Numerical range of holomorphic mappings and applications 9783030050191, 9783030050207

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Mark Elin Simeon Reich David Shoikhet

Numerical Range of Holomorphic Mappings and Applications

Mark Elin • Simeon Reich • David Shoikhet

Numerical Range of Holomorphic Mappings and Applications

Mark Elin Department of Mathematics ORT Braude College Karmiel, Israel

Simeon Reich Department of Mathematics The Technion - Israel Institute of Technology Haifa, Israel

David Shoikhet Department of Mathematics Holon Institute of Technology Holon, Israel Department of Mathematics ORT Braude College Karmiel, Israel

ISBN 978-3-030-05019-1 ISBN 978-3-030-05020-7 (eBook) https://doi.org/10.1007/978-3-030-05020-7 Library of Congress Control Number: 2019934203 Mathematics Subject Classification (2010): 46G20, 46T25, 47H20, 47D03 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated with deep appreciation to the memory of our friend and colleague Professor Jaroslav Zem´anek (1946–2017)

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1 Semigroups of Linear Operators 1.1 Linear operators. Spectrum and resolvent . . 1.2 Continuous semigroups and their generators . 1.3 Numerical range of linear operators . . . . . 1.4 Analytic semigroups . . . . . . . . . . . . . . 1.5 Ces` aro and Abel averages of linear operators 1.6 Abel averages: recent results . . . . . . . . .

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1 3 6 9 11 17

2 Numerical Range 2.1 Holomorphic mappings in Banach spaces . . . . . . 2.2 Spectrum and resolvent of holomorphic mappings . 2.3 Numerical range . . . . . . . . . . . . . . . . . . . . 2.4 Real part estimates . . . . . . . . . . . . . . . . . . 2.5 Holomorphically dissipative and accretive mappings 2.6 Growth estimates for the numerical range . . . . . . 2.7 Filtration of dissipative mappings . . . . . . . . . .

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21 25 26 32 41 49 54

3 Fixed Points of Holomorphic Mappings 3.1 Fixed points in the unit disk . . . . . . . . . . . . . . . . 3.2 Fixed points in the Hilbert ball . . . . . . . . . . . . . . . 3.3 Boundary fixed points and the horosphere function . . . . 3.4 Canonical representation of the fixed point set . . . . . . 3.5 Around the Earle–Hamilton fixed point theorem . . . . . 3.6 Inexact orbits of holomorphic mappings . . . . . . . . . . 3.7 The Bohl–Poincar´e–Krasnoselskii theorem . . . . . . . . . 3.8 Fixed points of pseudo-contractive holomorphic mappings

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64 69 71 76 78 83 85 90 vii

viii

Contents

4 Semigroups of Holomorphic Mappings 4.1 Generated semigroups . . . . . . . . . . . . . . . . . . . . . 4.2 Stationary points of semigroups . . . . . . . . . . . . . . . 4.3 Flow invariance conditions . . . . . . . . . . . . . . . . . . 4.4 Semi-complete vector fields on bounded symmetric domains 4.5 Rates of convergence of semigroups . . . . . . . . . . . . . 4.6 Semigroups and pseudo-contractive holomorphic mappings 4.7 Semigroups on the Hilbert ball . . . . . . . . . . . . . . . .

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97 100 102 107 110 115 121

5 Ergodic Theory of Holomorphic Mappings 5.1 General remarks . . . . . . . . . . . . . . . . 5.2 Power bounded holomorphic mappings . . . . 5.3 Ergodicity and fixed points . . . . . . . . . . 5.4 Numerical range and power boundedness . . 5.5 Dissipative and pseudo-contractive mappings 5.6 Examples . . . . . . . . . . . . . . . . . . . .

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129 131 133 142 153 162

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166 175 179 187 192 197 200

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6 Some Applications 6.1 Bloch radii . . . . . . . . . . . . . . . . . . . . . . . 6.2 Radii of starlikeness and spirallikeness . . . . . . . . 6.3 Analytic extension of one-parameter semigroups . . 6.4 Analytic extension of semigroups without stationary 6.5 Composition operators and semigroups . . . . . . . 6.6 Analytic semigroups of composition operators . . . . 6.7 Semigroups of composition operators on H p (Π) . . .

