ∥). Assume that \overline{B} is sequentially compact in the topology σ(X, N) and (X, ∥·∥) has the Kadec-Klee property with respect to σ(X, N). If {x_{k}} and {y_{k}} are sequences in B, {x_{k}} is strongly convergent to ξ ∈ ∂B, and

sup {k_{B} (x_{k}, y_{k}) : k = 1, 2, ...} = c < ∞,

then {y_{k}} is also strongly convergent to ξ.

Proof. We can assume that the sequence {y_{k}} is σ(X, N)-convergent to η. If ξ = η, then {y_{k}} is strongly convergent to ξ by the Kadec-Klee property of X. Assume ξ ≠ η. In this case by the strict convexity of B each point of the open segment

(ξ, η) = {z ∈ X : z = sξ + (1 − s) η for some 0 < s < 1}

lies in B. Next, for 0 < s, t < 1 the sequences {sx_{k} + (1 − s)y_{k}} and {tx_{k} + (1 − t)y_{k}} tend in σ(X, N) to sξ + (1 − s)η and tξ + (1 − t)η, respectively. Now, applying Lemma 3.3 and Theorem 3.7, we get

k_{B} (sξ + (1 − s) η, tξ + (1 − t) η) ≤ lim inf_{k→∞} k_{B} (sx_{k} + (1 − s) y_{k}, tx_{k} + (1 − t) y_{k})

≤ lim sup_{k→∞} k_{B} (x_{k}, y_{k}) ≤ c < ∞

for all s, t ∈ (0, 1). By Theorem 3.5 this means that (ξ, η) lies strictly inside B, which is impossible since ξ ∈ ∂B. ■

In a similar way we can prove the following lemma.

Lemma 3.11 [85]. Let D be a strictly convex bounded domain in a Banach space (X, ∥ · ∥). Let {x_{k}} and {y_{k}} be two sequences in D which converge in norm to ξ ∈ ∂D and to η ∈ \overline{D}, respectively. If

sup {k_{D} (x_{k}, y_{k}) : k = 1, 2, ...} < ∞,

then ξ = η.

To prove the last result of this section we will need still another lemma.

Lemma 3.12 [84]. Let X be a Banach space with the open unit ball B. Let {w_{k}} be an arbitrary sequence in B with lim_{k} w_{k} = ξ ∈ ∂B and w_{k} ≠ 0 for k = 1, 2, ... . Let 0 < α < 1, and let Proj_{k} denote, for each k, the projection of X onto the one-dimensional space lin(w_{k}) generated by w_{k} given by

Proj_{k} (x) = l_{k} (x) \frac{1}{∥w_{k}∥}w_{k}, x ∈ X,

where l_{k} is a linear functional with ∥l_{k}∥ = 1 and l_{k}(w_{k}) = ∥w_{k}∥. If

z_{k} = Proj_{k} (αξ) ,

then

z_{k} ∈ B, lim_{k→∞} z_{k} = αξ, lim_{k→∞} k_{B} (z_{k}, αξ) = 0 and lim_{k→∞} (z_{k}, w_{k}) = α,  (3.4)

where for w ∈ X, w ≠ 0, s_{1}, s_{2} ∈ ℂ and v_{1} = (s_{1}/∥w∥)w, v_{2} = (s_{2}/∥w∥)w, the scalar product (v_{1}, v_{2}) in lin(w) is defined by (v_{1}, v_{2}) = s_{1}\overline{s_{2}}.

Proof. It is obvious that z_{k} ∈ B. We also have

∥z_{k} − αξ∥ = ∥αl_{k} (ξ) \frac{w_{k}}{∥w_{k}∥} − αξ∥ ⟶_{k} 0.

Hence we get lim_{k→∞} k_{B}(z_{k}, αξ) = 0. Next, taking α_{k} = αl_{k}(ξ) we obtain

|α_{k} − α| = |αl_{k} (ξ) − αl_{k} (\frac{w_{k}}{∥w_{k}∥}) | ≤ ∥ξ − \frac{w_{k}}{∥w_{k}∥}∥ ⟶_{k} 0.

Therefore

(z_{k}, w_{k}) = α_{k} ∥w_{k}∥ ⟶_{k}α.

■

Theorem 3.13 [84]. For every ξ ∈ ∂B and 0 < α < 1 we have

lim_{w→ξ} [k_{B} (αξ, w) − k_{B} (0, w)] = −k_{B} (0, αξ) .  (3.5)

Proof. Let us observe that

k_{B} (0, w) = arg tanh ∥w∥ = \frac{1}{2} log \frac{1 + ∥w∥}{1 − ∥w∥} = \frac{1}{2} log \frac{(1 + ∥w∥)^{2}}{1 − ∥w∥^{2}}  (3.6)

for w ∈ B, and that

k_{B} (0, αξ) = arg tanh ∥αξ∥ = \frac{1}{2} log \frac{1 − α^{2}}{(1 − α)^{2}} .  (3.7)

Now let us take an arbitrary sequence {w_{k}} in B with w_{k} ≠ 0 for all k and lim_{k} w_{k} = ξ. For each k, let Proj_{k} denote the projection of X onto the one-dimensional space lin(w_{k}) given in Lemma 3.12. If z_{k} = Proj_{k}(αξ), then by (3.4) we get z_{k} ∈ B, lim_{k} z_{k} = αξ, and lim_{k} k_{B}(z_{k}, αξ) = 0. Therefore, in place of (3.5), we need to prove that

lim_{k→∞} [k_{B} (z_{k}, w_{k}) − k_{B} (0, w_{k})] = −k_{B} (0, αξ) .  (3.8)

Since for each k, the points z_{k} and w_{k} belong to lin(w_{k}), we have

k_{B} (z_{k}, w_{k}) = arg tanh (1 − \frac{(1 − ∥z_{k}∥^{2})(1 − ∥w_{k}∥^{2})}{|1 − (z_{k}, w_{k})|^{2}})^{\frac{1}{2}}

= \frac{1}{2} log \frac{\{|1 − (z_{k}, w_{k})| + [|1 − (z_{k}, w_{k})|^{2} − (1 − ∥z_{k}∥^{2})(1 − ∥w_{k}∥^{2})]^{\frac{1}{2}} \}^{2}}{(1 − ∥z_{k}∥^{2})(1 − ∥w_{k}∥^{2})} . (3.9)

By (3.6) and (3.9) we get

k_{B}(z_{k}, w_{k}) − k_{B}(0, w_{k})

= \frac{1}{2} log \frac{\{|1 − (z_{k}, w_{k})| + [|1 − (z_{k}, w_{k})|^{2} − (1 − ∥z_{k}∥^{2})(1 − ∥w_{k}∥^{2})]^{\frac{1}{2}} \}^{2}}{(1 − ∥z_{k}∥^{2})(1 + ∥w_{k}∥)^{2}} ,

and by (3.4) and (3.7) this inequality implies that

lim_{k→∞} [k_{B} (z_{k}, w_{k}) − k_{B} (0, w_{k}) ] = \frac{1}{2} log \frac{(1 − α)^{2}}{1 − α^{2}} = −k_{B} (0, αξ) .

Thus the formula (3.8) is indeed valid. ■

4. The Kobayashi distance on the Hilbert ball

Let B_{H} denote the open unit ball of a complex Hilbert space (H, (·, ·)). For each a ∈ B_{H} consider the Möbius transformation defined by

M_{a} (x) = \frac{1}{1 + (x, a)} (\sqrt{1 − ∥a∥}Q_{a} + P_{a}) (x + a) ,

where x ∈ B_{H}, P_{a} is the orthogonal projection of H onto lin (a) = {λa : λ ∈ ℂ} and Q_{a} = I −P_{a}. Each M_{a} is a k_{B_{H}} -isometry and has a norm continuous injective extension from \overline{B_{H}} onto itself. We also note that M_{a} (0) = a, M_{a}^{−1} = M_{−a}, M_{0} = I, and for every a, b ∈ B_{H} the mapping M_{b} ∘ M_{−a} takes a to b. Applying these properties of Möbius transformations we get the following explicit formula for k_{B_{H}}, namely,

k_{B_{H}} (x, y) = arg tanh (1 − σ (x, y))^{\frac{1}{2}} ,

where x, y ∈ B_{H} and

σ (x, y) = \begin{matrix}(1 − ∥x∥^{2}) & (1 − ∥y∥^{2}) \\ |1 − (x, y)| & 2 \end{matrix}

(see [60] , [61]).

Theorem 4.1 The metric space (B_{H}, k_{B_{H}}) has the following properties:

(i) [61] . For each pair of distinct points x, y ∈ B_{H}, there exist a unique geodesic line passing through them and a unique geodesic segment [x, y] joining them. For each 0 ≤ t ≤ 1 there is a unique point z = (1−t)x⊕ty satisfying k_{B_{H}} (x, z) = tk_{B_{H}} (x, y) and k_{B_{H}} (z, y) = (1 − t)k_{B_{H}} (x, y).

(ii) [60] . If x, y, w and z are in B_{H} and 0 ≤ t ≤ 1, then

k_{B_{H}} ((1 − t) x ⊕ tw, (1 − t) x ⊕ tz) ≤ tk_{B_{H}} (w, z)

and

k_{B_{H}} ((1 − t) x ⊕ tw, (1 − t) y ⊕ tz) ≤ (1 − t) k_{B_{H}} (x, y) + tk_{B_{H}} (w, z).

Strict inequalities occur if 0 < t < 1 and the relevant points do not lie on the same geodesic.

(iii) [60] . For any a ∈ B_{H} and R > 0, the open ball

B (a, R) = {x ∈ B_{H} : k_{B_{H}} (a, x) < R}

is the “ellipsoid”

\{x ∈ B_{H} : \frac{∥P_{a}x − c∥^{2}}{d^{2}m^{2}} + \frac{∥Q_{a}x∥^{2}}{d^{2}m} < 1\} ,

where

d = tanh R, c = \frac{(1 − d^{2})}{1 − d^{2} ∥a∥^{2}} a, and m = \frac{1 − ∥a∥^{2}}{1 − d^{2} ∥a∥^{2}} .

(iv) [121], [122] . If the sequences {x_{k}} and {y_{k}} of elements of B_{H} are weakly convergent to x ∈ B_{H}, then for any y ∈ B_{H} we have

LIM\frac{σ(y_{k}, y)}{σ(x_{k}, x)} = σ (x, y) LIM\frac{1 − ∥y_{k}∥^{2}}{1 − ∥x_{k}∥^{2}} ,

where LIM denotes any of the following limits (the same on both sides of the above equality),

lim inf \frac{σ(y_{k}, y)}{σ(x_{k}, x)}, \frac{lim inf σ(y_{k}, y)}{lim inf σ(x_{k}, x)}, lim sup \frac{σ(y_{k}, y)}{σ(x_{k}, x)}, ...,

whenever it makes sense.

(v) [61], [122] . All the holomorphic automorphisms of (B_{H}, k_{B_{H}}) are weakly continuous.

(vi) [61], [122] . The metric space (B_{H}, k_{B_{H}}) has the Opial property [145] with respect to the weak topology in H, i.e., if a bounded sequence {x_{k}} in (B_{H}, k_{B_{H}}) tends weakly to x, then

lim sup k_{B_{H}} (x_{k}, x) < lim sup k_{B_{H}} (x_{k}, y)

for each y ∈ B_{H} \ {x}.

(vii) [60], [61] . The metric space (B_{H}, k_{B_{H}}) is k_{B_{H}} -uniformly convex, i.e.,

\begin{matrix}k_{B_{H}} (a, x) ≤ R \\ k_{B_{H}} (a, y) ≤ R \\ k_{B_{H}} (x, y) ≥ ϵR \end{matrix} \} ⇒ k_{B_{H}} (a, \frac{1}{2}x ⊕ \frac{1}{2}y) ≤ (1 − δ (R, ϵ)) R,

where δ(ϵ, R) > 0 for all R > 0 and 0 < ϵ < 2, or equivalently,

\begin{matrix}k_{B_{H}} (a, x) ≤ R \\ k_{B_{H}} (a, y) ≤ R \\ k_{B_{H}} (x, y) ≥ ϵR \end{matrix} \} ⇒ k_{B_{H}} (a, \frac{1}{2}x ⊕ \frac{1}{2}y) ≤ arg tanh \sqrt{\frac{tanh^{2} (R) − tanh^{2}(\frac{ϵR}{2})}{1 − tanh^{2}(\frac{ϵR}{2})}}.

(viii) [122] . The metric space (B_{H}, k_{B_{H}}) is linearly uniformly convex, i.e.,

\begin{matrix}k_{B_{H}} (a, x) ≤ R \\ k_{B_{H}} (a, y) ≤ R \\ k_{B_{H}} (x, y) ≥ ϵR \end{matrix} \} ⇒ k_{B_{H}} (a, \frac{1}{2}x + \frac{1}{2}y) ≤ (1 − δ_{1} (a, R, ϵ)) R,

where δ_{1}(a, R, ϵ) > 0 for all R > 0 and 0 < ϵ < 2.

(ix) [150] . Let {x_{k}} and {z_{k}} be two sequences in B_{H} and let R ≥ 0. Suppose that for some y in B_{H}, lim sup_{k} k_{B_{H}} (x_{k}, y) ≤ R, lim sup_{k} k_{B_{H}} (z_{k}, y) ≤ R, and lim inf_{k} k_{B_{H}} ((x_{k} + z_{k})/2, y) ≥ R. Then lim_{k} ∥x_{k} − z_{k}∥ = lim_{k} k_{B_{H}} (x_{k}, z_{k}) = 0.

We say that a subset C of B_{H} is a k_{B_{H}} -Chebyshev set if to each point x in B_{H} there corresponds a unique point z in C such that

k_{B_{H}} (x, z) = inf {k_{B_{H}} (x, y) : y ∈ C} .

In this case we define the nearest point projection r_{C} : B_{H} → C by assigning z to x. The following result is an application of Theorem 4.1.

Theorem 4.2 (i) [122] . Every k_{B_{H}} -closed and convex subset of B_{H} is a k_{B_{H}} -Cheby-shev set.

(ii) [61] . Every k_{B_{H}} -closed and k_{B_{H}} -convex subset C of B_{H} is a k_{B_{H}} -Chebyshev set. Moreover, the nearest point projection r_{C} : B_{H} → C is a k_{B_{H}} -nonexpansive mapping and for each x ∈ B_{H} \ C, y ∈ [x, r_{C}(x)] \ {x} and z ∈ C we have r_{C}(x) = r_{C}(y) and k_{B_{H}} (z, y) < k_{B_{H}} (z, x).

The next lemma will be essential when we look for common fixed points of holomorphic mappings in \overline{B_{H}^{n}} (Section 7).

Lemma 4.3 [181] . Let {x_{α}}_{α∈J} be a k_{B_{H}} -unbounded net in B_{H} satisfying

sup_{α≤β} {k_{B_{H}} (x_{α}, x_{β}) − k_{B_{H}} (x, x_{β})} = R < ∞

for some x ∈ B. Then there is a point ξ ∈ ∂B such that ξ = lim_{α} x_{α}.

Proof. [28] . It is obvious that the point x can be replaced by the origin. We first note that lim_{α} k_{B_{H}} (0, x_{α}) = ∞. Otherwise, there would exist R_{1} > 0 and, for each index i ∈ J, an α_{i} ≥ i such that k_{B_{H}} (0, x_{α_{i}}) ≤ R_{1}. But then, for all i, we would have

k_{B_{H}} (0, x_{i}) ≤ k_{B_{H}} (0, x_{α_{i}}) + k_{B_{H}} (x_{α_{i}}, x_{i}) ≤ 2k_{B_{H}} (0, x_{α_{i}}) + R ≤ 2R_{1} + R,

contradicting the k_{B_{H}} -unboundedness of {x_{α}}_{α∈J}. Now, without loss of generality we can assume that x_{α} ≠ 0 for all α ∈ J. Then there exists 0 < r < 1 such that for each

z_{α} = −\frac{r}{∥x_{α}∥}x_{α}, α ∈ J,

we have k_{B_{H}} (z_{α}, x_{α}) = k_{B_{H}} (0, x_{α}) + R. This implies

k_{B_{H}} (M_{−z_{β}}x_{α}, M_{−z_{β}}x_{β}) = k_{B_{H}} (x_{α}, x_{β}) ≤ k_{B_{H}} (0, x_{β}) + R
= k_{B_{H}} (z_{β}, x_{β}) = k_{B_{H}} (0, M_{−z_{β}}x_{β})

for α ≤ β, or equivalently,

|1 − (M_{−z_{β}}x_{α}, M_{−z_{β}}x_{β})|^{2} ≤ 1 − ∥M_{−z_{β}}x_{α}∥^{2}

for α ≤ β. Next, we observe that for ∥x_{α}∥ ≥ r we have

min_{∥z∥=r} k_{B_{H}} (M_{−z}x_{α}, 0) = k_{B_{H}} (x_{α}, 0) − R

and therefore

lim_{α} ( min_{∥z∥= r} ∥M_{−z}x_{α}∥) = 1.

Now, let {x_{α_{j_{1}}} }_{j_{1}∈J_{1}} and {x_{β_{j_{2}}} }_{j_{2}∈J_{2}} be two subnets of {x_{α}}_{α∈J} which are weakly convergent to ξ_{1} and ξ_{2}, respectively. Then we obtain

1 = lim_{j_{1}} lim_{j_{2}} (M_{−z_{β_{j_{2}}}} x_{α_{j_{1}}} , M_{−z_{β_{j_{2}}}} x_{β_{j_{2}}} )

= lim_{j_{1}} lim_{j_{2}} \begin{matrix}(∥x_{β_{j_{2}}} ∥ + r) & [((x_{α_{j_{1}}} , x_{β_{j_{2}}} ) + r∥x_{β_{j_{2}}} ∥)] \\ (1 + r∥x_{β_{j_{2}}} ∥) & [r(x_{α_{j_{1}}} , x_{β_{j_{2}}} ) + ∥x_{β_{j_{2}}} ∥] \end{matrix} = \frac{(ξ_{1}, ξ_{2}) + r}{r (ξ_{1}, ξ_{2}) + 1},

and finally we have

0 = 1 − \frac{(ξ_{1}, ξ_{2}) + r}{r (ξ_{1}, ξ_{2}) + 1} = (1 − r) \frac{1 − (ξ_{1}, ξ_{2})}{r (ξ_{1}, ξ_{2}) + 1}

which implies ξ_{1} = ξ_{2} ∈ ∂B. This concludes the proof of Lemma 4.3. ■

5. Fixed points in Banach spaces

The basic fixed point theorem for holomorphic mappings in Banach spaces is due to C. J. Earle and R. S. Hamilton [46]. The one-dimensional and finite-dimensional cases were proved earlier by J. F. Ritt [171] and H. J. Reiffen [170], respectively.

Theorem 5.1 [46]. Let D be a bounded convex domain in a Banach space (X, ∥ · ∥). If a holomorphic f : D → D maps D strictly inside itself, then there exists 0 ≤ s < 1 such that

k_{D} (f(x), f(y)) ≤ sk_{D} (x, y)

for all x, y ∈ D, and therefore f has a unique fixed point. Moreover, for any x in D the sequence of iterates {f^{k}(x)} converges to this fixed point.

Proof. Suppose that ∥x∥ < R for all x ∈ D. Let ϵ > 0 be such that

dist_{∥·∥}(f(D), ∂D) > ϵ

and set t = ϵ/(2R). Choose x_{0} ∈ D and let g : D → X be defined by

g (y) = (1 + t) f(y) − tf (x_{0}) = f(y) + t(f(y) − f (x_{0}))

for each y ∈ D. Since for any y ∈ D,

t ∥f(y) − f (x_{0})∥ ≤ t(∥f(y)∥ + ∥f (x_{0})∥) ≤ 2tR = ϵ,

we see that g (D) ⊂ D. Hence by Lemma 3.2,

α_{D} (g (x) , g′ (x) v) ≤ α_{D} (x, v)

for all v ∈ X. Since

g (x_{0}) = f (x_{0}) and g′ (x_{0}) = (1 + t) f′ (x_{0}) ,

we also have by (3.2),

α_{D} (f (x_{0}) , f′ (x_{0}) v) = α_{D} (g (x_{0}) , \frac{1}{1 + t}g′ (x_{0}) v)

= \frac{1}{1 + t}α_{D}(g (x_{0}) , g′ (x_{0}) v) ≤ \frac{1}{1 + t}α_{D} (x_{0}, v) .

Since x_{0} ∈ D was chosen in an arbitrary way, the second definition of Kobayashi’s distance shows that for s=1/(1+t), we have

k_{D} (f(x), f(y)) ≤ sk_{D} (x, y)

for all x, y ∈ D. Therefore f has a unique fixed point by Banach’s fixed point theorem. Moreover, for any x in D, the sequence of iterates {f^{k}(x)} converges to this fixed point.

■

Remark 5.2 As a matter of fact, the proof of Theorem 5.1 shows that it holds for all bounded domains [46].

Corollary 5.3 [71]. Let D be a bounded convex domain in a Banach space (X, ∥ · ∥) with 0 ∈ D. Suppose f : D → D is holomorphic, and for α ∈ Δ let

z (α) = αf (z (α))

be the unique fixed point of αf. Then the mapping z : Δ → D is holomorphic.

Proof. It is sufficient to observe that

z (α) = lim_{k} (αf)^{k} (0) = lim_{k} z_{k} (α)

and that each z_{k} is holomorphic. ■

Before stating the next fixed point theorem we recall briefly the definitions and results of R. Ryan [175] which are needed in its proof. For 1 ≤ p < ∞ let L^{p}(T) denote the space of measurable complex-valued functions defined on the unit circle T = ∂Δ for which

N_{p} (f) = (\frac{1}{2π} ∫_{0}^{2π} |f (e^{iθ})|^{p} dθ)^{\frac{1}{p}} < ∞.

L^{∞}(T) denotes the space of measurable and essentially bounded complex-valued functions on T. If f ∈ L^{∞}(T), then N_{∞}(f) = ess sup |f(e^{iθ})|. If (X, ∥ · ∥) is a separable Banach space, then L_{X}^{p}(T) denotes the space of functions defined a.e. on T with values in X which are measurable in the norm topology of X and such that ∥f(·)∥ ∈ L^{p}(T). Next, for f ∈ L_{X}^{p}(T) denote

N_{p} (f) = N_{p} (∥f (·)∥) .

For x ∈ X and l ∈ X^{∗}, the dual of X, denote by ⟨x, l⟩ the value l(x) of the norm continuous linear functional l at the point x. Since X is separable, a function f from T to X is measurable if and only if ⟨f(·), l⟩ is measurable for every l ∈ X^{∗}. Let L_{X_{s}^{∗}}^{p} denote the space of functions which are defined a.e. on T with values in X^{∗}, and such that ⟨x, f(·)⟩ is measurable for each x ∈ X and ∥f(·)∥ ∈ L^{p}(T). Once more the separability of X implies that ∥f(·)∥ is measurable if ⟨x, f(·)⟩ is measurable for all x ∈ X. In general, L_{X^{∗}}^{p} ⊂ L_{X_{s}^{∗}}^{p}, but we have equality if X is reflexive because then X^{∗} is separable too. If f ∈ L_{X_{s}^{∗}}^{p}, then ⟨x, f(·)⟩ ∈ L^{p}(T) for each x ∈ X and the weak integral or Pettis integral of f is the bounded linear functional on X given by

⟨x, (P) ∫_{0}^{2π} f (e^{iθ}) dθ⟩ = ∫_{0}^{2π} ⟨x, f (e^{iθ})⟩ dθ.

The following two theorems are due to R. Ryan [175].

Theorem 5.4 [175]. Let (X, ∥ · ∥) be a separable Banach space and f ∈ L_{X^{∗}}^{p}(T) for 1 ≤ p ≤ ∞. If (P) ∫_{0}^{2π} e^{inθ}f(e^{iθ})dθ = 0 for n = 1, 2, ..., then N_{p}(f̃_{r}) ≤ N_{p}(f) for 0 < r < 1, where

f̃(z) = \frac{1}{2πi}(P) ∫_{|t|=1} \frac{f(t)}{t − z} dt

and f̃_{r} (e^{iθ}) = f̃(re^{iθ}) . Furthermore, N_{p}(f̃_{r} − f) → 0 as r → 1^{−} and f̃_{r}(e^{iθ}) → f(e^{iθ}) a.e. in the norm of X^{∗}. If f is norm continuous, then f̃_{r}(e^{iθ}) → f(e^{iθ}) uniformly for θ ∈ [0, 2π].