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Preface

The numerical range of (generally speaking, unbounded) linear operators plays a crucial role in linear semigroup theory because of the celebrated Lumer–Phillips Theorem. For holomorphic mappings a similar notion was introduced and studied by Harris [99]. Nowadays it arises in many aspects of nonlinear analysis, finite- and infinite-dimensional holomorphy, complex dynamical systems and nonlinear ergodic theory. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite- and infinite-dimensional Banach spaces, the study of complete and semi-complete vector fields and their applications to several classes of biholomorphic mappings, and in the study of Bloch (univalence) radii for locally biholomorphic mappings. The classical Lumer–Phillips Theorem states that a closed linear operator A on a Banach space X with a dense domain D ⊆ X generates a (pointwise) continuous semigroup {T (t) : t ≥ 0} of contractive operators on X if and only if it is dissipative and for some λ0 > 0 (hence, for all λ > 0), satisfies the range condition (λ0 I − A)D = X. (0.0.1) The first condition in this theorem means that the numerical range of the operator A lies in the closed left half-plane. If the second condition holds too, then the operator A is often referred to as an m-dissipative operator , and condition (0.0.1) just means that the resolvent (λ0 I−A)−1 is a bounded linear operator on X. A nonlinear analog of the Lumer–Phillips Theorem for holomorphic mappings and its applications was studied in [103]. It should be mentioned that a linear operator is holomorphic if and only if it is bounded. Therefore the above-mentioned nonlinear analog extends the Lumer–Phillips as well as the Hille–Yosida Theorems (see, for example, [240]). Moreover, it can be applied to locally uniformly continuous semigroups. Therefore a natural idea is to consider nonlinear mappings f (not necessarily holomorphic) the (nonlinear) resolvent of which exists and is nonexpansive with respect to the hyperbolic metric on the range of f. This leads us to nonlinear semigroup theory, which is not only of intrinsic interest, but is also important in the study of evolution problems. In recent years ix

x

Preface

many developments have occurred, in particular, in the area of nonexpansive semigroups in Banach spaces. As a rule, such semigroups are generated by dissipative operators and can be viewed as nonlinear analogs of the classical linear contraction semigroups. See, for example, [32], [17] and [174]. Another class of nonlinear semigroups consists of those semigroups generated by holomorphic mappings in complex manifolds and complex Banach spaces. Such semigroups appear in several diverse fields, including, for example, the theory of Markov stochastic branching processes [104], [202], Kre˘ın spaces [231, 232], the geometry of complex Banach spaces [13, 225], control theory and optimization [107]. As already mentioned above, these semigroups can be considered natural nonlinear analogs of the semigroups generated by bounded linear operators. These two distinct classes of nonlinear semigroups are related by the fact that holomorphic self-mappings are nonexpansive with respect to Schwarz–Pick pseudometrics. For the finite-dimensional case, Abate proved in [1] that each continuous semigroup of holomorphic mappings is everywhere differentiable with respect to its parameter, that is, it is generated by a holomorphic mapping. In addition, he established a criterion for a holomorphic mapping to be the generator of a one-parameter semigroup. Earlier, in the one-dimensional case, similar results were presented by Berkson and Porta in their study [19] of linear C0 -semigroups of composition operators on Hardy spaces. Vesentini investigated semigroups of those fractional-linear transformations which are isometries with respect to the infinitesimal hyperbolic metric on the unit ball of a Banach space (see [231, 232]). He used this approach to study several important problems in the theory of linear operators on indefinite metric spaces. Note that, generally speaking, such semigroups are not everywhere differentiable in the infinite-dimensional case. Since holomorphic self-mappings of a domain D in a complex Banach space are nonexpansive with respect to any pseudometric ρ assigned to D by a Schwarz–Pick system [100], it is natural to inquire whether mapping and fixed point theories analogous to the monotone and nonexpansive operator theories can be developed in the setting of those mappings. On the other hand, many evolution problems related to, for example, the asymptotic behavior of dynamical systems, rigidity properties, and common fixed points of commuting mappings may be studied by using the Earle–Hamilton Fixed Point Theorem and the generalized Schwarz–Pick Lemma in Banach and Hilbert spaces. Over the last thirty years these results have been developed in many directions. One of them concerns increasing the dimension of the underlying space. Finite-dimensional extensions are to be found, for instance, in the papers by Kubota [143], MacCluer [161], Chen [42], Abate [1, 2] and Mercer [168]. In this connection see also [6, 4, 159, 106] and [218]. Infinite-dimensional generalizations are due, for example, to Fan [75, 76], Wlodarczyk [236, 237, 238], Goebel [79], Vesentini [228, 229], Sine [211] and Mellon [167]. These authors used a variety of approaches and assumed diverse conditions on the mappings and the domains. Another direction is concerned with analogues of the classical Denjoy–Wolff Theo-