Theorem 5.5 [175]. Suppose that X is a separable Banach space with dual X^{∗}. Let f̃ be holomorphic in Δ with values in X^{∗} and assume N_{p}(f̃_{r}) ≤ 1 for 0 < r < 1, where 1 < p ≤ ∞ and f̃_{r}(e^{iθ}) = f̃(re^{iθ}). Then there exists a function f ∈ L_{X_{s}^{∗}}^{p}(T) such that

f̃(z) = \frac{1}{2πi}(P) ∫_{|t|=1} \frac{f(t)}{t − z} dt

and furthermore, N_{p}(f) ≤ 1, (P) ∫_{0}^{2π} e^{inθ}f(e^{iθ})dθ = 0 for n = 1, 2, ..., and f is uniquely determined modulo a set of measure zero.

Now we can state and prove the following remarkable theorem.

Theorem 5.6 [71]. Let B be the open unit ball of a separable and reflexive Banach space X and let f : \overline{B} → \overline{B} be norm continuous on \overline{B} and holomorphic in the open unit ball B. Then for almost all θ in [0, 2π] the mapping e^{iθ}f has a fixed point in \overline{B}.

Proof. For |α| < 1 let

z (α) = αf (z (α))

be the unique fixed point of αf. By Corollary 5.3 the function {z(α)}_{α∈Δ} is holomorphic. Since

∥z (re^{iθ})∥ ≤ r and N_{p} (z_{r}) = (\frac{1}{2π} ∫_{0}^{2π} ∥z (re^{iθ})∥^{p} dθ)^{\frac{1}{p}} ,

we have N_{p} (z_{r}) ≤ 1 for 0 ≤ r < 1 and p > 1. Since X is reflexive and since the function z has values in X, the dual of X^{∗}, by Theorem 5.5 there exists a function

y ∈ L_{X_{s}^{∗∗}}^{p} (T) = L_{X^{∗∗}}^{p} (T) = L_{X}^{p} (T)

such that

z (w) = \frac{1}{2πi} (P) ∫_{|t|=1} \frac{y (t)}{t − w} dt, N_{p} (y) ≤ 1 and (P) ∫_{0}^{2π} e^{inθ}y (e^{iθ}) dθ = 0

for n = 1, 2, ... . Now we can apply Theorem 5.4 to the function y ∈ L_{X}^{p} (T) and conclude that

z (re^{iθ}) → y (e^{iθ})

a.e. in the norm of X. We complete the proof by noting that the fixed points of re^{iθ}f are precisely z (re^{iθ}) and that these have a limit as r → 1^{−}, so that y (e^{iθ}) is a fixed point of e^{iθ}f. ■

The following simple example shows that Theorem 5.6 fails in the nonreflexive separable space c_{0}.

Example 5.7 [71]. Let X = c_{0} with the supremum norm. Let f : X → X be defined by

f (z_{1}, z_{2}, z_{3}, ...) = (\frac{1}{2}, z_{1}, z_{2}, z_{3}, ...) .

It is obvious that f is holomorphic, f(\overline{B}) ⊂ \overline{B} and that for no θ in [0, 2π] does the function e^{iθ}f have a fixed point.

Remark 5.8 It seems that in the setting of Theorem 5.6 no example is known of a fixed-point-free mapping f.

As we have already seen in Corollary 5.3, the Earle-Hamilton theorem (Theorem 5.1) is sometimes useful even if f does not map D strictly inside itself. Indeed, if f is any holomorphic self-mapping of a bounded convex domain D, then Theorem 5.1 shows that for each t ∈ [0, 1) one can define a holomorphic mapping h_{f}(t, ·) : D → D by

h_{f}(t, x) = (1 − t) x + t f (h_{f}(t, x)) , x ∈ D.  (5.1)

In other words, h_{f}(t, ·) coincides with the so-called nonlinear resolvent

J_{r} = (I + r (I − f))^{−1} ,  (5.2)

where r = \frac{t}{1−t}. These resolvents are used, for instance, in the proofs of the following results.

Theorem 5.9 [91] , [161]. Let D be a bounded convex domain in a Banach space X and let f : D → D be holomorphic. Suppose that f admits a uniformly norm continuous extension to \overline{D} which satisfies the following boundary condition:

∥x − f(x)∥ ≥ ϵ > 0

for all x ∈ ∂D. Then f has a unique fixed point.

See [91] for an application of this theorem to operator theory on spaces with an indefinite metric.

As a matter of fact, the resolvent method is also applicable to a wider class of holomorphic mappings which are not necessarily self-mappings of D. This wider class consists of those mappings of the form h = I − g, where g : D → X is a holomorphic mapping for which the resolvent J_{r} = (I + rg)^{−1} is a well-defined holomorphic selfmapping of D for each r ≥ 0.

It is shown in [161] (see also [162] and [164]) that for a bounded convex domain D, a holomorphic mapping g, which is bounded on each subset strictly inside D, has this property if and only if it is the infinitesimal generator of a semigroup of holomorphic self-mappings of D (see Section 15). When D is the open unit ball B in X, this class can be described in terms of the numerical range of holomorphic mappings defined by L. A. Harris [67]. More precisely, let D be a convex domain in a Banach space X and suppose D contains the origin. Let h : D → X be a holomorphic mapping. For x ∈ ∂D, let J(x) be the set of all norm continuous linear functionals which are tangent (in the real sense) to D at x, i.e.,

J (x) = {l ∈ X^{∗} : l(x) = 1, Re( l(y)) ≤ 1 for all y ∈ D} .

If h ∈ H(D, X) has a norm continuous extension to \overline{D}, then the numerical range of h is the set

V (h) = {l(h(x)) : l ∈ J(x), x ∈ ∂D} .

It is shown in [69] that for a holomorphic mapping h : B → X, the nonlinear resolvent J_{r} = (I + r (I − h))^{−1}, r ≥ 0, is a well-defined holomorphic self-mapping of B for each r ≥ 0 if and only if

L(h) = lim sup_{s→1^{−}} Re( V (h_{s})) ≤ 1,

where h_{s}(x) = h(sx) for s ∈ [0, 1) and x ∈ B.

Theorem 5.10 [69]. If h : B → X is holomorphic and L(h) < 1, then h has a unique fixed point.

Finally, we mention a related result in Hilbert space.

Theorem 5.11 [12], [161]. Let B_{H} be the open unit ball of a Hilbert space H and let h : B_{H} → H be a holomorphic mapping in B_{H} which has a norm continuous extension to \overline{B_{H}}. If L (h) ≤ 1, then h has at least one fixed point in \overline{B_{H}}. Moreover, if the set of interior fixed points of h is nonempty, then it is an affine subset of B_{H}.

The proof of this result is based on Theorem 6.4 in the next section which is devoted to a detailed study of fixed points in B_{H}.

6. Fixed points in the Hilbert ball

Let B_{H} be the open unit ball of a Hilbert space H. Applying the asymptotic center method due to M. Edelstein [48] and the properties of k_{B_{H}} we get the following fixed point theorem.

Theorem 6.1 [61] , [122] , [130]. Let f : B_{H} → B_{H} be a holomorphic mapping or more generally, a k_{B_{H}} -nonexpansive mapping. Then the following statements are equivalent:

(i) f has a fixed point;

(ii) there exists x ∈ B_{H} such that {f^{k}(x)} lies strictly inside B_{H} (this means that {f^{k}(x)} is k_{B_{H}} -bounded);

(iii) there exists a ball B(x, r) in (B_{H}, k_{B_{H}}) which is f-invariant;

(iv) there exists a nonempty, k_{B_{H}} -bounded, k_{B_{H}} -closed and k_{B_{H}} -convex subset of B_{H} which is f-invariant;

(v) there exists a nonempty, k_{B_{H}} -bounded, k_{B_{H}} -closed and convex subset of B_{H} which is f-invariant.

If f : B_{H} → B_{H} has a fixed point, then the approximating curve (cf. Corollary 5.3 and (5.1))

[0, 1] ∋ t ⟶ z_{0} (t) = tf (z_{0} (t))

behaves very well:

Lemma 6.2 [61] If a k_{B_{H}} -nonexpansive f : B_{H} → B_{H} has a fixed point, then the approximating curve z_{0}(t) = tf(z_{0}(t)), 0 ≤ t < 1, tends strongly, as t → 1^{−}, to the unique fixed point of f with the smallest norm.

We also know the structure of the fixed point set of a k_{B_{H}} -nonexpansive (holomorphic) mapping.

Theorem 6.3 [61], [173], [174].

(i) The fixed point set of a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H} is k_{B_{H}} -closed and k_{B_{H}} -convex and therefore it is a k_{B_{H}} -nonexpansive retract of B_{H}.

(ii) The fixed point set of a holomorphic mapping f : B_{H} → B_{H} is affine and therefore it is a holomorphic retract of of B_{H}.

Proof. (i) It is sufficient to apply Theorems 4.1 and 4.2.

(ii) Let C be a maximal affine subset of Fix(f). Suppose that z ∈ Fix(f) does not belong to C and let w ∈ C. Then the metric segment [w, z] is contained in Fix(f) by (i). Since f is holomorphic, it follows that the entire one-dimensional disc spanned by z and w in B_{H} is contained in Fix(f). Thus the affine set spanned by z and C is contained in Fix(f). This contradicts the maximality of C. Hence C = Fix(f) and the proof is complete. ■

When we study the possible existence of a fixed point of a norm continuous mapping f : \overline{B_{H}} → \overline{B_{H}} which is holomorphic inside B_{H}, then the following theorem will turn out to be very useful.

Theorem 6.4 [56] , [60] , [173] , [174] , [214]. If a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H} is fixed-point-free, then there exists a unique point ξ_{f} of norm one such that all the “ellipsoids”

E (ξ_{f}, R) = \{x ∈ B_{H} : \frac{|1 − (x, ξ_{f})|^{2}}{1 − ∥x∥^{2}} < R\} , R > 0,

are invariant under f, \overline{E(ξ_{f}, R)}∩∂B_{H} = \{ξ_{f}\} and for every x ∈ B_{H} there exists R > 0 such that x ∈ E(ξ_{f}, R). Moreover, all the approximating curves defined by

w_{a} (t) = (1 − t) a ⊕ tf(w_{a} (t)) and z_{a} (t) = (1 − t) a + tf (z_{a} (t)) ,

t ∈ [0, 1), a ∈ B_{H}, converge strongly to ξ_{f} as t → 1^{−}.

Remark 6.5 [56] , [174]. If

Φ_{ξ} (x) = \frac{|1 − (x, ξ)|^{2}}{1 − ∥x∥^{2}}

for x ∈ B_{H} and ξ ∈ ∂B_{H}, then for R > 0 the inequality Φ_{ξ} (x) ≤ R is equivalent to

\frac{∥P_{ξ} (x) − (1 − α) ξ∥^{2}}{α^{2}} + \frac{∥Q_{ξ} (x)∥^{2}}{α} ≤ 1,

where, R = α/(1 − α).

Lemma 6.6 [150]. Let the point ξ belong to the boundary of B_{H}, and let {x_{k}} and {z_{k}} be two sequences in B_{H}. Suppose that for R ≥ 0, we have lim sup_{k} Φ_{ξ}(x_{k}) ≤ R, lim sup_{k} Φ_{ξ}(z_{k}) ≤ R, and lim inf_{k} Φ_{ξ}((x_{k} + z_{k})/2) ≥ R. Then lim_{k} ∥x_{k} − z_{k}∥ = 0.

Proof. By Remark 6.5, we have

lim inf_{k} \{\frac{∥P_{ξ} (x_{k}) − (1 − α) ξ + P_{ξ} (z_{k}) − (1 − α) ξ∥^{2}}{4α^{2}} + \frac{∥Q_{ξ} (x_{k}) + Q_{ξ} (z_{k})∥^{2}}{4α} \} ≥ 1.

Using the parallelogram law, we see that

lim sup_{k} \{\frac{∥P_{ξ} (x_{k}) − P_{ξ} (z_{k})∥^{2}}{4α^{2}} + \frac{∥Q_{ξ} (x_{k}) − Q_{ξ} (z_{k})∥^{2}}{4α} \} ≤ 0.

Therefore lim_{k} ∥x_{k} − z_{k}∥ = 0, as claimed. ■

Remark 6.7 [56] , [60]. If a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H} is fixed-point-free, then Φ_{ξ_{f}} (f(x)) ≤ Φ_{ξ_{f}} (x) for each x ∈ B_{H}.

Theorem 6.8 [61]. If a norm continuous mapping f : \overline{B_{H}} → \overline{B_{H}} maps B_{H} into B_{H} and is k_{B_{H}} -nonexpansive on B_{H}, then it has a fixed point in \overline{B_{H}}.

Proof. If the k_{B_{H}} -nonexpansive mapping f_{|B_{H}} : B_{H} → B_{H} is fixed-point-free, then by Theorem 6.4 there exists a unique point ξ_{f} of norm one such that all the ellipsoids

E (ξ_{f}, R) = \{x ∈ B_{H} : \frac{|1 − (x, ξ_{f})|^{2}}{1 − ∥x∥^{2}} < R\} , R > 0,

are invariant under f_{|B_{H}},

\overline{E (ξ_{f}, R)} ∩ ∂B_{H} = \{ξ_{f}\} and ⋂_{R>0} \overline{E (ξ_{f}, R)} = \{ξ_{f}\} .

This means that ξ_{f} must be a fixed point of f. ■

Corollary 6.9 [60]. If a norm continuous f : \overline{B_{H}} → \overline{B_{H}} is k_{B_{H}} -nonexpansive on B_{H} and Fix(f) ∩ B_{H} ≠ ∅, then the set \overline{Fix(f) ∩ B_{H}} is weakly compact.

Observe that in the setting of Corollary 6.9 the set of fixed points in \overline{B_{H}} may be larger than the norm closure of the nonempty fixed point set of f in B_{H}.

Example 6.10 [60]. Consider H = ℂ^{2} and the mapping f : \overline{B_{H}} → \overline{B_{H}} given by f(z, w) = (z, w^{2}). In this case Fix(f) = {(z, 0) : |z| ≤ 1} ∪ {(0, 1)}.

Now we investigate the possible existence of common fixed points of commuting holomorphic mappings.

Theorem 6.11 [117] , [118] , [121]. If {f_{k}}_{k=1}^{m} is a finite family of commuting k_{B_{H}} - nonexpansive self-mappings of B_{H} and if Fix(f_{k}) is nonempty for each k = 1, 2, ...m, then ∩_{k=1}^{m} Fix(f_{k}) is a nonempty k_{B_{H}} -nonexpansive retract of B_{H}.

Proof. By Theorem 6.3, each Fix(f_{k}) is k_{B_{H}} -closed, k_{B_{H}} -convex, and f_{k}-invariant for 1 ≤ k ≤ m. To see that ∩_{k=1}^{m} Fix(f_{k}) is nonempty, it is now sufficient to apply Theorem 6.1 and induction with respect to m. Applying Theorem 4.2 we also see that this nonempty intersection is a k_{B_{H}} -nonexpansive retract of B_{H} ■

The next example shows that Theorem 6.11 is no longer valid if the family of commuting holomorphic mappings is infinite.

Example 6.12 [126]. Let B_{H} be the open unit ball in a separable Hilbert space H with the orthonormal basis {e_{k}}_{k∈ℕ}. With each affine subset Y = (x + Ỹ) ∩ B_{H} of B_{H}, where Ỹ is a closed subspace of H and x ∈ B_{H} is the unique point of least norm in Y, we associate the nearest point holomorphic retraction of B_{H} onto Y given by the formula

r = M_{x} ∘ ProjỸ ∘M_{−x},

where ProjỸ denotes the orthogonal projection onto Ỹ. We construct the family {f_{k}}_{k∈ℕ} in the following way: We choose a sequence of positive real numbers {α_{k}}_{k∈ℕ} such that this sequence is strictly decreasing, 0 < α_{k} < \frac{1}{2}π and lim_{k→∞} α_{k} = 0. We set

X_{0} = H, C_{0} = B_{H}, a_{0} = 0, and R_{0} = ∥e_{1} − a_{0}∥ .

For k = 1, the mapping r_{1} is the nearest point holomorphic retraction of B_{H} onto the affine set C_{1} in B_{H}, where

e′_{1} = \frac{1}{∥e_{1} − a_{0}∥} (e_{1} − a_{0}) ,
g_{1} = e_{2} − a_{0} − \frac{1}{∥e_{1} − a_{0}∥^{2}} (e_{2} − a_{0}, e_{1} − a_{0}) (e_{1} − a_{0}) ,
h_{1} = \frac{1}{∥g_{1}∥}g_{1},
a_{1} = a_{0} + R_{0} [(\frac{1}{2} + \frac{1}{2} cos α_{1}) e′_{1} + (\frac{1}{2} sin α_{1}) h_{1}] ,
R_{1} = ∥e_{1} − a_{1}∥ ,
X_{1} = {x ∈ X_{0} : x = a_{1} + y and (a_{1}, y) = 0}

and

C_{1} = X_{1} ∩ B.

If we already have R_{0}, R_{1}, ..., R_{k} ∈ ℝ, a_{0}, a_{1}, a_{2}, ..., a_{k} ∈ H, e′_{1}, e′_{2}, ..., e′_{k} ∈ H, g_{1}, g_{2}, ... ,g_{k} ∈ H, h_{1}, h_{2}, ..., h_{k} ∈ H, X_{0}, X_{1}, ..., X_{k}, C_{0}, C_{1}, ..., C_{k} and r_{1}, r_{2}, ..., r_{k}, where each X_{i} is an affine set in H, e_{1}, a_{i} ∈ C_{i} ⊂ X_{i}, e_{m} ∈ X_{i} for m ≥ i + 2 (i = 0, ..., k), X_{i+1} ⊂ X_{i} for i = 0, ..., k − 1 and each r_{i} is the nearest point holomorphic retraction of B_{H} onto C_{i} for i = 1, ..., k, then the next holomorphic retraction r_{k+1} is the nearest point holomorphic retraction onto

C_{k+1} = X_{k+1} ∩ B_{H},

where

e′_{k+1} = \frac{1}{∥e_{1} − a_{k}∥} (e_{1} − a_{k}) ,
g_{k+1} = e_{k+2} − a_{k} − \frac{1}{∥e_{1} − a_{k}∥^{2}} (e_{k+2} − a_{k}, e_{1} − a_{k}) (e_{1} − a_{k}) ,
h_{k+1} = \frac{1}{∥g_{k+1}∥}g_{k+1},
a_{k+1} = a_{k} + R_{k} [(\frac{1}{2} + \frac{1}{2} cos α_{k+1}) e′_{k+1} + (\frac{1}{2} sin α_{k+1}) h_{k+1}] ,
R_{k+1} = ∥e_{1} − a_{k+1}∥ ,

and

X_{k+1} = {x ∈ X_{k} : x = a_{k+1} + y and (a_{k+1}, y) = 0} .

Now it is sufficient to take

f_{k} = r_{k} ∘ r_{k−1} ∘ ... ∘ r_{1}

for k = 1, 2, ... to get

f_{k} ∘ f_{m} = f_{m} ∘ f_{k}

for every k, m ∈ ℕ,

Fix(f_{k}) = C_{k}

for each k ∈ ℕ,

lim_{k→∞} diam_{∥·∥} C_{k} = 0,

and

⋂_{k=1}^{∞} Fix (f_{k}) = ∅.

On the other hand, we have the following positive result when at least one of the holomorphic mappings is condensing.

Theorem 6.13 [126]. Let B_{H} be the open unit ball in a Hilbert space H and let {f_{α}}_{α∈J} be a family of commuting holomorphic self-mappings of B_{H}. If Fix(f_{α}) is nonempty for each α ∈ J and at least one fα̃ is condensing, then ∩_{α∈J} Fix(f_{α}) is nonempty.

Proof. It is sufficient to observe that Fix(f_{α}) is affine for each α ∈ J (see Theorem 6.3), ∩_{k=1}^{m} Fix(f_{α_{k}}) is nonempty and affine for an arbitrary choice of a finite number of indices and Fix(fα̃) is a finite-dimensional affine subset of B_{H} because fα̃ is condensing.

■

The next example shows that in Theorem 6.13 we cannot replace the holomorphic mappings by k_{B_{H}} -nonexpansive mappings.

Example 6.14 [126]. If B_{H} = Δ is the open unit disc in H = ℂ, let P_{C} be the projection of B_{H} onto the real segment C = (−1, 1) defined by P_{C}(x + iy) = x, where x, y ∈ ℝ and x + iy ∈ B_{H}. Next let Q_{k} : (−1, 1) → (−1, 1) be given by Q_{k}(x) = max\{x, 1 − \frac{1}{k}\} for x ∈ (−1, 1) and k ∈ ℕ. Then we set f_{k} = Q_{k}P_{C} for k ∈ ℕ, and we obtain f_{k} ∘ f_{m} = f_{m} ∘ f_{k} for every k, m ∈ ℕ, Fix(f_{k}) = (1 − \frac{1}{k}, 1) for each k ∈ ℕ, and ∩_{k=1}^{∞} Fix(f_{k}) = ∅.

We end this section by establishing a theorem about the existence of common fixed points of a commuting family of mappings on \overline{B_{H}}.

Theorem 6.15 [117] , [121]. Suppose {f_{α}}_{α∈J} is a commuting family of norm continuous mappings of \overline{B_{H}} into \overline{B_{H}} such that each f_{α} : \overline{B_{H}} → \overline{B_{H}} is either a constant mapping or maps B_{H} into B_{H} and is k_{B_{H}} -nonexpansive on B_{H}. Then ∩_{α∈J} Fix(f_{α}) ≠ ∅.

Proof. We consider three cases.

Case 1. Fix(f_{α_{k}}) ∩ B_{H} ≠ ∅ for each α ∈ J. In this case the result is clear because the set

⋂_{k=1}^{m} \overline{Fix f_{α_{k}} ∩ B_{H}}

is nonempty and weakly compact (see Corollary 6.9) for every choice of finite number of indices.

Case 2. One f_{β} is a constant mapping: f_{β} ≡ c, where c ∈ \overline{B_{H}}. Then the point c is the common fixed point of the whole family.

Case 3. Neither case 1 nor case 2 is valid. Then one mapping, say f_{β}, maps B_{H} into B_{H} and Fix(f_{β}) ∩ B_{H} = ∅. By Theorem 6.4 there exists ξ_{f_{β}} ∈ ∂B_{H} such that

f_{β} (E (ξ_{f_{β}}, R)) ⊂ E (ξ_{f_{β}}, R)

for each R > 0. Let

C_{α} = f_{α} (E (ξ_{f_{β}}, R))

for an arbitrary α ∈ J. Then

f_{β} (C_{α}) = f_{β} (f_{α} (E (ξ_{f_{β}}, R))) = f_{α} (f_{β} (E (ξ_{f_{β}}, R))) ⊂ f_{α} (E (ξ_{f_{β}}, R)) = C_{α}

and since Fix(f_{β}) = ∅, the set C_{α} is unbounded in (B_{H}, k_{B_{H}}). This implies that

ξ_{f_{β}} ∈ \overline{C_{α}} ⊂ \overline{B_{H}}

and therefore there exists a sequence {x_{k}} of points in E(ξ_{f_{β}}, R) such that f_{α}(x_{k}) → ξ_{f_{β}}. But then we also have x_{k} → ξ_{f_{β}} and hence f_{α}(ξ_{f_{β}}) = ξ_{f_{β}}. ■

7. Fixed points in finite powers of the Hilbert ball

It is not difficult to observe that the Kobayashi metric in the Cartesian product B_{H}^{n} of n open unit balls B_{H} is given by

k_{B_{H}^{n}} (x, y) = max_{1≤j≤n} k_{B_{H}} (x_{j}, y_{j}) .

In this section N(B_{H}^{n}) will denote the class of all k_{B_{H}^{n}} -nonexpansive self-mapping on B_{H}^{n}. The class of those mappings in N(B_{H}^{n}) which have a continuous (in norm) extension to \overline{B_{H}^{n}} will be denoted by CN(\overline{B_{H}^{n}}). It will also be convenient to consider the slightly more general class of mappings N(\overline{B_{H}^{n}}) which consists of all norm continuous mappings f : \overline{B_{H}^{n}} → \overline{B_{H}^{n}} such that tf_{|B_{H}^{n}} ∈ N(B_{H}^{n}) for all 0 < t < 1 [118].

Note that when f ∈ N(\overline{B_{H}^{n}}) it may happen that f (x) ∈ ∂(B_{H}^{n}) for x ∈ B_{H}^{n}. But then, if f (x) = v with

∥v_{j_{1}}∥ = ... = ∥v_{j_{r}}∥ = 1,

then

f(y)_{j_{1}} = v_{j_{1}}, ..., f(y)_{j_{r}} = v_{j_{r}}

for all y ∈ \overline{B_{H}^{n}}.

The next lemma is an extension of Lemma 4.3. It will be used in the proof of the main theorem of this section.

Lemma 7.1 [181] . Let {z_{α}}_{α∈J} be a k_{B_{H}^{n}} -unbounded net in B_{H}^{n} such that

sup_{α≤β} [k_{B_{H}^{n}} (z_{α}, z_{β}) − k_{B_{H}^{n}} (0, z_{β})] < ∞.

Then there are indices 1 ≤ j_{1} < i_{2} < ... < j_{r} ≤ n (1 ≤ r ≤ n) and points {e_{l}}_{l=1}^{r} on ∂B_{H} such that for any f ∈ CN(B_{H}^{n}) for which there is an index α_{0} with {k_{B_{H}^{n}} (z_{α}, f(z_{α}))}_{α≥α_{0}} bounded by 0 < R < ∞, the face

K = {y ∈ B_{H}^{n} : y_{j_{1}} = e_{1}, ..., y_{j_{r}} = e_{r}}

is f-invariant.

Proof. By passing to a subnet and reordering indices if necessary, we may assume that for some r with 1 ≤ r ≤ n we have

sup_{α} [k_{B_{H}^{n}} (0, z_{α}) − k_{B_{H}} (0, (z_{α})_{j})] < ∞  (7.1)

for 1 ≤ j ≤ r, and

lim_{α} [k_{B_{H}^{n}} (0, z_{α}) − k_{B_{H}} (0, (z_{α})_{j})] = ∞  (7.2)

for r + 1 ≤ j ≤ n. Then for 1 ≤ j ≤ r and β ≥ α we get

k_{B_{H}} ((z_{α})_{j}, (z_{β})_{j}) − k_{B_{H}} (0, (z_{β})_{j}) ≤ k_{B_{H}^{n}} (z_{α}, z_{β}) − k_{B_{H}^{n}} (0, z_{β}) + M

for some M. Hence by Lemma 4.3 there are e_{1}, ..., e_{r} in ∂B_{H} such that lim_{α}(z_{α})_{j} = e_{j} for j = 1, .., .r. If r = n, then

lim_{α} z_{α} = (e_{1}, ..., e_{n})

is a fixed point of f by Lemma 3.10. So we may assume that r < n. We will show that

K = {y ∈ B_{H}^{n} : y_{1} = e_{1}, ..., y_{r} = e_{r}}

is f-invariant. To this end, fix (x_{r+1}, ..., x_{n}) in B_{H}^{n−r} and for each α let

v_{α} = ((z_{α})_{1}, ..., (z_{α})_{r}, x_{r+1}, ..., x_{n}).

We have

lim_{α} v_{α} = (e_{1}, ..., e_{r}, x_{r+1}, ..., x_{n}) = v,

and hence

lim_{α} f (v_{α}) = f (v) = w

exists. We claim that w_{j} = e_{j} for j = 1, ..., r. Indeed, for α ≥ α_{0} and 1 ≤ j ≤ r we have

k_{B_{H}} (\frac{1}{2}f(v_{α})_{j} + \frac{1}{2}(z_{α})_{j}, (z_{α})_{j}) ≤ k_{B_{H}} ((f(v_{α})_{j}, (z_{α})_{j}) ≤ k_{B_{H}^{n}} (f(v_{α}), z_{α})  (7.3)

≤ k_{B_{H}^{n}} (f(v_{α}), f(z_{α})) + R

≤ max_{r+1≤k≤n} k_{B_{H}} (x_{k}, (z_{α})_{k}) + R.

Hence, if w_{j} ≠ e_{j} and w_{j} ∈ B_{H} for some 1 ≤ j ≤ r, then there would exist α_{1} such that

sup_{α̃≥α_{1}} k_{B_{H}} (0, f(vα̃)_{j}) < ∞

and therefore, by (7.1), (7.2) and (7.3), for α ≥ α_{1} we would get the following contradiction:

∞ ⟵_{α≥α_{1}} [k_{B_{H}} (0, z_{α}) − max_{r+1≤k≤n} k_{B_{H}} (x_{k}, (z_{α})_{k})]

≤ k_{B_{H}} (0, z_{α}) − k_{B_{H}} ((f(v_{α})_{j}, (z_{α})_{j}) + R

≤ sup_{α̃≥α_{1}} {[k_{B_{H}} (0, zα̃) − k_{B_{H}} (0, (zα̃)_{j})] + k_{B_{H}} (f(vα̃)_{j}, 0) + R} < ∞.

Next, if w_{j} ≠ e_{j} and w_{j} ∈ ∂B_{H}, then by (7.1), (7.2) and (7.3) this would lead to the following contradiction:

∞ ⟵_{α} [k_{B_{H}} (0, z_{α}) − max_{r+1≤k≤n} k_{B_{H}} (x_{k}, (z_{α})_{k})]

≤ k_{B_{H}} (0, z_{α}) − k_{B_{H}} (\frac{1}{2}f(v_{α})_{j} + \frac{1}{2}(z_{α})_{j}, (z_{α})_{j}) + R

≤ sup_{α̃} \{[k_{B_{H}} (0, zα̃) − k_{B_{H}} (0, (zα̃)_{j})]\} + k_{B_{H}} (\frac{1}{2}f(v_{α})_{j} + \frac{1}{2}(z_{α})_{j}, 0) + R

⟶_{α} sup_{α̃} \{[k_{B_{H}} (0, zα̃) − k_{B_{H}} (0, (zα̃)_{j})]\} + k_{B_{H}} (\frac{1}{2}w_{j} + \frac{1}{2}e_{j}, 0) + R < ∞.

Thus w_{j} = e_{j} for all 1 ≤ j ≤ r. ■

Remark 7.2 Note that the face K obtained in Lemma 7.1 depends only on the net {z_{α}}_{α∈J} and that its f-invariance results from the boundedness of{k_{B_{H}^{n}} (z_{α}, f(z_{α}))}_{α≥α_{0}}.

We begin to study the existence of fixed points in B_{H}^{n} with the following simple generalization of Theorem 6.1. Its proof uses induction with respect to n and is based on the asymptotic center method.

Theorem 7.3 [55] , [117] , [121]. Let f : B_{H}^{n} → B_{H}^{n} be a holomorphic mapping or more generally, a k_{B_{H}^{n}} -nonexpansive mapping. Then the following statements are equivalent:

(i) f has a fixed point;

(ii) there exists x ∈ B_{H}^{n} such that {f^{k}(x)} lies strictly inside B_{H}^{n} (this means that {f^{k}(x)} is k_{B_{H}^{n}} -bounded);

(iii) there exists a ball B(x, r) in (B_{H}^{n}, k_{B_{H}^{n}}) which is f-invariant;

(iv) there exists a nonempty, k_{B_{H}^{n}} -bounded, k_{B_{H}^{n}} -closed and convex subset of B_{H}^{n} which is f-invariant.

Remark 7.4 An analogous theorem holds in a more general situation, where f : C → C is a k_{B_{H}^{n}} -nonexpansive self-mapping of a nonempty convex and k_{B_{H}^{n}} -closed subset of B_{H}^{n}.

Here we quote a result from [118] (see also [117] and [119]).

Theorem 7.5 [118]. Let f_{1}, ..., f_{m} be m commuting mappings in N(B_{H}^{n}) (H(B_{H}^{n})) such that Fix(f_{j}) ≠ ∅ for 1 ≤ j ≤ m. Then ∩_{j=1}^{m} Fix(f_{j}) is a nonempty k_{B_{H}^{n}} - nonexpansive (holomorphic) retract of B_{H}^{n}.

We omit the proof of this theorem because the existence of such a k_{B_{H}^{n}} -nonexpansive retraction of B_{H}^{n} follows directly from a modification of Bruck’s method [27] and we will use similar modifications in the proofs of Theorems 11.3 and 14.5. A different proof of the existence of a common fixed point can be found in [117].

Theorem 7.6 [181] , [187]. Let C be a nonempty convex and k_{B_{H}^{n}} -closed subset of B_{H}^{n}. Let {f_{α}}_{α∈J} be a commuting family of k_{B_{H}^{n}} -nonexpansive mappings f_{α} : C → C with a k_{B_{H}^{n}} -bounded common invariant subset C_{1}. Then ∩_{α∈J} Fix(f_{α}) ≠ ∅.

Proof. We will use the asymptotic center method and induction on n. First we make the following observation. Let {g_{j}}_{j∈J′} denote the algebraic semigroup generated by {f_{α}}_{α∈J} via composition. Each j ∈ J′ may be identified with a function from J to the nonnegative integers which is zero except for a finite number of entries. This identification induces a natural order on J′. Fix x ∈ C_{1} and consider the function r(·, {g_{j}(x)}_{j∈J′}) : C → [0, ∞) defined by

r (y, {g_{j} (x)}_{j∈J′}) = lim_{j∈J′} sup_{j′≥j} k_{B_{H}^{n}} (y, g_{j′} (x))

for y ∈ C. It is easy to see that

r (f_{α}(y), {g_{j} (x)}_{j∈J′}) ≤ r (y, {g_{j} (x)}_{j∈J′})

for each α ∈ J and y ∈ C. In addition (see Lemma 3.3) we have

k_{B_{H}^{n}} (\frac{1}{2}y_{1} + \frac{1}{2}y_{2}, g_{j′} (x)) ≤ max [k_{B_{H}^{n}} (y_{1}, g_{j′} (x)), k_{B_{H}^{n}} (y_{2}, g_{j′} (x))]

for all y_{1}, y_{2} ∈ C. Therefore, if

r_{0} = inf {r (y, {g_{j} (x)}_{j∈J′}) : y ∈ C}

is the asymptotic radius of {g_{j}(x)}_{j∈J′}, then the asymptotic center of {g_{j}(x)}_{j∈J′},

{y ∈ C : r (y, {g_{j} (x)}) = r_{0}} ,

is a nonempty, closed and convex invariant subset for {f_{α}}_{α∈J}, which by Theorem 4.1 is either a singleton if n = 1 or, after a possible reordering of indices, of the form {x_{1}} × C̃, where C̃ is a nonempty k_{B_{H}^{n−1}}-bounded closed and convex subset of B_{H}^{n−1} if n > 1. In the first case we have found the desired common fixed point and in the second one we can use the induction hypothesis. ■

Now we state and prove the main theorem of this section.

Theorem 7.7 [181]. A commuting family of mappings \{f_{α}\}_{α∈J} in N(\overline{B_{H}^{n}}) has a common fixed point in \overline{B_{H}^{n}}.

Proof. We use induction on n. For n = 1 see Theorem 6.15. For n > 1 we consider three cases.

Case 1. There exists α_{0} ∈ J such that

f_{α_{0}} (B_{H}^{n}) ⊂ ∂(B_{H}^{n}).

This means that there exist e_{1}, ..., e_{r} ∈ ∂B_{H} and 1 ≤ j_{1} < j_{2} < ... < j_{r} ≤ n such that

f_{α_{0}} (\overline{B_{H}^{n}}) ⊂ K = \{y ∈ ∂(B_{H}^{n}) : y_{j_{1}} = e_{1}, ..., y_{j_{r}} = e_{r}\} .

We have

f_{α} (f_{α_{0}} (K)) = f_{α_{0}} (f_{α} (K)) ⊂ K

for each α ∈ J. If r = n, then f_{α_{0}} is a constant mapping and (e_{1}, ..., e_{n}) is a common fixed point of the family {f_{α}}_{α∈J}. If r < n, then the face K can be shown to be invariant under each f_{α} and the result follows by induction.

Case 2. Each f_{α} belongs to CN(B_{H}^{n}) and there exists f_{α_{0}} for which

Fix(f_{α_{0}}) ∩ B^{n} = ∅.

We construct a sequence {f_{α_{0}}^{k_{m}}(0)} = {z_{m}} in the following way:

k_{1} = 1

and

k_{m+1} = min {k > k_{m} : k_{B_{H}^{n}} (0, f_{α_{0}}^{k}(0)) = max_{1≤j≤k} k_{B_{H}^{n}} (0, f_{α_{0}}^{j}(0))} .

The sequence {z_{m}} is k_{B_{H}^{n}} -unbounded and for j ≥ m we have

k_{B_{H}^{n}} (z_{m}, z_{j}) − k_{B_{H}^{n}} (0, z_{j}) = k_{B_{H}^{n}} (f_{α_{0}}^{k_{m}}(0), f_{α_{0}}^{k_{j}}(0)) − k_{B_{H}^{n}} (0, f_{α_{0}}^{k_{j}}(0))

≤ k_{B_{H}^{n}} (0, f_{α_{0}}^{k_{j}−k_{m}}(0)) − k_{B_{H}^{n}} (0, f_{α_{0}}^{k_{j}}(0)) ≤ 0.

We also have for each α,

sup_{m} k_{B_{H}^{n}} (z_{m}, f_{α}(z_{m})) ≤ k_{B_{H}^{n}} (0, f_{α} (0)) < ∞.

Therefore by Lemma 7.1 we obtain a face of \overline{B_{H}^{n}} which is invariant under \{f_{α}\}_{α∈J} (see Remark 7.2) and we can use induction to get a common fixed point of {f_{α}}_{α∈J}.

Case 3. Each f_{α} belongs to CN(B_{H}^{n}) and

Fix (f_{α}) ∩ B^{n} ≠ ∅

for all α. Let S be the set of all finite subsets of J. The set S is directed by inclusion. Then by Theorem 7.5 for each s = (α_{1}, ..., α_{k}) ∈ S there exists a k_{B_{H}^{n}} -nonexpansive retraction

P_{s} : B_{H}^{n} → ⋂_{i=1}^{k} (Fix (f_{α_{i}}) ∩ B_{H}^{n}) .

Let us consider the net {P_{s}(0)}_{s∈S}. If {P_{s}(0)}_{s∈S} is k_{B_{H}^{n}} -bounded, then the asymptotic center

{y ∈ B_{H}^{n} : r (y, {P_{s} (0)}_{s∈S}) = r_{0}} ,

where

r (y, {P_{s} (0)}_{s∈S}) = lim_{s∈S} sup_{s′≥s} k_{B_{H}^{n}} (y, P_{s′} (0))

and

r_{0} = inf {r (y, {P_{s} (0)}_{s∈S}) : y ∈ C} ,

is a nonempty, k_{B_{H}^{n}} -bounded, closed and convex subset of B_{H}^{n}. Next, it is easy to observe that if α ∈ J and α ∈ s ∈ S, then

f_{α} (P_{s} (0)) = P_{s} (0)

and therefore the inequality

r (f_{α} (x) , {P_{s} (0)}) = lim sup_{s} k_{B_{H}^{n}} (f_{α} (x) , P_{s} (0))

≤ lim sup_{s} k_{B_{H}^{n}} (x, P_{s} (0))

≤ r (x, {P_{s} (0)})

is valid for each α ∈ J and each x ∈ B_{H}^{n}. This implies that the asymptotic center is f_{α}-invariant for each α. Hence by Theorem 7.6 there is a common fixed point in B_{H}^{n}.

Finally, let us assume that {P_{s}(0)}_{s∈J} is k_{B_{H}^{n}} -unbounded. For s′, s′′ ∈ S, s′ ≤ s′′, we have

k_{B_{H}^{n}} (P_{s′} (0) , P_{s′′} (0)) = k_{B^{n_{H}}} (P_{s′} (0) , P_{s′}P_{s′′} (0)) ≤ k_{B_{H}^{n}} (0, P_{s′′} (0)) .

Next, for any singleton s_{0} = {α_{0}} and for any s ≥ s_{0} it is obvious that

f_{α_{0}} (P_{s} (0)) = P_{s} (0)

and we may apply Lemma 7.1 to get an invariant face for {f_{α}}_{α∈J}. The result now follows from the induction hypothesis. This completes the proof of Theorem 7.7. ■

Corollary 7.8 [118], [131]. If f ∈ N(\overline{B_{H}^{n}}), then Fix(f) ≠ ∅.

Remark 7.9 Earlier and different proofs of the existence of a fixed point for a single f ∈ N(\overline{B_{H}^{n}}) can be found in [118] and [131].

Remark 7.10 When we look at Example 5.7 and Corollary 7.8 and compare them with Theorem 5.6, we see that these are the two extreme cases which may occur in infinite-dimensional Banach spaces.

Remark 7.11 At this point it is worthwhile recalling that there exist two continuous and commuting mappings f, g : [0, 1] → [0, 1] such that Fix(f) ∩ Fix(g) = ∅ ([24], [76]).

8. Isometries on the Hilbert ball and its finite powers

By an isometry on (B_{H}^{n}, k_{B_{H}^{n}}) we always mean a k_{B_{H}^{n}} -isometric automorphism of B_{H}^{n} onto B_{H}^{n}. If ∅ ≠ C ⊂ B_{H}^{n}, then a mapping f : C → B_{H}^{n} satisfying

k_{B_{H}^{n}} (f(x), f(y)) = k_{B_{H}^{n}} (x, y)

for all x, y ∈ B_{H}^{n} is called an isometric mapping. We begin this section with a complete characterization of k_{B_{H}} -isometric automorphisms.

Theorem 8.1 [68] , [70] , [79] , [124] , [146]

(i) Every isometry from B_{H} onto B_{H} is of the form I = U ∘ M_{a}, where M_{a} is a Möbius transformation with a ∈ B_{H} and U is either a unitary linear operator or an antiunitary linear operator from H onto H.

(ii) Every isometry from B_{H} onto B_{H} is of the form I = M_{a} ∘ U, where M_{a} is a Möbius transformation with a ∈ B_{H} and U is either a unitary linear operator or an antiunitary linear operator from H onto H.

In (B_{H}^{n}, k_{B_{H}^{n}} ), n ≥ 2, we also have a nice situation.

Theorem 8.2 [64] , [124]. If n ≥ 2 and I : B_{H}^{n} → B_{H}^{n} is a k_{B_{H}^{n}} -isometry, then there exist a permutation π of (1, ..., n) and n k_{B_{H}} -isometries I_{1}, ..., I_{n} such that

I(x_{1}, ..., x_{n}) = (I_{1}(x_{π(1)}), ..., I_{n}(x_{π(n)})).

More information on isometric mappings between finite powers of finite-dimensional open unit balls can be found in [82].

Corollary 8.3 [124]. Every k_{B_{H}} -isometry I : B_{H} → B_{H} transforms affine sets onto affine sets.

Corollary 8.4 [124]. Every k_{B_{H}^{n}} -isometry I : B_{H}^{n} → B_{H}^{n} has a unique norm continuous extension to \overline{B_{H}^{n}}. We will call this unique norm continuous extension to \overline{B_{H}^{n}} a k_{B_{H}^{n}} -isometry I : \overline{B_{H}^{n}} → \overline{B_{H}^{n}}.

Remark 8.5 [124]. As we already know, every k_{B_{H}^{n}} -isometry I : \overline{B_{H}^{n}} → \overline{B_{H}^{n}} is of the form

I(x_{1}, ..., x_{n}) = (I_{1}(x_{π(1)}), ..., I_{n}(x_{π(n)})) ,

where π is a permutation of (1, ..., n) and I_{1}, ..., I_{n} are k_{B_{H}} -isometries. If π consists of one cycle (this kind of an isometry is called a cyclic isometry), then, after reindexing if necessary, we can write

I(x_{1}, ..., x_{n}) = (I_{1}(x_{2}), I_{2} (x_{3}) , ..., I_{n−1} (x_{n}) , I_{n}(x_{1})).

Now we define the isometry

Ĩ = I_{1} ∘ I_{2} ∘ ⋯ ∘ I_{n} : \overline{B_{H}} → \overline{B_{H}}.

It is easy to see that a point (x_{1}, ..., x_{n}) ∈ \overline{B_{H}^{n}} is a fixed point of I if and only if x_{1} ∈ Fix(Ĩ) and (x_{1}, ..., x_{n}) = (x_{1}, (I_{2} ∘ ⋯ ∘ I_{n}) (x_{1}) , ..., I_{n} (x_{1})).

We now turn our attention to the structure of the fixed point sets of isometries.

Theorem 8.6 [70] , [124].

(i) If I : \overline{B_{H}} → \overline{B_{H}} is an isometry and Fix(I_{|B_{H}}) = ∅, then I has either one or two fixed points on ∂B_{H}.

(ii) If n ≥ 2 and I : \overline{B_{H}^{n}} → \overline{B_{H}^{n}} is a cyclic isometry with Fix(I_{|B_{H}^{n}}) = ∅, then I has either one or two fixed points on (∂B_{H})^{n}.

Proof. By Remark 8.5 it is sufficient to prove (i). Suppose that I has at least two fixed points on ∂B. Without loss of generality we may assume that they are ξ_{I} and −ξ_{I}. As we know, I is of the form I = M_{I(0)} ∘ U, where U is either a unitary operator or an antiunitary one. Since I transforms affine sets onto affine sets, the set B_{H} ∩ lin (ξ_{I}) is invariant under I and therefore I (0) = αξ_{I} with 0 < |α| < 1. If

v = βξ_{I} + w ∈ Fix(I),

where |β| ≤ 1 and (ξ_{I}, w) = 0, is another fixed point of I on ∂B_{H}, then (U (w) , ξ_{I}) = 0 and

βξ_{I} + w = v = I (v) = (M_{I(0)} ∘ U)(βξ_{I} + w)

= M_{I(0)}(U(βξ_{I}) + U(w))

= M_{I(0)}(U(βξ_{I})) + \frac{\sqrt{1 − ∥I (0)∥^{2}}}{1 + (U(βξ_{I}), I (0))}U (w)

= I(βξ_{I}) + \frac{\sqrt{1 − ∥I (0)∥^{2}}}{1 + (U(βξ_{I}), I (0))}U (w) .

This means that βξ_{I} ∈ Fix (I). Next, for γ ∈ \overline{Δ} we have

I_{|\overline{B}_{H}∩lin(ξ_{I})}(γξ_{I}) = \{  \begin{matrix}\frac{e^{iθ}γ+α}{1+e^{iθ}γ\overline{α}}ξ_{I}  if U is unitary, \\ \frac{e^{iθ}\overline{γ}+α}{1+e^{iθ}\overline{γ}\overline{α}}ξ_{I}  if U is antiunitary, \end{matrix}

where U(γξ_{I}) = e^{iθ}γξ_{I} if U is unitary and U(γξ_{I}) = e^{iθ}\overline{γ}ξ_{I} if U is antiunitary. Since I(ξ_{I}) = ξ_{I} and I(−ξ_{I}) = −ξ_{I} we get e^{iθ} = 1 and α = \overline{α} > 0 (the last inequality is a consequence of the definition of ξ_{I}). This implies that I_{|\overline{B}_{H}∩lin(ξ_{I})} has exactly two fixed points ξ_{I} and −ξ_{I}. Thus β ∈ \{−1, 1\} and therefore \overline{B}_{H} ∋ v = βξ_{I} + w = βξ_{I} ∈ {−ξ_{I}, ξ_{I}}. In other words, I has exactly two fixed points. ■

If we have a k_{B_{H}^{n}} -isometry I : \overline{B_{H}^{n}} → \overline{B_{H}^{n}}, then, as we know, it consists of cyclic isometries of

\overline{B_{H}^{k}} (1 ≤ k ≤ n)

and the fixed point set of I is equal to the Cartesian product of the fixed point sets of these cyclic isometries. As we saw above (Remark 8.5), each fixed point of a cyclic isometry I is of the form (a, (I_{2} ∘ ⋯ ∘ I_{n})(a), ..., I_{n}(a)), where a ∈ Fix(Ĩ) and Ĩ = I_{1} ∘ ⋯ ∘ I_{n}. Thus we only need to characterize the fixed point sets of k_{B_{H}} -isometries of \overline{B_{H}}.

Theorem 8.7 [124]. If I : \overline{B_{H}} → \overline{B_{H}} is a k_{B_{H}} -isometry and Fix(I_{|B_{H}}) ≠ ∅ is not a singleton, then for each a ∈ Fix(I_{|B_{H}}) there exist an orthonormal basis {e_{j}}_{j∈J} of H and ∅ ≠ J′ ⊂ J such that

Fix(I) = \{ \begin{matrix}M_{a} (\overline{B}_{H} ∩ lin (e_{j})_{j∈J′}) if M_{−a} ∘ I ∘ M_{a} is unitary \\ M_{a} (\overline{B}_{H} ∩ lin_{ℝ} (e_{j})_{j∈J′}) if M_{−a} ∘ I ∘ M_{a} is antiunitary , \end{matrix}

where lin_{ℝ}(e_{j})_{j∈J′} consists of all x in lin(e_{j})_{j∈J′} with real coefficients.

As an immediate corollary of Theorems 8.1, 8.2, and 8.7 we get the following fact.