Preface

xi

rem for continuous semigroups. This approach has been used by several authors to study the asymptotic behavior of solutions to Cauchy problems (see, for example, [19, 3, 46] and [186]). It also turns out that the asymptotic behavior of solutions to evolution equations can be used in the study of the geometry of certain domains in complex spaces. For example, a classical result, due to Nevanlinna (1921), states that if f is holomorphic in the open unit disk and satisfies f (0) = 0, f  (0) = 0, then f is univalent and maps the open unit disk onto a starlike domain (with respect to 0) if and only if Re[zf  (z)/f (z)] > 0 everywhere. This fact, as well as many other results in geometric function theory can easily be obtained by using a dynamical approach. It seems that the idea to use a dynamical approach was first suggested by Robertson [196, 197] and developed by Brickman [33], who introduced the concept of Φ-like functions as a generalization of starlike and spirallike functions (with respect to the origin) of a single complex variable. Suffridge [215, 216], Pfaltzgraff [176, 177] and Gurganus [94] developed a similar method to characterize starlike, spirallike (with respect to the origin), convex and closed-to-convex mappings in higher-dimensional settings, where the numerical range of nonlinear holomorphic mappings is essentially used. Since 1970, the list of papers on these subjects has become quite long (see, for example, the book [92]). In view of all of this, in this book we describe recent developments as well as a historical outline of this area. In the first chapter we present a brief survey of the theory of semigroups of linear operators including the Hille–Yosida and the Lumer–Phillips Theorems. We discuss in more detail the numerical range and the spectrum of closed densely defined linear operators. In addition, we provide an overview of ergodic theory including some classical and recent results on Ces`aro and Abel averages. The analytic extension of semigroups of linear operators is also discussed. We observe that the recent study of the numerical range of composition operators establishes a strong connection between semigroups on Hardy and Dirichlet spaces and nonlinear semigroups of holomorphic self-mappings of the unit disk. In the second chapter we first discuss the basic notions and facts in infinitedimensional holomorphy and hyperbolic geometry in Banach and Hilbert spaces. We then present and generalize Harris’ theory of the numerical range of holomorphic mappings and discuss the main properties of the so-called quasi-dissipative mappings and their growth estimates. In particular, we present lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. We end this chapter with a study of the filtration method for dissipative mappings. In Chapter 3 we discuss some geometric and quantitative analytic aspects of fixed point theory. In particular, we present some extensions of the Earle– Hamilton and Bohl–Poincar´e–Krasnoselskii Theorems including their connections with Schwarz–Pick systems of pseudometrics and pseudo-contractive mappings. We also present a solution to the so-called coefficient problem in branching stochas-