Theorem 8.8 [124]. If I : \overline{B_{H}^{n}} → \overline{B_{H}^{n}} is an isometry and Fix(I_{|B_{H}^{n}}) ≠ ∅, then

\overline{Fix(I_{|B_{H}^{n}})} = Fix(I)

Remark 8.9 [124]. Theorem 8.8 does not hold for all mappings in CN(B_{H}^{n}). Indeed, consider z → f(z) = z^{2} in \overline{Δ}. Then Fix(f_{|Δ}) = \{0\}, but Fix(f) = \{0, 1\}. See also Example 6.10.

Now let H_{ℝ} be a real Hilbert space and let B_{H_{ℝ}} be the open unit ball in H_{ℝ}. Then H_{ℝ} and B_{H_{ℝ}} can be identified in a natural way with a subset of a complex Hilbert space H and with a subset of the open unit ball B_{H}, respectively. Thus the Kobayashi distance k_{B_{H}} in B_{H} may be restricted to B_{H_{ℝ}}. We denote this restriction by k_{B_{H_{ℝ}}}. There are at least three reasons to investigate (B_{H_{ℝ}}, k_{B_{H_{ℝ}}}):

(a) there is an obvious connection between (B_{H_{ℝ}}, k_{B_{H_{ℝ}}}) and the Klein model of hyperbolic geometry;

(b) the distance k_{B_{H_{ℝ}}} in B_{H_{ℝ}} is a projective invariant [111];

(c) the distance k_{B_{H_{ℝ}}} in B_{H_{ℝ}} has some metric properties which are different from the properties of the distance k_{B_{H}} in B_{H}.

It is easy to observe that the ball B_{H_{ℝ}} is a k_{B_{H}} -closed and k_{B_{H}} -convex subset of (B_{H}, k_{B_{H}}) and that geodesics in (B_{H_{ℝ}}, k_{B_{H_{ℝ}}}) are simply linear segments with their endpoints on the boundary of B_{H_{ℝ}}.

As a direct consequence of Theorem 8.2 we get that each mapping U ∘ M_{a}, where U is a unitary operator from H_{ℝ} onto H_{ℝ} and M_{a} is a Möbius transformation with a ∈ B_{H_{ℝ}}, is an isometry in (B_{H_{ℝ}}, k_{B_{H_{ℝ}}}). In fact we know a bit more.

Theorem 8.10 [133].

(i) Every isometry from B_{H_{ℝ}} onto B_{H_{ℝ}} is of the form I = U ∘ M_{a}, where M_{a} is a Möbius transformation with a ∈ B_{H_{ℝ}} and U is a unitary linear operator from H_{ℝ} onto H_{ℝ}.

(ii) Every isometry from B_{H_{ℝ}} onto B_{H_{ℝ}} is of the form I = M_{a} ∘ U, where M_{a} is a Möbius transformation with a ∈ B_{H_{ℝ}} and U is a unitary linear operator from H_{ℝ} onto H_{ℝ}.

Corollary 8.11 [133]. If I is an isometry from from B_{H_{ℝ}} onto B_{H_{ℝ}}, then it has a natural norm continuous extension to \overline{B_{H_{ℝ}}}.

Corollary 8.12 [133]. If I is an isometry from from B_{H_{ℝ}} onto B_{H_{ℝ}} and has no fixed point in B_{H_{ℝ}}, then its norm continuous extension to \overline{B_{H_{ℝ}}} has either one or two fixed points on the boundary of B_{H_{ℝ}}.

Corollary 8.13 [133]. If I is an isometry from from B_{H_{ℝ}} onto B_{H_{ℝ}} with no fixed point in B_{H_{ℝ}} and if its norm continuous extension to \overline{B_{H_{ℝ}}} has two fixed points on the boundary of B_{H_{ℝ}}, then there exists a fixed point ξ_{I} ∈ ∂B_{H_{ℝ}} of this extension such that for each x ∈ B_{H_{ℝ}}, the sequence of iterates {(I)^{k}(x)} converges to ξ_{I}. This convergence is uniform on each subset of B_{H_{ℝ}} which lies strictly inside B_{H_{ℝ}}.

The form of an isometry in B_{H_{ℝ}}^{n} is analogous to the one in B_{H}^{n}.

Theorem 8.14 [124]. If n ≥ 2 and I : B_{H_{ℝ}}^{n} → B_{H_{ℝ}}^{n} is a k_{B_{H_{ℝ}}^{n}}-isometry, then there exist a permutation π of (1, ..., n) and n k_{B_{H_{ℝ}}}-isometries (I)_{1}, ..., (I)_{n} such that

I(x_{1}, ..., x_{n}) = ((I)_{1} (x_{π(1)}), ..., (I)_{n} (x_{π(n)})).

Theorem 8.15 [124]. If I : \overline{B_{H_{ℝ}}} → \overline{B_{H_{ℝ}}} is a k_{B_{H_{ℝ}}}-isometry and Fix(I_{|B_{H_{ℝ}}}) ≠ ∅ is not a singleton, then for each a ∈ Fix(I_{|B_{H_{ℝ}}}) there exist an orthonormal basis {e_{j}}_{j∈J} of H and ∅ ≠ J′ ⊂ J such that

Fix(I) = M_{a} (\overline{B}_{H} ∩ lin (e_{j})_{j∈J′}) .

9. The extension problem

We saw in the previous sections that some metric properties of (B_{H}, k_{B_{H}}) are close to the corresponding properties of the Hilbert space H. Therefore it is natural to ask whether the Kirszbraun-Valentine extension theorem ([52], [108], [177], [193]) is also valid in (B_{H}, k_{B_{H}}). In the present section we give the answer to this question.

Theorem 9.1 [132] . If dim H = 1, ∅ ≠ C ⊂ Δ and f : C → Δ is k_{Δ}-nonexpansive, then there exists a mapping f_{1} : Δ → Δ which is a k_{Δ}-nonexpansive extension of f.

Theorem 9.2 [124]. If dim H = 1, ∅ ≠ C ⊂ Δ and f : C → Δ is a k_{Δ}-isometric mapping, then there exists a k_{Δ}-isometry I : Δ → Δ which is an extension of f.

The following examples show that both Theorems 9.1 and 9.2 fail in higher dimensions.

Example 9.3 [124] , [132]. In H = ℂ^{2} consider the points a_{1} = (α, 0), a_{2} = (iα, 0), a_{3} = (−iα, 0), b_{1} = (α′, 0) , b_{2} = (0, α) and b_{3} = (0, −α), where α, α′ ∈ (0, 1) and

α′ = α\sqrt{\frac{1 + α^{2}}{1 + α^{4}}} .

For r = arg tanh α we have

k_{B_{H}} (a_{j}, a_{k}) = k_{B_{H}} (b_{j}, b_{k}), j, k = 1, 2, 0 ∈ ⋂_{k=1}^{3} \overline{B} (a_{k}, r) and ⋂_{k=1}^{3} \overline{B} (b_{k}, r) = ∅.

To obtain a counterexample it is sufficient to take C = {a_{1}, a_{2}, a_{3}} and f(a_{j}) = bj.

We can construct similar counterexamples in B_{H}^{2} = Δ × Δ ⊂ ℂ × ℂ = H^{2}. Example 9.4 [124] , [132]. If

a_{1} = (0, 0) , a_{2} = (α, 0) , a_{3} = (0, α) ,
b_{1} = (0, 0) , b_{2} = (α, 0) , b_{3} = (\frac{α (1 + α^{2})}{2} + i\frac{α\sqrt{4 − (1 + α^{2})^{2}}}{2} , 0) ,

and r = \frac{1}{2} arg tanh α, where 0 < α < 1, then we obtain k_{B_{H}} (b_{j}, b_{k}) ≤ k_{B_{H}} (a_{j}, a_{k}), j, k = 1, 2, 3 ,

0 ∈ ⋂_{k=1}^{3} \overline{B} (a_{k}, r) , and ⋂_{k=1}^{3} \overline{B} (b_{k}, r) = ∅.

Example 9.5 [124]. Take the points

a_{1} = (0, 0) , a_{2} = (\frac{1}{2}, 0) , a_{3} = (0, \frac{1}{2})

in B_{H}^{2} = Δ × Δ ⊂ ℂ × ℂ = H^{2}, put C = {a_{1}, a_{2}, a_{3}} and set

f(a_{1}) = (0, 0) , f(a_{2}) = (\frac{1}{2}, 0) , f(a_{3}) = (\frac{1}{2}, \frac{1}{2}) .

Then

k_{B_{H}} (a_{j}, a_{k}) = k_{B_{H}} (f(a_{j}), f(a_{k}))

for j, k = 1, 2, 3, but by Theorem 8.2 there is no k_{B_{H}^{2}} -isometry I such that I_{|C} = f.

We now turn our attention to k_{B_{H_{ℝ}}}. Theorem 9.8 below will be used in the construction of the counterexample we will present in the next section.

Our considerations are based on the following key result. Theorem 9.6 [133]. If a_{1}, ..., a_{m}, b_{1}, ..., b_{m}, p are points of B_{H_{ℝ}} such that

k_{B_{H_{ℝ}}} (b_{j}, b_{k}) ≤ k_{B_{H_{ℝ}}} (a_{j}, a_{k})

for j, k = 1, ..., m, then there exists a point p′ in B_{H_{ℝ}} such that

k_{B_{H_{ℝ}}} (b_{j}, p′) ≤ k_{B_{H_{ℝ}}} (a_{j}, p)

for j = 1, ..., m. Proof. For every sufficiently large μ ≥ 0 the set

P_{μ} = {q ∈ B_{H_{ℝ}} : k_{B_{H_{ℝ}}} (b_{j}, q) ≤ μk_{B_{H_{ℝ}}} (a_{j}, p)}

is nonempty, k_{B_{H_{ℝ}}}-bounded, k_{B_{H_{ℝ}}}-closed and convex, hence weakly compact. Moreover, 0 ≤ μ ≤ λ implies P_{μ} ⊂ P_{λ}. Let μ_{0} be the smallest nonnegative number for which the set P_{μ_{0}} is nonempty. If μ_{0} ≤ 1, then the proof is finished.

Suppose that μ_{0} > 1 and let q_{0} be an element of P_{μ_{0}}. Without loss of generality we may assume that p = q_{0} = 0,

k_{B_{H_{ℝ}}} (b_{j}, 0) > k_{B_{H_{ℝ}}} (a_{j}, 0) for j = 1, ..., l

and

k_{B_{H_{ℝ}}} (b_{j}, 0) ≤ k_{B_{H_{ℝ}}} (a_{j}, 0) for j = l + 1, ..., m.

Hence we have ∥b_{j}∥ > ∥a_{j}∥ for j = 1, ..., l. Now we may apply the method due to Schoenberg [177]. The element 0 has to lie in the k_{B_{H_{ℝ}}}-convex hull (equal to the linear convex hull) of the set {b_{1}, ..., b_{k}}. Therefore we get

∑_{j=1}^{l} β_{j}b_{j} = 0,

where β_{1}, ..., β_{k} ≥ 0 and ∑_{j=1}^{l} β_{j} = 1. Next, we have the inequalities

σ (a_{j}, a_{k}) ≤ σ (b_{j}, b_{k})

for j, k = 1, ..., l, which imply that

\begin{matrix}(1 − (b_{j}, b_{k})) & 2 \\ (1 − (a_{j}, a_{k})) & 2 \end{matrix} ≤ \begin{matrix}(1 − ∥b_{j}∥^{2}) & (1 − ∥b_{k}∥^{2}) \\ (1 − ∥a_{j}∥^{2}) & (1 − ∥a_{k}∥^{2}) \end{matrix} < 1

for j, k = 1, ..., l, and finally,

(a_{j}, a_{k}) < (b_{j}, b_{k})

for j, k = 1, ..., l. Therefore we obtain

0 ≤ ∥∑_{j=1}^{l} β_{j}a_{j}∥^{2} = ∑_{j,k=1}^{l} β_{j}β_{k} (a_{j}, a_{k}) < ∑_{j,k=1}^{l} β_{j}β_{k} (b_{j}, b_{k}) = ∥∑_{j=1}^{l} β_{j}b_{j}∥^{2} = 0.

This contradiction completes the proof of Theorem 9.6. ■

As a simple consequence of this theorem we obtain the following two equivalent theorems.

Theorem 9.7 [133]. Let \{\overline{B}(a_{α}, r_{α})\}_{α∈J} and \{\overline{B}(b_{α}, r_{α})\}_{α∈J} be two families of closed balls in (B_{H_{ℝ}}, k_{B_{H_{ℝ}}}). If

k_{B_{H_{ℝ}}} (b_{α}, b_{β}) ≤ k_{B_{H_{ℝ}}} (a_{α}, a_{β})

for all α, β ∈ J and the intersection ∩_{α∈J} \overline{B} (a_{α}, r_{α}) is nonempty, then so is the intersection ∩_{α∈J} \overline{B} (b_{α}, r_{α}).

Theorem 9.8 [133]. If ∅ ≠ C ⊂ B_{H_{ℝ}} and f : C → B_{H_{ℝ}} is a k_{B_{H_{ℝ}}}-nonexpansive mapping, then there exists a mapping f_{1} : B_{H_{ℝ}} → B_{H_{ℝ}} which is a k_{B_{H_{ℝ}}}-nonexpansive extension of f.

Now let us consider the case of (B_{H_{ℝ}}^{n}, k_{B_{H_{ℝ}}^{n}}) with n ≥ 2. Here we have two distinct situations. For n ≥ 2 and dim H_{ℝ} ≥ 2 Theorem 9.7 fails as the following example shows.

Example 9.9 [133]. It is sufficient to consider the case n = 2 and dim H_{ℝ} = 2. If 0 < α < 1,

a_{1} = (0, 0, 0, 0) , a_{2} = (α, 0, 0, 0), a_{3} = (0, 0, α, 0),
b_{1} = a_{1}, b_{2} = a_{2}, b_{3} = (\frac{1 − \sqrt{1 − α^{2}}}{α} , \frac{\sqrt{α^{4} − (1 − \sqrt{1 − α^{2}})^{2}}}{α} , 0, 0) ,

and

r = \frac{1}{2} arg tanh α,

then k_{B_{H_{ℝ}}^{2}} (b_{j}, b_{k}) = k_{B_{H_{ℝ}}^{2}} (a_{j}, a_{k}) for 1 ≤ j, k ≤ 3,

⋂_{k=1}^{3} \overline{B} (a_{k}, r) ≠ ∅ and ⋂_{k=1}^{3} \overline{B} (b_{k}, r) = ∅.

The case H_{ℝ} = ℝ and B_{H_{ℝ}}^{n} = (−1, 1)^{n} is different.

Theorem 9.10 [133]. If H_{ℝ} = ℝ, B_{H_{ℝ}}^{n} = (−1, 1)^{n} and ∅ ≠ C ⊂ B_{H_{ℝ}}^{n}, then every k_{B_{H_{ℝ}}^{n}}-nonexpansive mapping f : C → B_{H_{ℝ}}^{n} has a k_{B_{H_{ℝ}}^{n}}-nonexpansive extension f_{1} to all of B_{H_{ℝ}}^{n}.

The proof of this result is based on Helly’s theorem [73].

Finally, we consider k_{B_{H}^{n}} -isometric mappings and the problem of extending them to isometries. In this case we have a more complicated situation.

Theorem 9.11 [124]. If ∅ ≠ C ⊂ B_{H_{ℝ}} and f : C → B_{H_{ℝ}} ⊂ H_{ℝ} is a k_{B_{H_{ℝ}}}-isometric mapping, then there exists a real Hilbert space H_{1,ℝ} with the open unit ball B_{H_{1,ℝ}} such that H_{ℝ} ⊂ H_{1,ℝ}, B_{H_{ℝ}} ⊂ B_{H_{1,ℝ}}, k_{B_{H_{ℝ}}} = k_{B_{H_{1,ℝ}}} on B_{H_{ℝ}} and there exists a k_{B_{H_{1,ℝ}}}- isometry I : B_{H_{1,ℝ}} → B_{H_{1,ℝ}} with I_{|B_{H_{ℝ}}} = f.

Remark 9.12 The following example shows that to obtain extensions of k_{B_{H_{ℝ}}}-isometric mappings to k_{B_{H_{1,ℝ}}}-isometries the larger real Hilbert spaces B_{H_{1,ℝ}} are sometimes really needed.

Example 9.13 Consider the real Hilbert space l^{2} with the standard basis {e_{k}}. If C = \{\frac{1}{2}e_{k}\} ∪ \{0\} and the isometric mapping is the shift operator

f (0) = 0, f (\frac{1}{2}e_{k}) = \frac{1}{2}e_{k+1},

for k = 1, 2, 3, ... , then we can extend f to a k_{B_{l^{2}}} -isometric mapping (defined on all of B_{l^{2}}), but not to a k_{B_{l^{2}}} -isometry from B_{l^{2}} onto B_{l^{2}}.

10. Approximating sequences in the Hilbert ball

First we recall a theorem due to K. Goebel and W. A. Kirk [57] which generalizes and unifies earlier results of M. A. Krasnoselskii [113], M. Edelstein and R. C. O’Brien [49], S. Ishikawa [77] and of W. A. Kirk [107]. To state this theorem we need several definitions.

Let (M, d) be a metric space and let ℝ denote the real line. We say that a mapping c : ℝ → X is a metric embedding of ℝ into M if

d (c(s), c(t)) = |s − t|

for all real s and t. The image of ℝ under a metric embedding will be called a metric line. The image of a real interval [a, b] under such a mapping will be called a metric segment.

Assume that (M, d) contains a family L of metric lines such that for each pair of distinct points x and y in M there is a unique metric line in L which passes through x and y. This metric line determines a unique metric segment joining x and y. We denote this segment by [x, y]. For each 0 ≤ t ≤ 1 there is a unique point z in [x, y] such that

d (x, z) = t d (x, y) and d (z, y) = (1 − t) d (x, y) .

This point will be denoted (1 − t) x ⊕ ty.

We shall say that M, or more precisely (M, d, L), is a hyperbolic space if

d (\frac{1}{2}x ⊕ \frac{1}{2}y, \frac{1}{2}x ⊕ \frac{1}{2}z) ≤ \frac{1}{2}d (y, z)  (10.1)

for all x, y and z in M.

An equivalent requirement is that

d (\frac{1}{2}x ⊕ \frac{1}{2}y, \frac{1}{2}w ⊕ \frac{1}{2}z) ≤ \frac{1}{2}d (x, w) + \frac{1}{2}d (y, z)  (10.2)

for all x, y, z and w in M.

It is clear that all normed linear spaces are hyperbolic. A discussion of metric convexity and of more examples of hyperbolic spaces and, in particular, of the Hilbert ball can be found, for instance, in [57] , [60] , [105] , [107] , [160] and in references therein.

Proposition 10.1 [107]. Suppose (M, d) is a hyperbolic space, let 0 < λ < 1, and suppose the sequences {x_{k}} and {y_{k}} in M satisfy

(i) x_{k+1} = (1 − λ) x_{k} ⊕ λy_{k}, and

(ii) d(y_{k+1}, y_{k}) ≤ d(x_{k+1}, x_{k}),

for all k ∈ ℕ. Then

d(y_{k+m}, x_{k}) ≥ \frac{1}{(1 − λ)^{m}} [d(y_{k+m}, x_{k+m}) − d(y_{k}, x_{k})] + (1 + mλ) d(y_{k}, x_{k})

for all k, m ∈ ℕ.

Theorem 10.2 [57]. Let (M, d) be a hyperbolic space, x_{0} ∈ M, 0 < b < 1, and let {λ_{k}} ⊂ [0, b] satisfy ∑_{k=0}^{∞} λ_{k} = +∞. Let f : M → M be nonexpansive, and for each k = 0, 1, 2, ... , suppose that x_{k+1} = (1 − λ_{k})x_{k} ⊕ λ_{k}f(x_{k}). If {x_{k}} is bounded, then lim_{k} d(x_{k}, f(x_{k})) = 0.

Given a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H}, we associate with it two kinds of k_{B_{H}} -nonexpansive averaged mappings as follows:

f_{1,λ} = (1 − λ) I ⊕ λf

(an averaged mapping of the first kind) and

f_{2,λ} = (1 − λ) I + λf

(an averaged mapping of the second kind), where 0 < λ < 1. If Fix (f) ≠ ∅, then these averaged mappings have the same fixed point set as the mapping f. They also have several remarkable properties which are not shared by all k_{B_{H}} -nonexpansive mappings.

Corollary 10.3 [150]. If a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H} has a fixed point, then for each 0 < λ < 1 and each x ∈ B_{H} we have

lim_{k} k_{B_{H}} (f_{1,λ}^{k} (x) , f(f_{1,λ}^{k} (x) )) = lim_{k} k_{B_{H}} (f_{1,λ}^{k} (x) , f_{1,λ}^{k+1} (x)) = 0.

In other words, f_{1,λ} is asymptotically regular [25], [58].

It turns out that an averaged mapping of the second kind enjoys the same property.

Theorem 10.4 [150]. If a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H} has a fixed point and

f_{2,λ} = (1 − λ) I + λf,

where 0 < λ < 1, is an averaged mapping of the second kind associated with f, then f_{2,λ} is asymptotically regular.

Proof. Choose x ∈ B_{H} and denote f_{2,λ}^{k}(x) by x_{k} and f(x_{k}) by w_{k}. Let y be a fixed point of f (and f_{2,λ}). Assume without loss of generality that 0 < λ ≤ \frac{1}{2} and let

z_{k} = (1 − 2λ) x_{k} + 2λw_{k}.

Then

x_{k+1} = \frac{1}{2}x_{k} + \frac{1}{2}z_{k},
lim_{k} k_{B_{H}} (x_{k}, y) = L exists,
lim sup_{k} k_{B_{H}} (w_{k}, y) ≤ L,

and

lim_{k} k_{B_{H}} (\frac{1}{2}x_{k} + \frac{1}{2}z_{k}, y) = L.

Since any k_{B_{H}} -ball is convex, we also have

lim sup_{k} k_{B_{H}} (z_{k}, y) ≤ L.

By the linear uniform convexity of each k_{B_{H}} -ball (see Theorem 4.1 (viii) and (ix)) we conclude that

lim_{k} k_{B_{H}} (x_{k}, z_{k}) = 0.

Hence

lim_{k} k_{B_{H}} (x_{k}, w_{k}) = lim_{k} k_{B_{H}} (x_{k}, f (x_{k})) = 0

aswell. But this implies (see Lemma 3.3) that

lim_{k} k_{B_{H}} (x_{k}, f_{2,λ} (x_{k})) = 0,

or equivalently,

lim_{k} k_{B_{H}} (f_{2,λ}^{k} (x) , f_{2,λ}^{k+1} (x)) = 0.

■

Theorem 10.5 [150]. Let f_{j,λ} (j = 1, 2, 0 < λ < 1) be an averaged mapping associated with a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H}. If f has a fixed point, then for each x ∈ B_{H} the sequence of iterates {f_{j,λ}^{k}(x)}_{k=1}^{∞} converges weakly to a fixed point of f.

Proof. We know (see Theorem 4.1) that (B_{H}, k_{B_{H}}) has the Opial property with respect to the weak topology in H. By Corollary 10.3 and Theorem 10.4 we also have

lim_{k} k_{B_{H}} (f_{j,λ}^{k} (x) , f(f_{j,λ}^{k} (x) )) = 0.

It follows that the weak limit point z of every weakly convergent subsequence

{f_{j,λ}^{k_{m}}(x)}_{m=1}^{∞}

of the sequence {f_{j,λ}^{k}(x)}_{k=1}^{∞} is a fixed point of f (and f_{j,λ}). Hence

lim_{m} k_{B_{H}} (f_{j,λ}^{k_{m}} (x) , z) = lim_{k} k_{B_{H}} (f_{j,λ}^{k} (x) , z)

and by the Opial property of (B_{H}, k_{B_{H}}) with respect to the weak topology the whole sequence {f_{j,λ}^{k}(x)}_{k=1}^{∞} converges weakly to z, as claimed. ■

A modification of the Genel-Lindenstrauss example [54] shows that the convergence established in Theorem 10.5 is not strong in general.

Example 10.6 [133]. Let H_{ℝ} be the real space l_{ℝ}^{2} with the standard orthonormal basis {e_{m}}_{m=1}^{∞}. First we define inductively sequences {x_{k}} and {f(x_{k})} which satisfy

x_{k+1} = \frac{1}{2}x_{k} + \frac{1}{2}f (x_{k})

for k = 1, 2, ... . We start the construction of the sequence {x_{k}} by picking

x_{1} = \frac{1}{2}e_{1}.

Let m_{1} and φ_{1} satisfy the conditions

ℕ ∋ m_{1} > 10, φ_{1} = \frac{π}{3 (m_{1} − 1)} and \frac{1}{2} (cos φ_{1})^{m_{1}} > \frac{3}{8} = \frac{\frac{1}{4} + \frac{1}{2}}{2} .

Next, the points x_{k}, k = 2, ..., m_{1} and f (x_{k}), k = 1, ..., m_{1} − 1 will be chosen in the plane P_{1} = lin (e_{1}, e_{2}) = lin (x_{1}, e_{2}) according to the following rules:

∥x_{k}∥ = ∥f (x_{k})∥ , (x_{k}, f (x_{k})) = ∥x_{k}∥^{2} cos (2φ_{1}) , and x_{k+1} = \frac{1}{2}x_{k} + \frac{1}{2}f (x_{k})

for k = 1, ..., m_{1} − 1. It is clear that for every 1 ≤ j, k ≤ m_{1} − 1 we have

k_{B_{H_{ℝ}}}(f(x_{j}), f(x_{k})) = k_{B_{H_{ℝ}}}(x_{j}, x_{k})

and

∥x_{k}∥ ≥ \frac{3}{8} = \frac{\frac{1}{4} + \frac{1}{2}}{2} .

At this point we modify the Genel-Lindenstrauss example [54]. We choose the point f(x_{m_{1}}) in the following way. Let y_{1} be the next point after x_{m_{1}} (in the plane P_{1}) chosen according to the above rules. This means that

∥y_{1}∥ = ∥x_{m_{1}}∥ , and (x_{m_{1}}, y_{1}) = ∥x_{m_{1}}∥^{2} cos (2φ_{1}) .

We set

z_{1} = \frac{1}{2}x_{m_{1}} + \frac{1}{2}y_{1} and f(x_{m_{1}}) = z_{1} + (∥x_{m_{1}}∥ sin φ_{1})e_{3}.

Hence we get

∥x_{m_{1}}∥ = ∥f (x_{m_{1}})∥ and k_{B_{H_{ℝ}}}(f(x_{m_{1}}), f(x_{k})) < k_{B_{H_{ℝ}}}(x_{m_{1}}, x_{k})

for k = 1, ..., m_{1} − 1 because

cos((m_{1} − k)φ_{1}) < cos φ_{1} cos((m_{1} − k − 1)φ_{1})

and

σ(f(x_{m_{1}}), f(x_{k})) = \begin{matrix}(1 − ∥f(x_{m_{1}})∥^{2}) & (1 − ∥f(x_{k})∥^{2}) \\ [1 − ∥f(x_{m_{1}})∥ ∥f(x_{k})∥ cos φ_{1} cos((m_{1} − k − 1)φ_{1})] & 2 \end{matrix}

> \begin{matrix}(1 − ∥x_{m_{1}}∥^{2}) & (1 − ∥x_{k}∥^{2}) \\ [1 − ∥x_{m_{1}}∥ ∥x_{k}∥ cos((m_{1} − k)φ_{1})] & 2 \end{matrix} = σ(x_{m_{1}}, x_{k})

for k = 1, ..., m_{1} − 1. Now we put

x_{m_{1}+1} = \frac{1}{2}x_{m_{1}} + \frac{1}{2}f (x_{m_{1}}) .

The point x_{m_{1}+1} belongs to P_{2} = lin(x_{m_{1}}, e_{3}) (and so will all the points {x_{k}}_{k=m_{1}+2}^{m_{2}} which we will construct next) and

∥x_{m_{1}+1}∥ ≥ ∥z_{1}∥ ≥ \frac{3}{8}.

Since the angle between the half-planes

{λx_{m_{1}+1} + μ(f(x_{m_{1}}) − x_{m_{1}}) : λ ∈ ℝ, 0 < μ ∈ ℝ}

and

Q_{2} = {λx_{m_{1}+1} + μe_{3} : λ ∈ ℝ, 0 < μ ∈ ℝ}

is acute, the orthogonal projections of f(x_{m_{1}}) and x_{m_{1}} on P_{2} show that there exists an angle ψ_{2} > 0 such that for each u ∈ Q_{2} with

∥u∥ = ∥x_{m_{1}}∥ and (u, x_{m_{1}}) > ∥x_{m_{1}}∥^{2} cos ψ_{2}

we have

k_{B_{H_{ℝ}}} (u, f(x_{m_{1}})) < k_{B_{H_{ℝ}}} (x_{m_{1}+1}, x_{m_{1}}) .

Analogously, applying the orthogonal projection of

u′ ∈ \{u ∈ \overline{Q_{2}} : (u, x_{m_{1}+1}) ≥ ∥u∥ ∥x_{m_{1}+1}∥ cos \frac{π}{3} , \frac{1}{2} ≥ ∥u∥ ≥ \frac{\frac{1}{4} + \frac{3}{8}}{2} = \frac{5}{16}\}

onto lin(x_{m_{1}+1}, f(x_{m_{1}}) − x_{m_{1}}) we get

k_{B_{H_{ℝ}}} (u′, f(x_{m_{1}})) < k_{B_{H_{ℝ}}} (u′, x_{m_{1}}) .

Next, taking

w_{1} = \frac{1}{2}z_{1} + \frac{1}{2}x_{m_{1}}

and observing that the angle between w_{1} and f(x_{k}) is strictly less than the angle between w_{1} and x_{k} (k = 1, ..., m_{1} − 1), we obtain

k_{B_{H_{ℝ}}}(λw_{1}, f(x_{k})) < k_{B_{H_{ℝ}}}(λw_{1}, x_{k})

for 0 < ∥λw_{1}∥ < 1, λ > 0 and k = 1, ..., m_{1} − 1. Hence for each

u ∈ {λw_{1} + μe_{3} : λ > 0, μ ≥ 0}

with ∥u∥ < 1 we have

k_{B_{H_{ℝ}}}(u, f(x_{k})) < k_{B_{H_{ℝ}}}(u, x_{k})

for k = 1, ..., m_{1} − 1. Therefore the number

ϵ_{2} = min {k_{B_{H_{ℝ}}}(u′, x_{k}) − k_{B_{H_{ℝ}}}(u′, f(x_{k})) : u′ ∈ R_{2}, k = 1, ..., m_{1} − 1} ,

where

R_{2} = \{u′ = λw_{1} + μe_{3} : μ ≥ 0, \frac{5}{16} ≤ ∥u∥ ≤ \frac{1}{2}, (u, w_{1}) ≥ ∥u∥ ∥w_{1}∥ cos (\frac{2}{5}π)\} ,

is positive. It is clear that we can find m_{2} and φ_{2} which satisfy

ℕ ∋ m_{2} − m_{1} > 10,
φ_{2} = \frac{π}{3 (m_{2} − m_{1} − 1)} < \frac{ψ_{2}}{2} ,
\frac{3}{8} (cos φ_{2})^{m_{2−}m_{1}} > \frac{\frac{1}{4} + \frac{3}{8}}{2} = \frac{5}{16}

and

arg tanh (1 − (1 − \frac{1}{4} cos (2φ_{2}))^{2}^{(\frac{3}{4})2} ) < ϵ_{2}.

Now we can repeat the procedure we have just used to construct {x_{k}}_{k=1}^{m_{1}} by starting with x_{m_{1}+1} and rotating always in the plane P_{2} by the fixed angle 2φ_{2}. We have, of course, to check whether f is k_{B_{H_{ℝ}}}-nonexpansive on its domain of definition up to now, i.e., whether the inequalities

k_{B_{H_{ℝ}}}(f(x_{j}), f(x_{k})) ≤ k_{B_{H_{ℝ}}}(x_{j}, x_{k}), 1 ≤ j, k ≤ m_{2} − 1,

are valid. For m_{1} + 1 ≤ j, k ≤ m_{2} − 1 we have them by the same reason as in the first case. Next, applying the orthogonal projection of f(x_{m_{1}}) and x_{m_{1}} onto P_{2} we obtain

k_{B_{H_{ℝ}}} (f(x_{m_{1}}), f(x_{k})) < k_{B_{H_{ℝ}}}(x_{m_{1}}, x_{k})

for m_{1} + 2 ≤ k ≤ m_{2} − 1. By the choice of ψ_{2} and φ_{2} we also have

k_{B_{H_{ℝ}}}(f(x_{m_{1}}), f(x_{m_{1}+1})) < k_{B_{H_{ℝ}}}(x_{m_{1}}, x_{m_{1}+1}).

For 1 ≤ j ≤ m_{1} − 1 and m_{1} + 1 ≤ k ≤ m_{2} − 1 we get

k_{B_{H_{ℝ}}}(f(x_{j}), f(x_{k})) ≤ k_{B_{H_{ℝ}}}(f(x_{j}), x_{k}) + k_{B_{H_{ℝ}}}(x_{k}, f(x_{k}))
< k_{B_{H_{ℝ}}}(f(x_{j}), x_{k}) + ϵ_{2}
≤ k_{B_{H_{ℝ}}}(x_{j}, x_{k}) − ϵ_{2} + ϵ_{2} = k_{B_{H_{ℝ}}}(x_{j}, x_{k}).

All other cases were considered earlier. Now it is clear how to continue the inductive definition of {x_{k}} and {f(x_{k})}. The sequence {x_{k}} is k_{B_{H_{ℝ}}}-bounded from above by arg tanh(\frac{1}{2}) and from below by arg tanh(\frac{1}{4}). It does not converge strongly to 0, but it does tend weakly to the origin. Next we use the extension property of (B_{H_{ℝ}}, k_{B_{H_{ℝ}}}) to obtain a k_{B_{H_{ℝ}}}-nonexpansive mapping f : B_{H_{ℝ}} → B_{H_{ℝ}}. It is easy to see that

\{f_{1, \frac{1}{2}}^{k} (x_{1})\}_{k=1}^{∞} and \{f_{2, \frac{1}{2}}^{k} (x_{1})\}_{k=1}^{∞}

tend only weakly to 0. Since there is a k_{B_{H}} -nonexpansive retraction of B_{H} onto B_{H_{ℝ}}, an analogous example of a k_{B_{H}} -nonexpansive f : B_{H} → B_{H} can be constructed in (B_{H}, k_{B_{H}}). Note, however, that in this example f is not a holomorphic mapping.

The next theorem is of particular interest when f is fixed-point-free. It can be considered an extension of Corollary 10.3. See [23] and [160] for more general results (which extend, inter alia, Theorem 10.2) and [154] for a continuous analog.

Theorem 10.7 [150]. Let f : B_{H} → B_{H} be a k_{B_{H}} -nonexpansive mapping, 0 < λ < 1, and let f_{1,λ} be an averaged mapping of the first kind associated with f. Then for each x in B_{H} and all m ≥ 1,

lim_{k} k_{B_{H}} (f_{1,λ}^{k+1}(x), f_{1,λ}^{k}(x)) = lim_{k} \frac{k_{B_{H}} (f_{1,λ}^{k+m}(x), f_{1,λ}^{k}(x))}{m} = lim_{k} \frac{k_{B_{H}} (f_{1,λ}^{k}(x), x)}{k}.

Proof. In order to prove the first equality, we fix x ∈ B_{H} and m ≥ 1. Since f_{1,λ} is k_{B_{H}} -nonexpansive, the limits

L = lim_{k} k_{B_{H}} (f_{1,λ}^{k} (x) , f_{1,λ}^{k+1} (x)) and R = lim_{k} k_{B_{H}} (f_{1,λ}^{k} (x) , f_{1,λ}^{k+m} (x))

exist. To see that R ≤ mL, we note that

k_{B_{H}} (f_{1,λ}^{k}(x), f_{1,λ}^{k+m}(x)) ≤ ∑_{j=1}^{m} k_{B_{H}} (f_{1,λ}^{k+j−1}(x), f_{1,λ}^{k+j}(x)) ≤ mk_{B_{H}} (f_{1,λ}^{k}(x), f_{1,λ}^{k+1}(x)) .

To show that R ≥ mL, we first recall that

f_{1,λ} = (1 − λ) I ⊕ λf.

Applying Proposition 10.1, we see that

k_{B_{H}} (f(f_{1,λ}^{k+m}(x)), f_{1,λ}^{k}(x))

≥ \frac{1}{(1 − λ)^{m}} [k_{B_{H}} (f(f_{1,λ}^{k+m}(x)), f_{1,λ}^{k+m}(x)) − k_{B_{H}} (f(f_{1,λ}^{k}(x)), f_{1,λ}^{k}(x))]

+ (1 + mλ) k_{B_{H}} (f(f_{1,λ}^{k}(x)), f_{1,λ}^{k}(x)) .

Now we observe that

k_{B_{H}} (f(f_{1,λ}^{k+m}(x)), f_{1,λ}^{k}(x))
≤ k_{B_{H}} (f(f_{1,λ}^{k+m}(x)), f(f_{1,λ}^{k}(x))) + k_{B_{H}} (f(f_{1,λ}^{k}(x)), f_{1,λ}^{k}(x))
≤ k_{B_{H}} (f_{1,λ}^{k+m}(x), f_{1,λ}^{k}(x)) + k_{B_{H}} (f(f_{1,λ}^{k}(x)), f_{1,λ}^{k}(x)) .

Hence

k_{B_{H}} (f_{1,λ}^{k+m}(x), f_{1,λ}^{k}(x))

≥ \frac{1}{(1 − λ)^{m}} [k_{B_{H}} (f(f_{1,λ}^{k+m}(x)), f_{1,λ}^{k+m}(x)) − k_{B_{H}} (f(f_{1,λ}^{k}(x)), f_{1,λ}^{k}(x))]

+ (mλ)k_{B_{H}} (f(f_{1,λ}^{k}(x)), f_{1,λ}^{k}(x)) .

Since

k_{B_{H}} (f(f_{1,λ}^{k}(x)), f_{1,λ}^{k}(x)) = \frac{1}{λ}k_{B_{H}} (f_{1,λ}^{k+1}(x), f_{1,λ}^{k}(x)) ,

we see that

k_{B_{H}} (f_{1,λ}^{k+m}(x), f_{1,λ}^{k}(x))

≥ \frac{1}{λ (1 − λ)^{m}} [k_{B_{H}} (f_{1,λ}^{k+m+1}(x), f_{1,λ}^{k+m}(x)) − k_{B_{H}} (f_{1,λ}^{k+1}(x), f_{1,λ}^{k}(x))]

+ mk_{B_{H}} (f_{1,λ}^{k+1}(x), f_{1,λ}^{k}(x)) .

Letting k → ∞, we obtain R ≥ mL, as claimed.

To establish the second equality we first note that

k_{B_{H}} (f_{1,λ}^{k+m}(x), f_{1,λ}^{k}(x)) ≤ k_{B_{H}} (f_{1,λ}^{m}(x), x) .

Therefore the first equality shows that

lim_{k} k_{B_{H}} (f_{1,λ}^{k+1} (x) , f_{1,λ}^{k} (x)) ≤ \frac{k_{B_{H}} (f_{1,λ}^{m} (x) , x)}{m}

for all m ≥ 1. Hence

lim_{k} k_{B_{H}} (f_{1,λ}^{k+1} (x) , f_{1,λ}^{k} (x)) ≤ lim inf_{m} \frac{k_{B_{H}} (f_{1,λ}^{m} (x) , x)}{m}

≤ lim sup_{m} \frac{k_{B_{H}} (f_{1,λ}^{m} (x) , x)}{m}

≤ lim sup_{m} \frac{1}{m} \{∑_{j=1}^{m} k_{B_{H}} (f_{1,λ}^{j} (x) , f_{1,λ}^{j−1} (x))\}

= lim_{k} k_{B_{H}} (f_{1,λ}^{k+1} (x) , f_{1,λ}^{k} (x)) .

Thus

lim_{k} \frac{k_{B_{H}} (f_{1,λ}^{k} (x) , x)}{k}

exists and equals

lim_{k} k_{B_{H}} (f_{1,λ}^{k+1} (x) , f_{1,λ}^{k} (x)) .

This completes the proof of Theorem 10.7. ■

To end this section we mention two strong convergence results. The first one verifies a conjecture formulated in [60] and is used in the proof of the second one. It is also used [93] in the study of the asymptotic behavior of the nonlinear resolvent J_{r} defined by (5.2).

Theorem 10.8 [182]. Let f : B_{H} → B_{H} be a k_{B_{H}} -nonexpansive mapping with a nonempty fixed point set Fix(f). For each 0 ≤ λ < 1 define the mapping h_{λ} : B_{H} → B_{H} by

h_{λ} (x) = (1 − λ) x + λf(h_{λ} (x)), x ∈ B_{H}.

Then the strong lim_{λ→1^{−}} h_{λ}(x) = q(x) exists for each x in B_{H} and belongs to Fix(f).

The mapping q : B_{H} → Fix(f) is the unique retraction of B_{H} onto Fix(f) which is firmly k_{B_{H}} -nonexpansive of the second kind (Definition 12.13). See [60] for an analogous result involving the approximating curve {g_{t}}_{0≤t<1} defined in Theorem 12.14.

Theorem 10.9 [157]. Let f : B_{H} → B_{H} be a k_{B_{H}} -nonexpansive mapping with a fixed point, x_{0} a point in B_{H}, and {λ_{k}} a real sequence in [0, 1) satisfying the following three conditions (where μ_{k} = 1 − λ_{k}):

lim_{k} λ_{k} = 1, ∑_{k=1}^{∞} μ_{k} = ∞ and lim_{k} \frac{μ_{k−1} − μ_{k}}{μ_{k}^{2}} = 0.

Then the sequence {x_{k}} defined by

x_{k} = (1 − λ_{k})x_{0} + λ_{k}f (x_{k−1}) , k ≥ 1,

converges strongly to a fixed point of f.

Remark 10.10 The conditions required of the sequence {λ_{k}} in Theorem 10.9 are satisfied, for example, if λ_{k} = 1 − k^{−β} with 0 < β < 1. Analogous results for nonexpansive mappings in Banach and Hilbert spaces can be found in [149] and references therein.

11. Fixed points in infinite powers of the Hilbert ball

Let J be a set of indices,

l^{∞} (H) = {x = {x_{j}}_{j∈J} ∈ ∏_{j∈J} H : sup_{j∈J} ∥x_{j}∥ < ∞} ,

and B_{H}^{∞} the open unit ball in l^{∞}(H) with the supremum norm.

The Kobayashi distance in B_{H}^{∞} is given by

k_{B_{H}^{∞}} (x, y) = sup_{j∈J} k_{B_{H}} (x_{j}, y_{j})

[129]. The proof of this equality is based on the fact that for each a = {a_{j}}_{j∈J}, the mapping

M_{a} = {M_{a_{j}}}_{j∈J} : B_{H}^{∞} ⟶ B_{H}^{∞},

where each M_{a_{j}} is a Möbius transformation, is biholomorphic.

Theorem 11.1 [128]. (Generalized Hartogs’ Theorem). Let X be a Banach space and D a nonempty open subset of X. Let {Y_{j}} be a family of Banach spaces. If f : D ⟶ l^{∞}(Y_{j}) is locally bounded, then the following statements are equivalent:

(i) f = {f_{j}} is holomorphic;

(ii) each f_{j} : D ⟶ Y_{j} is holomorphic.

Theorem 11.2 [128]. Let f : B_{H}^{∞} → B_{H}^{∞} be a holomorphic mapping. Then the following statements are equivalent:

(i) f has a fixed point;

(ii) there exists a ball B (x, r) in (B_{H}^{∞}, k_{B_{H}^{∞}}) which is f-invariant;

(iii) there exists an f-invariant, k_{B_{H}^{∞}}-bounded product ∏_{j∈J} C_{j} of closed convex subsets of B_{H}.

Proof. The implications (i)⇒(ii)⇒(iii) are obvious. The implication (iii)⇒(i) is proved by a simple modification of the classical proof of Kirk’s theorem [103]: we can find a set which is minimal with respect to inclusion, admissible in (∏_{j∈J} C_{j}, k_{B_{H}^{∞}}) and f-invariant, and after applying Theorems 2.4 and 4.1 we see that this minimal set is a singleton. ■

Remark 11.3 [128]. One can show that the Kobayashi distance in B_{H}^{∞} ∩c_{0}(H), where c_{0}(H) = {x = {x_{j}}_{j∈ℕ} ∈ ∏_{j=1}^{∞} H : lim_{j} x_{j} = 0}, is given by

k_{B_{H}^{∞}∩c_{0}(H)} (x, y) = sup_{j∈N} ρ (x_{j}, y_{j})

[129], but it is obvious that conditions (i) and (ii) are not equivalent in this case. To see this, it is sufficient to consider the mapping

f (\{x_{j}\}) = \{\frac{1}{2}, x_{1}, x_{2}, ...\} .

Remark 11.4 [128]. We observe that in contrast with the case of the open unit ball B_{H}, there exists in B_{H}^{∞} a holomorphic fixed-point-free self-mapping f with a k_{B_{H}^{∞}}- bounded iteration {f^{k}(x)} for each x.

Example 11.5 [128]. Let C be Cayley’s transform:

C : Δ → Π^{+} = {z ∈ ℂ : Im (z) > 0}

[30]. For H = ℂ and J = ℕ, f : B_{H}^{∞} → B_{H}^{∞} is defined in the following way:

B ∋ {z_{j}} ⟶ f ({z_{j}}) = {C^{−1} [1 + LIM {C (z_{j})}] , z_{1}, z_{2}, ...} ,

where LIM is a fixed Banach limit ([39], [41]). Then

(a) f is a holomorphic isometric mapping (with respect to k_{B_{H}^{∞}}) from B_{H}^{∞} into B_{H}^{∞};

(b) f is fixed-point-free;

(c) {f^{k}x} is k_{B_{H}^{∞}}-bounded for every x ∈ B_{H}^{∞}.

Now we are prepared to prove an analog of Theorems 6.3 and 7.5. Note that in the case of reflexive Banach spaces P. Mazet and J.-P. Vigué [142] obtained a holomorphic retraction onto the fixed point set of a holomorphic self-mapping by using complexanalytic arguments, the function h_{λ}(u) = (1 − λ)u + λf(h_{λ}(u)), and its convergence to fixed points when λ → 1^{−}. They also showed that their method failed in the case of the open unit ball in l^{∞}. Therefore our approach shows the strength of Bruck’s metric method ([26], [27]).

Theorem 11.6 [129]. If f : B_{H}^{∞} → B_{H}^{∞} is holomorphic (k_{B_{H}^{∞}}-nonexpansive), then Fix(f) is either empty or a holomorphic (k_{B_{H}^{∞}}-nonexpansive) retract of B_{H}^{∞}.

Proof. We prove this result only in the holomorphic case. Our metric approach works equally well in the k_{B_{H}^{∞}}-nonexpansive case. Set

N_{∞} = {g : g is a holomorphic self-mapping of B_{H}^{∞}, Fix (f) ⊂ Fix (g)} .

Fix x_{0} = {x_{0j}} ∈ Fix (f) . Note that

N_{∞} ⊂ ∏_{x∈B_{H}^{∞}} ∏_{j∈J} {y ∈ B : k_{B_{H}} (y, x_{0j}) ≤ k_{B_{H}^{∞}} (x, x_{0})} = ∏_{x∈B_{H}^{∞}} ∏_{j∈J} C_{xj}.

If each C_{xj} is equipped with the weak topology, then each C_{xj} is weakly compact and by Tychonoff’s Theorem [50] ∏_{x∈B_{H}^{∞}} ∏_{j∈J} C_{xj} is compact in the product topology. The set N_{∞} is closed in this topology, i.e., in the topology of coordinate pointwise weak convergence (see Theorems 2.10 and 3.7). This allows us to apply Bruck’s method [27].

Preorder N_{∞} by setting g ≤ h if and only if

k_{B_{H}^{∞}} (g (x) , w) ≤ k_{B_{H}^{∞}} (h (x) , w)

for all w ∈ Fix(f) and x ∈B_{H}^{∞}. Let us choose a descending chain

{g_{λ}}_{λ∈Λ} = {{g_{λj}}_{j∈J}}_{λ∈Λ}

in (N_{∞}, ≤) . By the compactness of ∏_{x∈B_{H}^{∞}} ∏_{j∈J} C_{xj}, {G_{λ}} has a subnet {g_{λ′}}_{λ′∈Λ′} for which Λ′ is an ultranet and

w- lim_{λ′} g_{λ′j} (x) = g_{j} (x) , x ∈ B_{H}^{∞} and j ∈ J.

The mapping g = {g_{j}}_{j∈J} is holomorphic (see Theorem 11.1). Since by Theorem 3.7, k_{B_{H}} is weakly lower semicontinuous, the following inequalities are valid for each w ∈ Fix(f) and x ∈ B_{H}^{∞}:

k_{B_{H}^{∞}} (g (x) , w) ≤ lim_{λ′} k_{B_{H}^{∞}} (g_{λ′} (x) , w) ≤ k_{B_{H}^{∞}} (g_{λ} (x) , w) , λ ∈ Λ,

and this means that g is a lower bound of the chain. So Zorn’s Lemma implies that N_{∞} contains a minimal element r.

Now we need to show that r maps B_{H}^{∞} onto Fix(f). Suppose there exists y ∈ B_{H}^{∞} such that r(y) ∉ Fix(f). Since r ∘ r ≤ r and r is minimal,

k_{B_{H}^{∞}} (r (y_{0}) , w) = k_{B_{H}^{∞}} (y_{0}, w) > 0

for y_{0} = r(y) and all w ∈ Fix(f). Since for each j ∈ J, after interchanging j-coordinate functions between two arbitrarily chosen mappings from N_{∞} we also get a mapping from N_{∞}, and since g, h ∈ N_{∞} and 0 ≤ β ≤ 1 imply that βg + (1 − β)h ∈ N_{∞} too, the set N_{∞} is equal to ∏_{j∈J} D_{j}, where each D_{j} is convex and weakly compact. Let

C = {(g ∘ r)(y_{0}) : g ∈ N_{∞}} .

Using the same arguments as above we see that C is k_{B_{H}^{∞}}-bounded and C = ∏_{j∈J} C_{j}, where each C_{j} is convex and weakly compact. The definition of N_{∞} implies that C is f-invariant and therefore by Theorem 11.2, C ∩ Fix(f) ≠ ∅. Let us choose an arbitrary point (g ∘ r)(y_{0}) ∈ C ∩ Fix(f). Since g ∘ r ≤ r and r is minimal, we get the following contradiction:

0 = k_{B_{H}^{∞}}( (g ∘ r) (y_{0}) , (g ∘ r) (y_{0}) ) = k_{B_{H}^{∞}}(r (y_{0}) , (g ∘ r) (y_{0}) ) > 0.

This completes the proof of Theorem 11.6. ■

12. The Denjoy-Wolff theorem in theHilbert ball and its powers

We begin by recalling the original Denjoy-Wolff theorem.

Theorem 12.1 [40] , [212] , [213]. Let Δ be the open unit disc in the complex plane ℂ. If f ∈ H(Δ) is not the identity and is not an automorphism of Δ with exactly one fixed point in Δ, then there is a unique point α in the closed unit disc \overline{Δ} such that the iterates {f^{k}} of f converge to α, uniformly on compact subsets of Δ.

In the next two sections we will only consider the case where f ∈ H(Δ) does not have fixed points in Δ. This means that we will actually deal with the following simpler version of the above theorem.

Theorem 12.2 [40], [212], [213]. Let Δ be the open unit disc in the complex plane ℂ. If f ∈ H (Δ) with no fixed point in Δ, then there is a unique point ξ_{f} ∈ ∂Δ such that the iterates {f^{k}} of f converge to ξ_{f}, uniformly on compact subsets of Δ.

The usual proof of Theorem 12.2 is divided into the following four steps [31].

Step 1. There are a point ξ_{f} ∈ ∂Δ and f-invariant horocycles at ξ_{f} with radii R > 0 (cf. Theorem 6.4),

E (ξ_{f}, R) = \{z ∈ Δ : \frac{|1 − z\overline{ξ_{f}} |^{2}}{1 − |z|^{2}} < R\} ,

in Δ. Here ξ_{f} is obtained as the limit point of any convergent subsequence of the approximating sequence

z_{j} = (1 − \frac{1}{j}) f(z_{j})

(the existence of such a sequence is guaranteed by Rouché’s theorem [30] applied to the mapping I − (1 − j^{−1})f). The above horocycles are obtained by applying the Schwarz-Pick Lemma. Observe that E(ξ_{f}, R) ⊂ Δ is a disc with the point ξ_{f} on its boundary which is internally tangent (in ℝ^{2}) to ∂Δ at ξ_{f}.

Step 2. By Montel’s theorem and the Open Map Theorem [30] each convergent subsequence {f^{k_{m}}} of the sequence of iterates {f^{k}} has a constant function g with |g| = 1 as its limit.

Step 3. For each z ∈ Δ there is R > 0 such that z ∈ E(ξ_{f}, R).

Step 4. According to steps 1 and 3 for every z ∈ Δ there is R > 0 such that each point f^{k}(z) lies in E(ξ_{f}, R) and therefore f^{k}(z) → ξ.

Somewhat more complicated arguments yield the following generalization of the Denjoy-Wolff theorem in ℂ^{n}.

Theorem 12.3 [74] , [116] , [139]. Let B be the open unit ball in the complex space ℂ^{n}. If f : B → B is holomorphic with no fixed point in B, then there is a unique point ξ_{f} ∈ ∂B such that the iterates {f^{k}} of f converge to ξ_{f}, uniformly on compact subsets of B.

If we wish to extend the above convergence results to other domains in ℂ^{n} or more generally, in a complex Banach space X, then we should keep in mind the following three negative examples.

Example 12.4 [36]. If ℂ^{2} is equipped with the maximum norm, then its open unit ball B is equal to the Cartesian product Δ × Δ. Let us choose any fixed-point-free h : Δ → Δ. If we define f : Δ × Δ → Δ × Δ by

f (z, w) = (iz, h (w))

for z, w ∈ Δ, then f is also fixed-point-free and it is clear that its sequence of iterates {f^{k}} does not converge to any boundary point on ∂(Δ × Δ). As we will see, the main reason for such a behavior of the sequence of iterates is the lack of strict convexity of the closed unit ball \overline{B} = \overline{Δ × Δ}.

To construct the next example we need the following two facts.

1) If B_{l^{2}} is the open unit ball in the complex infinite-dimensional Hilbert space l^{2} with the standard inner product and if e is a point in l^{2} of norm 1, then B_{l^{2}} is biholomorphically equivalent to the domain

Ω = {(λ, w) ∈ ℂ × l^{2} : Im λ > ∥w∥}

under the Cayley transform C : B → Ω given by

B_{l^{2}} ∋ z → C (z) = (λ, w) = (i\frac{1 + (z, e)}{1 − (z, e)}, \frac{i}{1 − (z, e)} (z − (z, e) e)) ∈ Ω.

The domain Ω is the analog of the upper half-plane [60].

2) In [47] M. Edelstein shows that there exists a fixed-point-free isometry g̃ of the complex infinite-dimensional Hilbert space l^{2} such that the sequence of iterates {g^{k}(0)} is unbounded, but a suitably chosen subsequence {g^{k_{m}}(0)} is convergent to 0. We recall this isometry:

g̃(w) = g̃(\{w_{j}\}) = \{e^{\frac{2πi}{j!}} (w_{j} − 1) + 1\} .

One can compute that for k_{m} = m! (m = 1, 2, ...) the subsequence {g^{k_{m}}(0)} tends to 0 and for

k̃_{m} = \frac{1}{2} ∑_{p=1}^{m} (2^{p}m)!

(m = 1, 2, ...) we have ∥gk̃_{m}(0)∥ → ∞. We also have that

g̃(w) = g (w) + a,

where a = g̃(0) and g : l^{2} → l^{2} is a linear isometry.

Now we are ready to present the second example in which we use Edelstein’s isometry g̃ = g + a.

Example 12.5 [188]. In the complex infinite-dimensional Hilbert space l^{2} convergence fails even for biholomorphic self-maps. To see this, consider the above-mentioned domain Ω. For the mapping φ : Ω → Ω defined by

φ (λ, w) = (λ + i ∥a∥^{2} + 2i (g (w) , a) , g (w))

we have

lim_{m→∞} φ^{k_{m}} (i, 0) = 0 for k_{m} = m! (m = 1, 2, ...) and lim sup_{k→∞} ∥φ^{k} (i, 0)∥ = ∞.

Setting f = C^{−1} ∘ φ ∘ C : B_{l^{2}} → B_{l^{2}} we see that f is a biholomorphic automorphism of B_{l^{2}}, Fix(f) = ∅, and

lim sup_{k→∞} ∥f^{k} (0)∥ = 1 > 0 = lim inf_{k→∞} ∥f^{k} (0)∥ .

In the third example we show that there is a complex Banach space X and a fixedpoint-free holomorphic self-mapping f of the open unit ball B in X such that

sup_{k} ∥f^{k} (x)∥ < 1

for each x ∈ B.

Example 12.6 [128]. Consider the complex Banach space of all bounded sequences l^{∞} = l^{∞}(ℂ) and let

C : Δ → {z ∈ ℂ : Im z > 0}

be the Cayley transform

C (z) = i\frac{(1 + z)}{1 − z}

for z ∈ Δ [30]. If B_{ℂ}^{∞} is the open unit ball in l^{∞}, LIM is a fixed Banach limit on l^{∞}, and f : B_{ℂ}^{∞} → B_{ℂ}^{∞} is given by the formula

f (x) = f ({x_{j}}) = { C^{−1} (1 + LIM ({C (x_{j})})) , x_{1}, x_{2}, ...} ,

then f is holomorphic, Fix(f) = ∅, and for each x ∈ B_{ℂ}^{∞} the sequence of its iterates {f^{k}(x)} lies strictly inside B_{ℂ}^{∞}, i.e.,

sup_{k} ∥f^{k} (x)∥ < 1.

The above examples show that in order to obtain a generalization of the Denjoy-Wolff theorem, not only do we need additional properties of the boundary of the open unit ball B, but we also have to impose some restrictions on the holomorphic self-mapping f : B → B.

We begin our discussion with a special class of k_{B_{H}} -nonexpansive self-mappings of the Hilbert ball B_{H}. As we already know (see Example 12.5), the Denjoy-Wolff theorem is false in general in B_{H}. However, one can prove that this theorem is valid for averaged mappings. To accomplish this we need two preliminary lemmas.

Lemma 12.7 If a k_{B_{H}} -nonexpansive f : B_{H} → B_{H} is fixed-point-free and f_{j,λ} (j = 1, 2 and 0 < λ < 1) is an averaged mapping associated with f (see Section 10), then the mapping f_{j,λ} is also fixed-point-free and ξ_{f} = ξ_{f_{j,λ}}.

Proof. It is obvious that f_{j,λ} is fixed-point-free. Next it is sufficient to observe that the ellipsoid E(ξ_{f}, R) is convex, k_{B_{H}} -convex and f-invariant. ■

Lemma 12.8 [150]. If a k_{B_{H}} -nonexpansive mapping f : B_{H} → B_{H} is fixed-point-free and lim_{k} ∥f^{k}(x)∥ = 1 for some x in B_{H}, then the strong lim_{k} f^{k}(y) = ξ_{f} for all y in B_{H}.

Proof. For this x there exists R > 0 such that x ∈ E(ξ_{f}, R). The ellipsoid E(ξ_{f}, R) is f-invariant and therefore lim_{k} f^{k}(x) = ξ_{f}. Next we apply Lemma 3.10 to get lim_{k} f^{k}(y) = ξ_{f} for all y in B_{H}. ■

Theorem 12.9 [150]. Let f : B_{H} → B_{H} be a k_{B_{H}} -nonexpansive mapping. If f is fixed-point-free, then for each x in B_{H} the sequence of iterates {f_{j,λ}^{k}(x)} of the averaged mapping f_{j,λ} (j = 1, 2 and 0 < λ < 1) converges strongly to ξ_{f}, a point on the boundary of B.

Proof. The case of f_{1,λ}. Denote f_{1,λ}^{k}(x) by x_{k} and assume that {x_{k}} has a k_{B_{H}} - bounded subsequence {x_{k_{m}}}. Then

lim_{k} k_{B_{H}} (f_{1,λ}^{k+1} (x) , f_{1,λ}^{k} (x)) = lim_{k} k_{B_{H}} \frac{(f_{1,λ}^{k} (x) , x)}{k} = 0

by Theorem 10.7 and hence lim_{k} k_{B_{H}} (f_{1,λ}(x_{k}), x_{k}) = 0. It follows that the asymptotic center x̃ of {x_{k_{m}}} is a fixed point of f_{1,λ} and therefore x̃ is a fixed point of f, a contradiction. Hence lim_{k} ∥x_{k}∥ = 1 and the result follows from Lemmas 12.7 and 12.8.

The case of f_{2,λ}. Denote f_{2,λ}^{k}(x) by x_{k} and f(x_{k}) by w_{k}. Assume without loss of generality that 0 < λ ≤ \frac{1}{2}, and let

z_{k} = (1 − 2λ) x_{k} + 2λw_{k}.

Since

Φ_{ξ_{f}} (f(y)) ≤ Φ_{ξ_{f}} (y) and Φ_{ξ_{f}} (f_{2,λ}(y)) ≤ Φ_{ξ_{f}} (y)

for all y in B_{H}, the sequence {Φ_{ξ_{f}} (x_{k})} decreases to a limit R and

lim sup_{k} Φ_{ξ_{f}} (w_{k}) ≤ lim_{k} Φ_{ξ_{f}} (x_{k}) = R.

The convexity of the ellipsoids E(ξ_{f}, R) now implies that

lim sup_{k} Φ_{ξ_{f}} (z_{k}) ≤ R

too. Since

lim_{k} Φ_{ξ_{f}} (\frac{1}{2}x_{k} + \frac{1}{2}z_{k}) = lim_{k} Φ_{ξ_{f}} (x_{k+1}) = R,

we can apply Lemma 6.6 and conclude that lim_{k} ∥x_{k} − z_{k}∥ = 0. Hence

lim_{k} ∥x_{k} − f (x_{k})∥ = 0

aswell. Since f does not have a fixed point, this implies that {x_{k}} cannot have a k_{B_{H}} -bounded subsequence. Thus

lim_{k} ∥x_{k}∥ = 1

and once again the result follows from Lemmas 12.7 and 12.8. ■

Now we prove the Denjoy-Wollf theorem for a certain class of k\overline{B_{H}^{n}} -isometries.

Theorem 12.10 [124].

(i) Assume that I : \overline{B_{H}} → \overline{B_{H}} is a k_{B_{H}} -isometry. If

Fix (I_{|B_{H}} ) = ∅

and I has two fixed points {ξ, ξ_{I}} on the boundary ∂B_{H}, then the sequence of iterates \{(I)^{k}(x)\} tends to ξ_{I} for each x ∈ \overline{B_{H}} \ \{ξ\}.

(ii) Let I : \overline{B_{H}^{n}} → \overline{B_{H}^{n}} be a cyclic k_{B_{H}^{n}} -isometry (n ≥ 2) without fixed points inside B_{H}^{n} and with exactly two fixed points on the boundary ∂(B_{H}^{n}). Then there exists ξ_{I} ∈ Fix(I) such that {(I)^{k}(x)} tends to ξ_{I} for each x ∈ B_{H}^{n}.

Proof. (i) Without loss of generality we may assume that Fix(I) = {ξ_{I}, −ξ_{I}}. Then we have

lim_{k} (I)^{k} (0) = ξ_{I}

(see the proof of Theorem 8.6) and by Lemma 3.10 (I is a k_{B_{H}} -isometry) we get

lim_{k} (I)^{k} (x) = ξ_{I}

for each x ∈ B_{H}. Next, it is easy to see that for

x ∈ (∂B_{H} ∩ lin(ξ_{I})) \ {−ξ_{I}}

we also have

lim_{k} (I)^{k} (x) = ξ_{I}.

Finally, for k = 1, 2, ... and for every x = αξ_{I} + u ∈ ∂B_{H} with (ξ_{I}, u) = 0, we get

∥(I)^{k} (x)∥ = 1 and (I)^{k} (x) = (I)^{k} (αξ_{I}) + u_{k}, with (ξ_{I}, u_{k}) = 0.

Therefore \{(I)^{k}(x)\} converges to ξ_{I} for each x ∈ \overline{B_{H}} \ \{−ξ_{I}\}.

(ii) Let

I(x_{1}, ..., x_{n}) = ( (I)_{1} (x_{2}) , ..., (I)_{n−1} (x_{n}) , (I)_{n} (x_{1}) )

and

Ĩ = (I)_{1} ∘ (I)_{2} ∘ ⋯ ∘ (I)_{n} .

Let x ∈ B_{H}^{n}, k = jn + m, j = 1, 2, ..., and 0 ≤ m < n, then we have

(I)^{k}(x_{1}, ..., x_{n}) = ((Ĩ)^{j}(x_{1}), ((I)_{2} ∘ ⋯ ∘ (I)_{n} ∘ (Ĩ)^{j−1} ∘ (I)_{1})(x_{2}), …
…, ((I)_{n} ∘ (Ĩ)^{j−1} ∘ (I)_{1} ∘ ⋯ ∘ (I)_{n−1})(x_{n})),

if m = 0, and

I^{k}(x_{1}, ..., x_{n}) = (((Ĩ)^{j} ∘ (I)_{1} ∘ ⋯ ∘ (I)_{m})(x_{m+1}),
((I)_{2} ∘ ⋯ ∘ (I)_{n} ∘ (Ĩ)^{j−1} ∘ (I)_{1} ∘ ⋯ ∘ (I)_{m+1})(x_{m+2}), …
…, ((I)_{n−m} ∘ ⋯ ∘ (I)_{n} ∘ (Ĩ)^{j−1} ∘ (I)_{1} ∘ ⋯ ∘ (I)_{n−1})(x_{n}),
((I)_{n+1−m} ∘ ⋯ ∘ (I)_{n} ∘ (Ĩ)^{j})(x_{1}),
((I)_{n+2−m} ∘ ⋯ ∘ (I)_{n} ∘ (Ĩ)^{j} ∘ (I)_{1})(x_{2}), …
…, ((I)_{n} ∘ (Ĩ)^{j} ∘ (I)_{1} ∘ ⋯ ∘ (I)_{m−1})(x_{m})),

if 0 < m < n. By (i), {(I)^{k}(x_{1}, ..., x_{n})} tends to

ξ_{I} = (ξĨ, ((I)_{2} ∘ ⋯ ∘ (I)_{n})(ξĨ), …, (I)_{n}(ξĨ)) ∈ Fix(I).
■

Next we consider firmly k_{B_{H}} -nonexpansive mappings of the first and second kinds.

Definition 12.11 [60] .A mapping f : B_{H} → B_{H} is said to be firmly k_{B_{H}} -nonexpansive of the first kind if for each x and y in B_{H}, the function φ : [0, 1] → [0, ∞) defined by

φ (s) = k_{B_{H}} ((1 − s) x ⊕ sf (x) , (1 − s) y ⊕ sf(y))

is decreasing.

Remark 12.12 [60]. The nearest point projection r_{C} of Theorem 4.2 (ii) is firmly k_{B_{H}} -nonexpansive of the first kind.

Definition 12.13 [60] .A mapping f : B_{H} →B_{H} is said to be a firmly k_{B_{H}}-nonexpansive mapping of the second kind if for each x and y in B_{H}, the function ψ : [0, 1] → [0, ∞) defined by

ψ (s) = k_{B_{H}} ((1 − s) x + sf (x) , (1 − s) y + sf(y))

is decreasing.

Theorem 12.14 [60]. Let f : B_{H} → B_{H} be a k_{B_{H}} -nonexpansive mapping. Then, for each 0 ≤ t < 1, the mapping g_{t} : B_{H} → B_{H} (h_{t} : B_{H} → B_{H}) implicitly defined by

g_{t} (x) = (1 − t) x ⊕ tf(g_{t} (x)) (h_{t} (x) = (1 − t) x + tf(h_{t} (x)))

for all x in B_{H} is a firmly k_{B_{H}} -nonexpansive mapping of the first kind (a firmly k_{B_{H}} - nonexpansive mapping of the second kind).

Proof. Let z = (1 − s)x ⊕ sg_{t}(x) and w = (1 − s)y ⊕ sg_{t}(y). Noting that there exists 0 ≤ p ≤ 1 such that

g_{t} (x) = (1 − p)z ⊕ pf(g_{t} (x)) and g_{t} (y) = (1 − p)z ⊕ pf(g_{t} (y)),

we obtain

φ (1) = k_{B_{H}} (g_{t} (x) , g_{t}(y)) = k_{B_{H}} (g_{p} (z) , g_{p} (w)) ≤ k_{B_{H}} (z, w) = φ (s) .

Since φ is convex, the first result follows.

Now denote h_{t} = h (t, f) by h. The mapping h is k_{B_{H}} -nonexpansive. Let 0 ≤ r < s ≤ 1,

u = (1 − r) x + rh (x) , v = (1 − s) x + sh (x) , w = (1 − r) y + rh (y) ,

and

z = (1 − s) y + rh (y) .

We have to show that k_{B_{H}} (v, z) ≤ k_{B_{H}} (u, w). If s = 1, then

v = h (x) = h(p, f) (u) and z = h(y) = h(p, f) (w) ,

where

p = p (r) = \frac{t (1 − r)}{1 − tr}.

Therefore

k_{B_{H}} (v, z) = k_{B_{H}} (h(p, f) (u) , h(p, f) (w)) ≤ k_{B_{H}} (u, w) ,

as required. If s < 1, then

h (x) = h(p (s) , f) (v) and h(y) = h(p (s) , f) (z) .

We also have

v = (1 − q)u + qh (x) and z = (1 − q)w + qh (y) ,

with

q = \frac{s − r}{1 − r} .

Hence

v = h(q, h(p(s), f))(u) and z = h(q, h(p(s), f))(w),

so that once again, k_{B_{H}} (v, z) ≤ k_{B_{H}} (u, w). ■

Remark 12.15 If f : B_{H} → B_{H} is holomorphic, then the mappings {h_{t}}_{0≤t<1} defined in Theorem 12.14 (see also (5.1)) are holomorphic too. More information on firmly nonexpansive mappings of both kinds can be found, for example, in [59], [60], [153],

[159] and [160]. Here we only mention the following result.

Theorem 12.16 [59] , [60] , [153] , [159]. If f : B_{H} → B_{H} is a firmly k_{B_{H}} -nonexpansive mapping of the first or the second kind and Fix(f) = ∅, then f^{k}(x) → ξ_{f} for each x ∈ B_{H}.

We conclude this section with the k_{B_{H}} -nonexpansive version of the Chu-Mellon theorem [36].

Theorem 12.17 [115]. For each compact, k_{B_{H}} -nonexpansive and fixed-point-free mapping f : B_{H} → B_{H} there exists ξ_{f} ∈ ∂B_{H} such that the sequence {f^{k}} of the iterates of f converges to the constant map taking the value ξ_{f}, uniformly on each ball strictly inside B_{H}.

13. The Denjoy-Wolff theorem in Banach spaces

We begin by recalling a useful result of P. Yang [215] concerning a characterization of the horocycle in terms of the Poincaré distance on Δ. To wit, he established the following formula:

lim_{Δ∋w→ξ} [k_{Δ} (z, w) − k_{Δ} (0, w)] = \frac{1}{2} log \frac{|1 − z\overline{ξ}|^{2}}{1 − |z|^{2}} ,  (13.1)

where z ∈ Δ and ξ ∈ ∂Δ. In terms of the horocycle E(ξ, R) in Δ this provides the following equivalent definition:

E (ξ, R) = \{z ∈ Δ : lim_{Δ∋w→ξ} [k_{Δ} (z, w) − k_{Δ} (0, w)] < \frac{1}{2} log R\} .

Since the Kobayashi distance can be defined on each bounded domain in ℂ^{n}, one can try to extend this formula and use it as a definition of the horosphere in a domain in ℂ^{n}. Unfortunately, the limit analogous to the one in (13.1) does not exist in general. Therefore, the new idea of M. Abate [1] was to study two kinds of horospheres. More precisely, for x ∈ D, ξ ∈ ∂D, R > 0, x_{n} ∈ D, n = 1, 2, ..., and lim_{n→∞} x_{n} = ξ, the small horosphere E_{x}(ξ, R) and the big horosphere F_{x}(ξ, R) of center x and radius R are defined by

E_{x} (ξ, R) = \{y ∈ D : lim sup_{w→ξ} [k_{D} (y, w) − k_{D} (x, w)] < \frac{1}{2} log R\}

and

F_{x} (ξ, R) = \{y ∈ D : lim inf_{w→ξ} [k_{D} (y, w) − k_{D} (x, w)] < \frac{1}{2} log R\} .

It is obvious that

E_{x} (ξ, R) ⊂ F_{x} (ξ, R)

and the horospheres E_{x}(ξ, R) and F_{x}(ξ, R) are convex. Moreover, for each y ∈ D there exists R > 0 such that y ∈ E_{x}(ξ, R). Thus every assertion which proclaims for a bounded convex domain D in ℂ^{n} and every holomorphic self-mapping f : D → D the existence of a point ξ ∈ ∂D such that

f^{k} (E_{x} (ξ, R)) ⊂ F_{x} (ξ, R)

for all x ∈ D, R > 0 and k = 1, 2, ..., and

\overline{E_{x} (ξ, R)} ∩ ∂D = \overline{F_{x} (ξ, R)} ∩ ∂D = \{ξ\} ,

is a generalization of the Denjoy-Wolff theorem. This is true, for example, for a bounded strongly convex C^{2} domain D in ℂ^{n}.

Theorem 13.1 [1]. Let D be a bounded strongly convex C^{2} domain in ℂ^{n} and let f : D → D be a holomorphic mapping without fixed points. Then the sequence of iterates of f converges to a unique point ξ_{f} on the boundary ∂D.

Although Abate’s proof does not have a metric character, an application of Theorem 2.5 yields the following result.

Theorem 13.2 [135] Let D be a bounded strongly convex C^{2} domain in ℂ^{n} and let f : D → D be a k_{D}-nonexpansive mapping without fixed points. Then the sequence of iterates of f converges to a unique ξ_{f} on the boundary ∂D.

The proof of the latter theorem included the following simple calculation. Since the k_{D}-nonexpansive mapping f is fixed-point-free, the Earle-Hamilton theorem (Theorem 5.1) and Całka’s theorem (Theorem 2.5) show that there is a sequence

{x_{m}} = {h_{f} (s_{m}, z_{m})} = {f_{s_{m},z_{m}} (x_{m})} = {(1 − s_{m}) z_{m} + s_{m}f (x_{m})}

(z_{m} ∈ D and 0 < s_{m} < 1 for m = 1, 2, ..., and lim_{m→∞} s_{m} = 1) which is convergent to ξ_{f} ∈ ∂D. Let us observe that for arbitrary y ∈ D and k ∈ ℕ,

lim inf_{D∋w→ξ_{f}} [k_{D}(f^{k}(y), w) − k_{D} (x, w)]

≤ lim inf_{m→∞} [k_{D}(f^{k}(y), x_{m}) − k_{D} (x, x_{m})]

= lim inf_{m→∞} [k_{D} (f^{k} (y) , f_{s_{m},z_{m}}^{k} (y)) + k_{D} (f_{s_{m},z_{m}}^{k} (y) , f_{s_{m},z_{m}}^{k} (x_{m})) − k_{D} (x, x_{m})]

≤ lim inf_{m→∞} [k_{D} (y, x_{m}) − k_{D} (x, x_{m})]

≤ lim sup_{D∋w→ξ_{f}} [k_{D} (y, w) − k_{D} (x, w)] .

The above calculation guarantees that

f^{k} (E_{x} (ξ_{f}, R)) ⊂ F_{x} (ξ_{f}, R)

for all x ∈ D, R > 0 and k = 1, 2, ... . This calculation is also the basis for our subsequent studies of the behavior of the sequence of iterates in the open unit ball of a strictly convex Banach space.

Remark 13.3 [84]. If X is a Hilbert space H, then the limit

lim_{w→ξ} [k_{B_{H}} (z, w) − k_{B_{H}} (0, w)] = \frac{1}{2} log \frac{|1 − (z, ξ)|^{2}}{(1 − ∥z∥^{2})}

exists for every z ∈ B_{H} and ξ ∈ ∂B_{H}, and the ellipsoid

E (ξ, R) = \{z ∈ B_{H} : \frac{|1 − (z, ξ)|^{2}}{1 − ∥z∥^{2}} < R\}

coincides with the Abate horospheres E_{0} (ξ, R) = F_{0} (ξ, R) , where lim_{m} w_{m} = ξ.

Note that even in the case of a strictly convex Banach space X and its open unit ball B we do not know if

\overline{E_{x} (ξ, R)} ∩ ∂B = \overline{F_{x} (ξ, R)} ∩ ∂B = \{ξ\} .

Therefore the inclusions f^{k} (E_{x} (ξ, R)) ⊂ F_{x} (ξ, R) , k = 1, 2, ..., do not necessarily imply the convergence of the sequence of iterates {f^{k}(y)} even if we know that

lim_{k} ∥f^{k}(y)∥ = 1.

So we need to introduce a new kind of horospheres. To this end, let B be the open unit ball of a complex Banach space X. Let x ∈ B, ξ ∈ ∂B, R > 0, x_{m} ∈ D, m = 1, 2, ..., and lim_{m→∞} x_{m} = ξ. Fix a Banach limit LIM ∈ (l^{∞})* ([39], [41]). The new horosphere H(x, ξ, R, {x_{m}}) in B is defined as follows:

H (x, ξ, R, \{x_{m}\}) = \{y ∈ B : LIM [k_{B} (y, x_{m}) − k_{B} (x, x_{m})] < \frac{1}{2} log R\}

[85]. Since

|k_{B} (y, x_{m}) − k_{B} (x, x_{m})| ≤ k_{B} (y, x)

for each m = 1, 2, ... , the horospheres H(x, ξ, R, {x_{m}}) are well defined and have the following properties.

Theorem 13.4 [85] . Let X be a Banach space with the open unit ball B. Let ξ and {x_{m}} be fixed. Then the horospheres H(ξ, R, x, {x_{m}}) have the following properties:

(i) H (ξ, R, x, {x_{m}}) is convex and nonempty;

(ii) for every 0 < R_{1} < R_{2} we have

[\overline{H (ξ, R_{1}, x, \{x_{m}\})} ∩ B] ⊂ H (ξ, R_{2}, x, \{x_{m}\}) ;

(iii) ⋃_{R>0}H (ξ, R, x, {x_{m}}) = B and ⋂_{R>0}H (ξ, R, x, {x_{m}}) = ∅;

(iv) ξ ∈ ⋂_{R>0}\overline{H (ξ, R, x, \{x_{m}\})} ⊂ ∂B;

(v) if X is strictly convex, then ⋂_{R>0}\overline{H (ξ, R, x, \{x_{m}\})} = \{ξ\} .

Proof. (i) Observe that by Theorem 3.13,

LIM [k_{B}(sξ, x_{m}) − k_{B}(x, x_{m})] = LIM [k_{B}(sξ, x_{m}) − k_{B}(0, x_{m})] + LIM [k_{B}(0, x_{m}) − k_{B}(x, x_{m})]

= lim_{m} [k_{B}(sξ, x_{m}) − k_{B}(0, x_{m})] + LIM [k_{B}(0, x_{m}) − k_{B}(x, x_{m})]

≤ −k_{B}(0, sξ) + k_{B}(0, x)

for each 0 < s < 1. This shows that the horosphere H(ξ, R, x, {x_{m}}) is nonempty. Its convexity follows from Lemma 3.3.

(ii) and (iii) are obvious.

(iv) See the proof of (i).

(v) This assertion follows directly from (i) and (iv). ■

Theorem 13.5 [84]. Let X be a strictly convex Banach space with the open unit ball B. Let f : B → B be a compact k_{B}-nonexpansive map with no fixed point in B. Then there exists ξ_{f} ∈ ∂B such that the sequence {f^{k}} of iterates of f converges to the constant map taking the value ξ_{f}, uniformly on each ball strictly inside B.

Proof. Since f is fixed-point-free, the Earle-Hamilton theorem (Theorem 5.1) and the compactess assumption yield a sequence

{x_{m}} = {h_{f} (s_{m}, z_{m})} = {f_{s_{m},z_{m}} (x_{m})} = {(1 − s_{m}) z_{m} + s_{m}f (x_{m})}

(z_{m} ∈ B and 0 < s_{m} < 1 for m = 1, 2, ..., and lim_{m→∞} s_{m} = 1) which is convergent to ξ_{f} ∈ ∂B.

Let us observe that for arbitrary y ∈ B,

LIM [k_{B} (f (y) , x_{m}) − k_{B} (0, x_{m})]
= LIM [k_{B} (f (y) , f_{s_{m},z_{m}} (y)) + k_{B} (f_{s_{m},z_{m}} (y) , f_{s_{m},z_{m}} (x_{m})) − k_{B} (0, x_{m})]
≤ LIM [k_{B} (y, x_{m}) − k_{B} (0, x_{m})] .

The above inequality implies that

f (H (ξ, R, 0, {x_{m}})) ⊂ H (ξ, R, 0, {x_{m}})

for arbitrary R > 0. Fix z ∈ B. Using Całka’s theorem (Theorem 2.5) and the convexity of k_{B}-balls we can apply the asymptotic center method to conclude that

lim_{k→∞} ∥f^{k} (z)∥ = 1.  (13.2)

Let A ⊂ ∂B denote the set of all accumulation points of the sequence {f^{k}(z)}. By the compactness of f and (13.2) the set A is nonempty and by Lemma 3.11 it is independent of the choice of z. Theorem 13.4 now implies that

∅ ≠ A ⊂ ⋂_{R>0} \overline{H (ξ, R, 0, \{x_{m}\})} ∩ ∂B = \{ξ\} .

Hence lim_{k} f^{k}(z) = ξ and by Lemma 3.11 and the compactness of f the sequence {f^{k}} converges to the constant map ξ_{f}, uniformly on each ball strictly inside B. ■

Now we examine mappings f : B → B which are k_{B}-nonexpansive and condensing with respect to the Kuratowski measure of noncompactness α_{∥·∥}. We begin with the following simple lemma.

Lemma 13.6 [85]. Let D be a nonempty bounded subset of a metric space (M, d). If f : D → D is condensing with respect to the Kuratowski measure of noncompactness α_{d} and {x_{m}} is a sequence of elements of D such that d(x_{m}, f(x_{m})) → 0, then the set of all elements of the sequence {x_{m}} is totally bounded in (M, d).

Proof. Since f is condensing with respect to α_{d} and

α_{d} ({x_{m} : m = 1, 2, ...}) = α_{d} ({f (x_{m}) : m = 1, 2, ...}) ,

the set {x_{m} : m = 1, 2, ...} must be totally bounded in (M, d). ■

Corollary 13.7 [85]. Let D be a bounded convex domain in a Banach space (X, ∥·∥). If f : D → D is k_{D}-nonexpansive and condensing with respect to α_{∥·∥}; C is a nonempty, k_{D}-closed, and f-invariant subset of D; {s_{m}} is a sequence such that lim_{m} s_{m} = 1, 0 < s_{m} < 1; and {z_{m}} is a sequence of elements of C, then the sequence {x_{m}} implicitly defined by

x_{m} = h_{f} (s_{m}, z_{m}) = f_{s_{m},z_{m}} (x_{m}) = (1 − s_{m}) z_{m} + s_{m}f (x_{m})

for each m contains a norm-convergent subsequence.

Proof. It is obvious that f(x_{m}) − x_{m} → 0. Thus it is sufficient to apply Lemma 13.6. ■

We now quote a convergence theorem for fixed-point-free condensing mappings. In addition to our previous arguments its proof also requires an application of Theorem 2.7 and Corollary 13.7.

Theorem 13.8 [85]. If B is the open unit ball of a strictly convex Banach space (X, ∥·∥) and f : B → B is k_{B}-nonexpansive, condensing with respect to the Kuratowski measure of noncompactness α_{∥·∥}, and fixed-point-free, then there exists ξ_{f} ∈ ∂B such that the sequence {f^{k}} of iterates of f converges in the compact-open topology to the constant map taking the value ξ_{f}.

In some Banach spaces we are able to show that the convergence provided by Theorem 13.8 is actually uniform on each ball strictly inside B.

Theorem 13.9 [83]. Let N be a norming set for a strictly convex Banach space (X, ∥· ∥), B the open unit ball in (X, ∥ · ∥), and assume that \overline{B} is sequentially compact in the topology σ(X, N). If (X, ∥ · ∥) has the Kadec-Klee property with respect to σ(X, N) and f : B → B is k_{B}-nonexpansive, condensing with respect to α_{∥·∥} and fixed-point-free, then there exists ξ_{f} ∈ ∂B such that the sequence {f^{k}} of iterates of f converges to the constant map taking the value ξ_{f}, uniformly on each ball strictly inside B.

Proof. By Theorem 13.8 the sequence of iterates {f^{k}(0)} converges to ξ_{f} ∈ ∂B. Let us choose an arbitrary sequence {f^{k_{m}}(x_{m})} with sup_{m} ∥x_{m}∥ = c_{1} < 1. Since

sup_{m} k_{B} (f^{k_{m}}(0), f^{k_{m}}(x_{m})) ≤ sup_{j} k_{B} (0, x_{m})
= sup_{m} [ arg tanh ∥x_{m}∥] = arg tanh c_{1} = c < ∞,

Lemma 3.10 yields the strong convergence of {f^{k_{m}}(x_{m})} to ξ_{f}. ■

Remark 13.10 [29]. In the case of separable Banach spaces one can get analogous results to Theorems 3.4 and 13.6 by using total sets [15] instead of norming sets.

Corollary 13.11 [85]. Let X be a uniformly convex Banach space with the open unit ball B. Let f : B → B be k_{B}-nonexpansive, condensing with respect to α_{∥·∥}, and fixedpoint-free. Then there exists ξ_{f} ∈ ∂B such that the sequence {f^{k}} of iterates of f converges on B to the constant map taking the value ξ_{f}, uniformly on each ball strictly inside B.

Let f : B → B be a k_{B}-nonexpansive mapping and 0 < s < 1. As before, we define h_{f}(s, ·) : B → B by

h_{f} (s, z) = f_{s,z}(h_{f} (s, z)) = (1 − s) z + sf(h_{f} (s, z)) = lim_{k} f_{s,z}^{k} (0)

for z ∈ B. This mapping is k_{B}-nonexpansive and holomorphic if f is holomorphic by Corollary 2.11. The method of proof of Theorem 13.9 combined with Corollary 13.7 and Lemma 3.11 yield the following proposition.

Proposition 13.12 [85]. Let (X, ∥ · ∥) be a strictly convex Banach space with the open unit ball B. If f : B → B is k_{B}-nonexpansive, condensing with respect to α_{∥·∥} and fixed-point-free, then there exists ξ_{f} ∈ ∂B such that {h_{f}(s, ·)} converges uniformly on B as s → 1^{−} to the constant map taking the value ξ_{f}. This ξ_{f} is the unique limit point of the sequence of iterates {f^{k}}.

Remark 13.13 In view of A. Stachura’s example (Example 12.5) and Theorem 6.4, the approximating curves {h_{f}(·, x)} of fixed-point-free k_{B}-nonexpansive self-mappings of the Hilbert ball B_{H} behave much better than the sequences of iterates {f^{k}(x)}. Namely, they always converge to the so-called “sink point” ξ_{f} on the boundary of B_{H} even when the sequences of iterates do not.

We now present two examples of nonreflexive Banach spaces to which Theorem 13.9 can be applied.

Example 13.14 [83]. Let l^{1} be furnished with the norm

∥x∥ = (∥x∥_{1}^{2} + ∑_{j=1}^{∞} \frac{x_{j}^{2}}{2^{j}} )^{2} ,

where ∥ · ∥_{1} is the usual norm of l^{1} and x = {x_{j}} ∈ l^{1}. If we take

N = c_{0} ⊂ (l^{1})^{∗}

(i.e., u(x) = ∑_{j=1}^{∞} u_{j}x_{j} for u = {u_{j}} ∈ N = c_{0} and x = {x_{j}} ∈ l^{1}), then we can observe that N is a norming set for the strictly convex Banach space (l^{1}, ∥ · ∥), the closed unit ball \overline{B} in (l^{1}, ∥ · ∥) is sequentially compact in the topology σ(l^{1}, N), and (l^{1}, ∥ · ∥) has the Kadec-Klee property with respect to σ(l^{1}, N).

Example 13.15 The James quasi-reflexive space J consists of all null sequences x = {x^{i}} = ∑_{i=1}^{∞} x^{i}e_{i} ( {e_{i}} is the standard basis in c_{0}) for which the squared variation

sup_{m, p_{1}<···<p_{m}} [∑_{j=2}^{m} |x^{p_{j}} − x^{p_{j−1}}|^{2}]^{\frac{1}{2}}  (13.3)

is finite, with the norm ∥ · ∥_{J} given by (13.3) (see [80] and [81]). A predual Banach space Ĩ to (J, ∥ · ∥_{J}) is generated by the biorthogonal functionals u_{i} to the basis {e′_{i}} = {e_{1} + ⋯ + e_{i}} (see [81] for details).The space (J, ∥ · ∥_{J}) has the uniform Opial property (see [147] for the definition) [125] and the uniform Kadec-Klee property, both with respect to the w^{∗}-topology ([125] combined with [88]). Let J be furnished with the new norm:

∥x∥ = (∥x∥_{J}^{2} + ∑_{i=1}^{∞} \frac{(u_{i} (x))^{2}}{2^{i}} )^{2} .

If we take N = Ĩ, then we can observe that N is a norming set for the strictly convex Banach space (J, ∥ · ∥), the closed unit ball \overline{B} in (J, ∥ · ∥) is sequentially compact in the topology σ(J, N), and (J, ∥ · ∥) has the Kadec-Klee property with respect to σ(J, N) [29].

14. Retractions onto fixed point sets

In order to formulate an extension of the Denjoy-Wolff theorem (Theorem 12.1) in the case where the mapping has a fixed point we need the following two definitions.

Definition 14.1 [53] , [79] . Let D be a domain in a complex Banach space (X, ∥·∥). A net {f_{j}}_{j∈J} ⊂ H(D, X) is said to converge to a mapping f ∈ H(D, X) in the topology of locally uniform convergence over D (or briefly, T -converge) if for every ball B_{∥·∥} strictly inside D,

lim_{j∈J} sup_{x∈B_{∥·∥}} ∥f_{j} (x) − f (x)∥ = 0.

In this case we write f = T -lim_{j∈J} f_{j}.

If D is a bounded domain in a finite-dimensional X, then its T -topology is equivalent to the compact open topology on D, i.e., the topology of uniform convergence on compact subsets of D.

Definition 14.2 A mapping f ∈ H(D) is said to be power convergent on D (with respect to the T -topology) if the sequence of iterates {f^{k} : f^{0} = I, f^{k+1} = f ∘ f^{k}, k = 0, 1, 2, ...} T -converges to a mapping f̃ ∈ H(D, X).

So, in these terms the classical Denjoy-Wolff theorem (Theorem 12.1) states that a holomorphic self-mapping of the unit disc Δ is power convergent if and only if it is not an elliptic automorphism of Δ. In other words, if f ∈ H(Δ) is not the identity and has a fixed point α ∈ Δ, then it is power convergent if and only if |f′(α)| < 1.

Theorem 14.3 [198] , [200] . Let D be a bounded convex domain in a complex Banach space X and let f ∈ H(D) have a fixed point. Then f is power convergent if and only if there exists α ∈ Fix(f) such that the linear operator A = f′(α) satisfies the following conditions:

(i) the spectrum σ(A) ⊂ Δ ∪ {1}, and

(ii) 1 is a pole of at most the first order for the resolvent of A.

Note that in this case the limit mapping r = T -lim_{k→∞} f^{k} is a holomorphic retraction of D onto Fix(f), i.e., r^{2} = r and Fix(f) = r(D). Therefore Fix(f) ⊂ D is a holomorphic retract of D. Note also that once the existence of a holomorphic retraction onto Fix(f) is established, it follows that Fix(f) is a complex-analytic submanifold of D [33]. Thus it is natural to ask if Fix(f) is a holomorphic retract of D even if f is not power convergent. If it is, then it would be of interest to construct a retraction of D onto Fix(F).

To formulate some results in this direction we will use the following definitions.

Definition 14.4 Let h be a holomorphic mapping from a domain D ⊂ X into X, and suppose that Fix(h) ≠ ∅. Set g = I − h. A point α ∈ Fix(h) is said to be quasi-regular if the following condition holds:

Ker g′ (α) ⊕ Im g′ (α) = X.  (14.1)

If, in addition, Ker g′(α) = 0, then we say that α is a regular fixed point of h.

Note that condition (14.1) is, in fact, equivalent to condition (ii) of Theorem 14.3 ([62], [191]). However, if condition (i) of this theorem does not hold, then f is not power convergent. Nevertheless, it is shown in [141] that if f ∈ H(D) has at least one quasiregular fixed point, then there is another mapping φ ∈ H(D) with Fix(φ) = Fix(f) which is power convergent. More precisely, P. Mazet and J.-P. Vigué considered the Cesàro averages

C_{m} = \frac{1}{m} ∑_{k=0}^{m−1} f^{k}  (14.2)

and proved the following result.

Theorem 14.5 [141] . Let D be a bounded convex domain in X, and let f ∈ H(D) have a fixed point. If Fix(f) contains at least one quasi-regular point, then there is an integer p such that the mapping C_{p} defined by (14.2) is power convergent to a holomorphic retraction of D onto Fix(f).

The following result shows, in particular, that in Theorem 14.5 the Cesàro averages defined by (14.2) are, in fact, power convergent for all p = 2, 3, ... .

Theorem 14.6 [166]. Let D be a bounded convex domain in X, and let f ∈ H(D) have a fixed point. If Fix(f) contains a quasi-regular point α ∈ D, then each mapping of the form

φ = ∑_{k=0}^{m} β_{k}f^{k},

where ∑_{k=0}^{m} β_{k} = 1 and 0 ≤ β_{k} ≠ 1 for all 0 ≤ k ≤ m, is power convergent.

Now we tackle the retraction problem using metric methods.

Theorem 14.7 [44], [126]. Let D ⊂ X be a bounded convex domain in X. If f : D → D is a compact holomorphic (k_{D}-nonexpansive) mapping and Fix(f) ≠ ∅, then Fix(f) is relatively compact in X and a holomorphic (k_{D}-nonexpansive) retract of D.

Proof. Since f is compact, for each x ∈ D the sequence {h_{n}(x)} defined by

h_{m} (x) = \frac{1}{m}x + (1 − \frac{1}{m}) f (h_{m} (x))

for m = 1, 2, ..., is relatively compact in X by Corollary 13.7, and since Fix(f) ≠ ∅, it lies strictly inside D. Let {m_{α}}_{α∈J} be an ultranet in ℕ ([14] , [50], [87]). Then the ultranet {h_{m_{α}}}_{α} is pointwise convergent to a compact holomorphic (k_{D}-nonexpansive) retraction r : D → Fix(f). ■

A more complicated proof is required in the case of α_{∥·∥}-condensing mappings. However, by using Bruck’s method [27] (which can be applied by Theorem 3.7), we can obtain the following result.

Theorem 14.8 [126]. Let D be a bounded convex domain in a reflexive Banach space X. If f : D → D is a condensing holomorphic (k_{D}-nonexpansive) mapping and Fix(f) ≠ ∅, then Fix(f) is relatively compact in X and a holomorphic (k_{D}- nonexpansive) retract of D.

To prove Theorem 14.8 we need three auxiliary lemmas. In all these lemmas we assume that X is a reflexive Banach space and that D ⊂ X is a bounded convex domain.

Lemma 14.9 [126]. Suppose A is a nonempty subset of D and let

N (A) = {g : D → D :g is holomorphic (k_{D}-nonexpansive)
and g(x) = x for all x ∈ A}.

Then N (A) is compact in the topology of weak pointwise convergence.

Proof. Let x_{0} be a point in A. Then

N (A) ⊂ ∏_{x∈D} \overline{B_{k_{D}}} (x_{0}, k_{D} (x, x_{0})) ,

where \overline{B_{k_{D}}}(y, r) denotes a closed ball in the metric space (D, k_{D}). Since each \overline{B_{k_{D}}}(y, r) is convex (see Lemma 3.3), hence weakly compact, the set

∏_{x∈D} \overline{B_{k_{D}}} (x_{0}, k_{D} (x, x_{0}))

is compact in the topology of weak pointwise convergence. Now to obtain our claim it is sufficient to observe that N (A) is closed in this topology (see Corollary 2.11 and Theorem 3.7). ■

Lemma 14.10 [126]. If A is a nonempty subset of D, then there exists an r in N (A) such that each g ∈ N (A) acts as an isometric mapping on the range of r. In other words,

k_{D} (g(r (x)), g(r(y))) = k_{D} (r (x) , r(y))

for all x, y ∈ D.

Proof. We introduce the following order ≤ on N (A):

g ≤ h ≡ ( ∀_{x,y∈D}k_{D}(g(x), g(y)) ≤ k_{D}(h(x), h(y))).

It is easy to observe (see the proof of Lemma 14.9) that the initial segment

I (h) = {g ∈ N (A) : g ≤ h}

is compact in the topology of weak pointwise convergence. By the Kuratowski-Zorn Lemma, (N (A), ≤) contains a minimal element r and for this element we have

∀_{g∈N (A)}g ∘ r ∈ N (A) .

The minimality of r in N (A) now implies that

∀_{g∈N (A)} ∀_{x,y∈D}k_{D}(g(r(x)), g(r(y))) = k_{D}(r(x), r(y)).

■

Lemma 14.11 [126]. If A is a nonempty subset of D and for every y ∈ D there exists g in N (A) such that g(y) ∈ A, then A is a holomorphic (k_{D}-nonexpansive) retract of D.

Proof. We will show that the mapping r obtained in Lemma 14.10 is a retraction of D onto A. To this end, it suffices to prove that r(x) ∈ A for each x ∈ D. Indeed, by the assumption of the lemma applied to the point r(x), there exists g ∈ N (A) such that g(r(x)) ∈ A. Applying now Lemma 14.10 to x and z = g(r(x)) we get

k_{D}(g(r(x)), g(r(z))) = k_{D}(r(x), r(z)).

However, since z ∈ A we have z = g(r(z)) = r(z), and therefore g(r(x)) = g(r(z)) and r(x) = r(z) = z ∈ A. ■

We are now ready to prove Theorem 14.8.

Proof of Theorem 14.8. It is obvious that Fix(f) is relatively compact in X. Let A = Fix(f). To apply Lemma 14.11 we have to show that for each x ∈ D there exists g ∈ N (A) such that g(x) ∈ A. When we fix x ∈ D, the x-th coordinate projection of N (A) is equal to

C = {h (x) : h ∈ N (A)}

and C is nonempty, weakly compact and convex. Since f ∘ h ∈ N (A) whenever h ∈ N (A), we also have f(C) ⊂ C. Using the assumption that f is α_{∥·∥} -condensing we see that f has a fixed point in C. This means that there exists g ∈ N(A) with g(x) ∈ A = Fix(f), as required. ■

Remark 14.12 In [141] and [142] P. Mazet and J.-P. Vigué prove that in a reflexive Banach space X, if D ⊂ X is a bounded convex domain and f : D → D is holomorphic with Fix(f) ≠ ∅, then Fix(f) is a holomorphic retract of D. However, their methods of proof use complex-analytic arguments and are not applicable to k_{D}-nonexpansive mappings.

For our next result we need another lemma.

Lemma 14.13 [126]. Let X be a reflexive Banach space and let D ⊂ X be a bounded convex domain. If f : D → D is a holomorphic (k_{D}-nonexpansive) mapping with Fix(f) ≠ ∅, and the set A is relatively compact in X, invariant under f, and a holomorphic (k_{D}-nonexpansive) retract of D, then Fix(f) ∩ A is a nonempty, relatively compact in X and holomorphic (k_{D}-nonexpansive) retract of D.

Proof. Let r be a holomorphic (k_{D}-nonexpansive) compact retraction of D onto A. Since f ∘ r : D ⟶ A, (f ∘ r)_{|A} = f_{|A}, Fix(f) ≠ ∅, and f : D → D is a holomorphic (k_{D}-nonexpansive) mapping, there exists a k_{D}-bounded, convex and ∥ · ∥-closed subset C ⊂ D which is (f ∘ r)-invariant. Next, we observe that A is relatively compact in X and

(f ∘ r) (C) ⊂ C ∩ A.

This implies that the set Fix(f ∘ r) is nonempty. It is easy to see that

Fix (f) ∩ A = Fix (f ∘ r)

and therefore Fix(f ∘ r) is relatively compact in X. Now we show that Fix(f ∘ r) is a holomorphic (k_{D}-nonexpansive) retract. Let A_{1} = Fix(f) ∩ A. To apply Lemma 14.11 we need to prove that for each x ∈ D, there exists g ∈ N (A_{1}) such that g(x) ∈ A_{1}. Indeed, when we fix x ∈ D, the x -th coordinate projection of N (A_{1}) is equal to

C_{1} = {h (x) : h ∈ N (A_{1})}

and C_{1} is nonempty, weakly compact and convex. Since (f ∘ r) ∘ h ∈ N (A_{1}) whenever h ∈ N (A_{1}), we also have (f ∘ r)(C_{1}) ⊂ C_{1}. We now observe that

(f ∘ r) (C_{1}) ⊂ C_{1} ∩ A

and therefore f ∘ r has a fixed point in C_{1}. This means that there exists g ∈ N (A_{1}) with g(x) ∈ A_{1} = Fix(f ∘ r). ■

Theorem 14.14 [126]. Let X be a reflexive Banach space and let D ⊂ X be a bounded convex domain. If for each j = 1, 2, ..., n, f_{j} : D → D is a holomorphic (k_{D}-nonexpansive) mapping with Fix(f_{j}) ≠ ∅, at least one of these mappings is condensing and all the mappings f_{j}, j = 1, 2, ..., n, commute, then ∅ ≠ ∩_{j=1}^{n} Fix(f_{j}) is relatively compact in X and a holomorphic ( k_{D}-nonexpansive) retract of D.

Proof. We apply mathematical induction with respect to n, Theorem 14.8 and Lemma 14.13. ■

We conclude this section with another existence result for a possibly infinite family of commuting mappings.

Theorem 14.15 [126]. Let X be a reflexive strictly convex Banach space and let B be the open unit ball in X. If f_{j} : \overline{B} → \overline{B}, j ∈ J, is a family of norm continuous condensing mappings which are holomorphic (each f_{j} ∈ N(\overline{B})) in B, Fix(f_{j}) ≠ ∅ for each j, and all the mappings f_{j} commute, then ∩_{j∈J} Fix(f_{j}) is nonempty.

Proof. We consider three cases.

Case 1. There exists j_{0} ∈ J such that f_{j_{0}}(\overline{B}) ⊂ ∂B. Then f_{j_{o}} is a constant mapping (see Corollary 2.17, the definition of N(\overline{B}) in Section 7 and Lemma 3.11) and f_{j_{0}} ≡ x_{0} for some x_{0} ∈ ∂B. Hence we obtain

f_{j} (x_{0}) = f_{j} (f_{j_{0}} (x_{0})) = f_{j_{0}} (f_{j} (x_{0})) = x_{0}

for each j ∈ J. In other words,

x_{0} ∈ ⋂_{j∈J} Fix(f_{j}).

Case 2. Fix(f_{j|B}) ≠ ∅ for each j ∈ J. Then by Theorem 14.14 the set

∩_{k=1}^{m}\overline{Fix(f_{j_{k}|B})} ⊂ ∩_{k=1}^{m} Fix(f_{j_{k}})

is nonempty and compact in X for every choice {j_{1}, ..., j_{m}} of indices. Hence we obtain

∅ ≠ ⋂_{j∈J} \overline{Fix(f_{j|B})} ⊂ ⋂_{j∈J} Fix(f_{j}).

Case 3. f_{j}(B) ⊂ B for each j ∈ J and Fix(f_{j_{0}|B}) = ∅ for some j_{0} ∈ J. Then there exists ξ_{f_{j_{0}}} ∈ ∂B such that

f_{j_{0}}^{k} (x) →_{k} ξ_{f_{j_{0}}}

for every x ∈ B (see Theorem 13.8). Hence we get

ξ_{f_{j_{0}}} = lim_{k} f_{j_{0}}^{k} (f_{j} (0)) = f_{j} (lim_{k} f_{j_{0}}^{k} (0)) = f_{j} (ξ_{f_{j_{0}}} )

for each j ∈ J, and therefore

ξ_{f_{j_{0}}} ∈ ⋂_{j∈J} Fix(f_{j}).

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15. Fixed points of continuous semigroups

In this section we examine continuous semigroups, their fixed points and their orbits. Continuous semigroups of holomorphic mappings arise in several diverse fields, including, for instance, Markov stochastic branching processes, Krein spaces, and the geometry of complex Banach spaces (see [161] and [164] for relevant references). We begin with the following definitions.

Definition 15.1 A family S = {f_{t}}, where either t ∈ ℝ^{+} = {s ∈ R : s ≥ 0} or t ∈ ℕ ∪ {0}, of self-mappings of a metric space (M, d) is called a (one-parameter) semigroup if f_{s+t} = f_{s} ∘ f_{t}, s, t ∈ ℝ^{+} (s, t ∈ ℕ ∪ {0}), and f_{0} = I. A semigroup S = {f_{t}}, t ∈ ℝ^{+}, is said to be (strongly) continuous if the function f_{(·)}(x) : ℝ^{+} → M is continuous in t for each x ∈ M. If t ∈ ℕ ∪ {0} we say that the semigroup S is discrete . In other words, a discrete semigroup S = {f_{t}}, t ∈ ℕ ∪ {0}, is the family of iterates of a self-mapping f = f_{1} : M → M.

Definition 15.2 Let S = {f_{t}}, t ∈ ℝ^{+} = {s ∈ ℝ : s ≥ 0} (or t ∈ ℕ ∪ {0}) be a continuous (or discrete) semigroup acting on a metric space M. A point x ∈ M is said to be a fixed (or a stationary) point of S if f_{t}(x) = x for all t ∈ ℝ^{+} (or t ∈ ℕ). A point x ∈ M is said to be a periodic point of S if there is t_{0} ∈ ℝ^{+} (or t_{0} ∈ ℕ), such that f_{t_{0}}(x) = x.

Note that in the discrete case a stationary point x ∈ M of S is simply a fixed point of f_{1}, and a periodic point x ∈ M of S is a fixed point of f_{1}^{m}, where m ∈ ℕ. In the sequel, the mappings f_{t}, t ∈ ℝ^{+} (or t ∈ ℕ), will often be denoted by f(t).

Theorem 15.3 [126]. Let S = {f(t)} be a continuous or discrete semigroup of k_{B_{H}^{n}} - nonexpansive self-mappings of B_{H}^{n}. Then the following statements are equivalent:

(i) there is t_{0} > 0 such that f(t_{0}) has a fixed point in B_{H}^{n};

(ii) f(t) has a fixed point for all t;

(iii) there is a stationary point of the semigroup S.

Proof. It is clear that (iii)⇒(ii)⇒(i). Therefore it is enough to show that (i)⇒(iii). Indeed, let x ∈ B_{H}^{n} be a fixed point of f(t_{0}) for some t_{0}. Then x is a periodic point of S, i.e.,

f (t + t_{0}) (x) = f (t) (f (t_{0}) (x)) = f (t) (x)

for all t. But the set {f(t)(x) : 0 ≤ t ≤ t_{0}} is a compact subset of B_{H}^{n} and therefore it is k_{B_{H}^{n}} -bounded. By Theorem 7.6 this implies the existence of a common fixed point of S. ■

Corollary 15.4 [126]. Let S be a continuous semigroup of holomorphic mappings of B_{H} such that for some t_{0} ∈ (0, ∞), f_{t_{0}} = I. Then S is a group of automorphisms of B_{H} of elliptic type, i.e. f (t) = M_{−a}^{−1} ∘ e^{tA} ∘ M_{−a}, where A is a linear conservative operator (Re (Ax, x) = 0 for all x in H), a ∈ B_{H} and M_{−a} is a Möbius transformation of B_{H}.

Proof. Since f_{t_{0}} = I, Theorem 15.3 shows that S has a common fixed point a ∈ B_{H}. Consider the semigroup f̃(t) = M_{−a}∘f_{t}∘M_{−a}^{−1} defined on B_{H}. It is clear that f̃(t)(0) = 0 for all t ≥ 0 and that f̃(t_{0}) = I. Since

f̃(t) ∘ f̃(t_{0} − t) = f̃(t_{0}) = I

for all 0 ≤ t ≤ t_{0}, we see that for all 0 ≤ t ≤ t_{0}, f̃^{−1}(t) = f̃(t_{0} − t) and f̃(t) is an automorphism of B_{H}. Hence f̃(t) is the restriction to B_{H} of a linear isometry of H onto itself for each t ≥ 0 (see Theorem 8.1). This implies our assertion (see [114], [204]).

■

Definition 15.5 We will say that a class G of k_{B}-nonexpansive mappings on B_{H} has the Denjoy-Wolff property if whenever f ∈ G has no fixed point in B_{H} the sequence of iterates {f^{k}} strongly converges to a point ξ_{f} on the boundary ∂B_{H} of B_{H}, uniformly on each compact subset of B_{H}.

Proposition 15.6 [126]. Let G be a class of k_{B_{H}} -nonexpansive mappings on B_{H} which has the Denjoy-Wolff property. Let S = {f(t)}_{t≥0} be a one-parameter continuous semigroup of k_{B_{H}} -nonexpansive mappings on B_{H} which has no common fixed point in B_{H}. If f(t_{0}) ∈ G for at least one t_{0} > 0, then there is a point ξ ∈ ∂B_{H} such that S converges to ξ, as t tends to infinity, uniformly on each compact subset of B_{H}.

Proof. First we note that by Theorem 15.3, f(t_{0}) = f_{t_{0}} has no fixed point in B_{H}. Therefore there is a point ξ_{f_{t_{0}}} ∈ ∂B_{H} such that f_{kt_{0}} = f_{t_{0}}^{k} converges to ξ_{f_{t_{0}}} , uniformly on each compact subset of B_{H}. Let C̃ be a compact subset of B_{H}. Since the semigroup S = {f(t)}_{t≥0} is continuous, the set

C = {z_{t} = f (t) z̃: z̃ ∈ C̃, 0 ≤ t ≤ t_{0}}

is also a compact subset of B_{H}. Therefore, using the notation z_{t} = f(t)z̃, for each ϵ > 0 one can find k_{0} ∈ ℕ such that

sup_{z∈C} ∥f_{t_{0}}^{k} (z) − ξ_{f_{t_{0}}} ∥ = sup_{z̃∈C̃} sup_{0≤t≤t_{0}} ∥f_{kt_{0}} (z_{t}) − ξ_{f_{t_{0}}} ∥ = sup_{z̃∈C̃} sup_{0≤t≤t_{0}} ∥f_{kt_{0}+t} \binom{}{z} − ξ_{f_{t_{0}}} ∥ < ϵ

for all k ≥ k_{0}. It now follows that

∥f_{t} \binom{}{z} − ξ_{f_{t_{0}}} ∥ < ϵ

for each t ≥ k_{0}t_{0} and z̃ ∈ C̃. ■

Remark 15.7 [126]. The following classes of self-mappings of B_{H} are known to enjoy the Denjoy-Wolff property:

(1) the class G_{1} consisting of the condensing k_{B_{H}} -nonexpansive mappings (see Theorem 13.8);

(2) the class G_{2} consisting of the firmly k_{B_{H}} -nonexpansive mappings of the first kind (see Theorem 12.16);

(3) the class G_{3} consisting of the firmly k_{B_{H}} -nonexpansive mappings of the second kind (see Theorem 12.16);

(4) the class G_{4} consisting of the averaged mappings of the first kind, i.e., f_{1,λ} = (1 − λ)I ⊕ λf, where f is k_{B_{H}} -nonexpansive and λ ∈ (0, 1) (see Theorem 12.9);

(5) the class G_{5} consisting of the averaged mappings of the second kind, i.e., f_{2,λ} = (1 − λ)I + λf, where f is k_{B_{H}} -nonexpansive and λ ∈ (0, 1) (see Theorem 12.9).

Thus if a semigroup S = {f(t)}_{t≥0} contains at least one element f(t_{0}), t_{0} > 0, of one of the classes G_{i}, 1 ≤ i ≤ 5, then we have the conclusion of Proposition 15.6.

Corollary 15.8 [126]. Let f̃ be a k_{B_{H}} -nonexpansive self-mapping of B_{H}.

1) The Cauchy problem

\{ \begin{matrix}\frac{∂u(t,x)}{∂t} + u (t, x) − f̃(u (t, x)) = 0 \\ u (0, x) = x \end{matrix}

has a unique solution {u(t, x) : t ∈ ℝ^{+}} ⊂ B_{H} for each x ∈ B_{H}.

2) The semigroup {f(t)}_{t≥0} defined by the formula

f (t) x = u (t, x)

consists of k_{B_{H}} -nonexpansive mappings f(t), t ≥ 0.

3) If f̃ has no fixed point in B_{H}, then f(t) is also fixed-point-free for all t > 0, and f(t)x converges, as t → ∞, to a point ξ on the boundary of B_{H} for all x ∈ B_{H}.

Proof. First we note that the curve g_{t} = (1 − t)I + tf̃ converges to I as t → 0^{+} and

\frac{1}{t} (I − g_{t}) = I − f̃.

Since

k_{B_{H}} (g_{t} (x) , g_{t}(y)) ≤ max [k_{B_{H}} (x, y) , k_{B_{H}} (f̃(x), f̃(y))] ,

each g_{t} is k_{B}-nonexpansive. Hence it follows by the product formula (see [167]) that for each t > 0 the mapping f(t) given by

f (t) = lim_{k→∞} gk_{\frac{t}{k}}

is the solution operator for the Cauchy problem in 1) and is also a k_{B_{H}} -nonexpansive mapping on B_{H}. If now f̃ has no fixed point in B_{H}, then it follows that

⋂_{t>0} Fix (f (t)) = ∅.

By Theorem 15.3 we now see that each f_{t} has no fixed point in B_{H}.

At the same time direct calculations show that the following formula holds

f (t) (x) = e^{−t}x + ∫_{0}^{t} e^{−(t−s)}f̃(f (s) (x) )ds.

Since

∥f̃(f (s) x)∥ < 1

for all s ∈ [0, t] and x ∈ B_{H}, we can write

f (t) = e^{−t}I + (1 − e^{−t}) g, where g = \frac{1}{1 − e^{−t}} ∫_{0}^{t} e^{−(t−s)} (f̃∘ f (s)) ds

is a k_{B_{H}} -nonexpansive mapping on B_{H} (cf. [18]). Hence f_{t} is an averaged mapping of the second kind and the result follows by Proposition 15.6 and Remark 15.7. ■

Remark 15.9 Similar results for other k_{B_{H}} -nonexpansive semigroups can be found in [154] and [160]. The paper [154] also contains a continuous analog of Theorem 10.5.

A continuous one-dimensional analog of the classical Denjoy-Wolff Theorem (Theorem 12.1) was established by E. Berkson and H. Porta [26] in their study of composition operators on Hardy spaces of the unit disc. M. Abate [2] generalized their result to the finite-dimensional case. In order to consider the infinite-dimensional case, we need the following definition.

Definition 15.10 Let D be a domain in a Banach space X. A one-parameter semigroup S = {f_{t}}_{t≥0} ⊂ H(D) is said to be locally uniformly continuous (or briefly, T -continuous) if

T - lim_{t→0+} f_{t} = I.

Proposition 15.11 [167] Let D be a bounded domain in X and let S = {f_{t}}_{t≥0} ⊂ H(D) be a strongly continuous semigroup. The following conditions are equivalent:

(a) S is a T -continuous semigroup;

(b) the differences

g_{t} = \frac{1}{t} (I − f_{t})

are uniformly bounded on each subset strictly inside D;

(c) for each x ∈ D, there exists the strong limit

lim_{t→0+} \frac{1}{t} (I − f_{t}) (x) = g (x)

which is bounded on each subset strictly inside D.

Since it is clear that if (c) holds, then the mapping g belongs to H(D, X) (see Corollary 2.11), it is called the holomorphic infinitesimal generator of the semigroup S. As a matter of fact,

g = T - lim_{t→0+} \frac{1}{t} (I − f_{t}) ,

i.e., the convergence in (c) is actually locally uniform convergence over D [161].

Moreover, condition (c) means that a T -continuous semigroup is right-differentiable with respect to the parameter t at the origin. This implies, in turn, that for each x ∈ D, f_{t}(x) is differentiable at each point t ∈ ℝ^{+} and that the function u : ℝ → X defined by

u (t) = f_{t}(x)

is the solution of the Cauchy problem ([161])

\{ \begin{matrix}u′ (t) + g (u (t)) = 0, \\ u (0) = x. \end{matrix}

Since in the finite-dimensional case a continuous semigroup of holomorphic selfmappings is T -continuous, it follows that such a semigroup always has a holomorphic infinitesimal generator (see also [22] and [6]).

The uniqueness of the solution to the Cauchy problem shows that

F = ⋂_{t≥0} Fix (f_{t}) = Null(g) = {x ∈ D : g (x) = 0} ,

i.e., the stationary point set of a T -continuous semigroup S = {f_{t}}_{t≥0} on D coincides with the null point set of its generator g. This property of T -continuous semigroups allows us to study the structure of the stationary point set F as well as the asymptotic behavior of the semigroup S by using the local and global properties of the analytic set Null(g). It turns out that under certain conditions this set is, in fact, a complex-analytic submanifold of D.

Theorem 15.12 [161] , [167]. Let D be a bounded convex domain in X, g ∈ H(D, X) the generator of a one-parameter continuous semigroup S of holomorphic self-mappings of D and h = I − g. If Fix(h) contains a quasi-regular point α ∈ D, then it is a holomorphic retract of D, hence a complex-analytic submanifold of D tangent to Ker g′(a). In particular, if α ∈ Fix(h) is regular, then it is the unique stationary point of S.

Remark 15.13 Note that since condition (14.1) always holds in the finite-dimensional case, so does Theorem 15.12.

However, simple examples show that even in the finite-dimensional case, and even if F = ∩_{t≥0} Fix(f_{t}) is not empty, the net {f_{t}(x)}_{t≥0} might not converge, as t → ∞, to a point of F for all x ∈ D. Take, for example, the semigroup (even group) f_{t} = e^{it}I, where I is the identity mapping on the open unit ball B ⊂ X. Nevertheless, regarding the T -convergence of a generated (T -continuous) semigroup of holomorphic self-mappings of D, the situation can be completely described by using the spectral properties of the linear operator obtained by differentiation at a stationary point. We again denote the spectrum of a linear operator A : X → X by σ(A).

Theorem 15.14 [99] Let D be a bounded convex domain in a complex Banach space X and let S = {f_{t}(x)}_{t≥0} ⊂ H(D) be a one-parameter T -continuous semigroup such that F = ∩_{t≥0} Fix(f_{t}) is not empty. Then the following assertions are equivalent:

(1) the semigroup {f_{t}(x)}_{t≥0} is T -convergent, as t → ∞, to a holomorphic mapping r which is a retraction of D onto F;

(2) there exists a quasi-regular point α ∈ F of g = lim_{t→0+} \frac{1}{t}(I − f_{t}) such that

{σ (g′ (a))} ∩ {λ ∈ ℂ : Re λ = 0, λ ≠ 0} = ∅.

Note that even in the one-dimensional case, when there is a fixed point, both versions of the Denjoy-Wolff theorem (discrete and continuous) establish the convergence of the corresponding semigroup in all cases except for automorphisms with exactly one fixed point. For higher dimensions, the scope of counterexamples is wider.

However, it turns out that for an arbitrary holomorphic self-mapping, one can construct an associated continuous semigroup which has the same fixed point set and which converges to a retraction onto this set provided a rather simple condition (which always holds in the finite-dimensional case) is satisfied.

More precisely, let D be a bounded convex domain in X and let f ∈ H(D). It was shown in [161] that g = I − f generates a T -continuous one-parameter semigroup S = {f_{t}(x)}_{t≥0} ⊂ H(D). We refer to this semigroup as the semigroup associated with f.

Theorem 15.15 [99] Let D be a bounded convex domain in a complex Banach space X and let f ∈ H(D). Suppose that f has a fixed point α in D such that Im (I −f′(α)) is closed. Then the associated semigroup {f_{t}}_{t≥0} is T -convergent, as t → ∞, to a holomorphic mapping r ∈ H(D) which is a retraction of D onto the fixed point set Fix(f).

This theorem is, in fact, a consequence of Theorem 15.14 because the fixed point α is quasi-regular and the semigroup associated with a holomorphic self-mapping always satisfies the spectral condition (2) of that theorem.

16. Final notes and remarks

For more information on Kobayashi distances and Schwarz-Pick systems see [42], [68], [78], [110], [112], [194], [195], [196] and [197].

Applications of the metric approach to the study of holomorphic inequivalence of domains can be found in [60] and [179] (see also [63] and [86]).

Different approaches to the study of k_{B_{H}} can be found in [65], [178], [182] and [183]. More information on fixed and null points of holomorphic mappings and their ap-

plications is in [8], [9], [12], [42], [44], [72], [90], [93], [94], [95], [100], [102], [127], [140], [141], [155], [157], [163], [186], [196], [199], [205], [206], [207], [208], [209] and [211].

The approximate fixed point property in the Hilbert ball and in more general hyperbolic spaces is investigated in [60], [154] and [180].

For results on commuting families of holomorphic mapping see [3], [7], [51], [134], [185], [189] and [210].

Iterates of holomorphic and ρ-nonexpansive mappings, approximating sequences, approximating curves and ergodic theory are also studied in [4], [5], [21], [23], [31], [35], [97], [120], [122], [136], [143], [151], [158], [159], [160], [165], [166], [169], [184], [190],

[198] and [200].

Semigroups of holomorphic and ρ-nonexpansive mappings are examined, for example, in [2], [6], [10], [11], [89], [90], [92], [94], [95], [96], [98], [101], [152], [156], [162], [164], [167], [168], [201], [202] and [203]. We note, in particular, that the related concept of k_{B_{H}} -monotonicity is studied in [10] and [164].

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