xii

Preface

tic processes by using the growth estimates for the numerical range of holomorphic mappings. Chapter 4 is devoted to semigroups of holomorphic mappings. We study, inter alia, the relations between such semigroups and the class of pseudo-contractive mappings. These mappings are not necessarily holomorphic, but have holomorphic resolvents. This class seems to be new and is further studied in Chapter 5, which is mainly devoted to the ergodic theory of holomorphic mappings. Chapter 6 is devoted to several applications of the numerical range to diverse geometric, analytic and dynamical problems. In particular, one of the general problems pertinent to our discussion is the following one: Given a quasi-dissipative mapping h on the open unit ball B, find r ∈ (0, 1) (if it exists) such that h is dissipative on the ball Br . Since every holomorphic mapping h which is defined in a domain D is locally Lipschitz, it follows that the Cauchy problem ⎧ ⎨ dx(t) = h(x(t)) dt ⎩x(0) = x 0

has a unique continuous solution x (t) defined on the interval [0, T ] , where T depends on the initial value x0 ∈ D. Recall that a mapping h is said to be a semi-complete vector field on D if for each x0 ∈ D, this solution is well defined on the right half-axis [0, ∞) and the values of x (t) belong to D for each initial datum x0 ∈ D. In this situation, h generates a one-parameter semigroup of holomorphic self-mappings of D. Note that it may happen that h is not semi-complete on all of the domain D, but is semi-complete on some open subset of D. In this case we say that h is a locally semi-complete vector field. It is known (see, for example, [191]) that if h is holomorphic on BR , then it is semi-complete on BR if and only if it is dissipative on BR . It turns out that if the numerical range of h (say, with respect to the open unit ball B of X) is not the whole complex plane, then for each r ∈ (0, 1), there is a real number ω = ω(r) such that the perturbed mapping h − ωI is holomorphically dissipative and semicomplete on the ball Br . The question is how this number ω depends on r and how to find the minimal value of the function ω(r) with respect to r ∈ (0, 1) (if this minimum is finite). By using the exponential formula for semigroups of holomorphic mappings (see, for example, [191]), one can see that this problem is equivalent to the following one. Find a function ω(r) on the interval (0, 1) such that the nonlinear resolvent −1 (λI − h) is well defined on the ball Bρ of radius ρ = λ − ω(r)r for all λ ≥ ω(r) and maps this ball into Br .

Preface

xiii

In this case it is also of interest, in analogy with the linear theory, to determine if the associated resolvent mapping (λI − h)−1 ◦ (λ − ω(r))I can be extended to a sector with vertex at ω(r) in the complex plane and to estimate the angle of its opening. Another circle of interesting problems is connected to the following notion. For a holomorphic mapping h : B → X, one says that it has unit radius of boundedness if it is bounded on each subset strictly inside B ([78, 99]; see also [25, 27]). It follows from a result of Harris [99] that a holomorphic mapping in B has unit radius of boundedness if and only if its numerical radius |VD (h)| is bounded with respect to any convex subset in B. Moreover, it was shown in [103] that this is equivalent to a formally weaker condition, namely,    lim sup sup Re λ : λ ∈ VBR (hs ) ≤ 0. s→1−

The problem of verifying whether a holomorphic mapping has unit radius of boundedness, as well as the general study of its numerical ranges, arises in many aspects of infinite-dimensional holomorphy (see, for example, [78, 99]) and complex dynamical systems [9, 191]. In particular, they play a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings [186, 187], the study of flow invariance and range conditions in nonlinear analysis [103, 185], and geometric function theory in finite- and infinite-dimensional Banach spaces [191]. They were specifically mentioned for the class of semi-complete vector fields (or, infinitesimal generators) in their application to the study of starlike and spirallike mappings [191], and the Bloch (univalence) radii [103] for locally biholomorphic mappings. Observe also that the concept of unit radius of boundedness for holomorphic mappings is a specific phenomenon in the infinite-dimensional case because in a finite-dimensional Banach space each holomorphic mapping on the open unit ball is bounded on each subset strictly inside the ball. This is no longer true in the general case. Relevant examples can be found in [40]. In addition, we show in Chapter 6 how the notion of the numerical range can be exploited in the study of analytic extensions of linear and nonlinear operator semigroups with respect to their parameter to a domain in the complex plane. For continuous semigroups of bounded linear operators on a complex Banach space, this problem goes back to the groundbreaking results in [111] and [240], where some criteria for the existence of analytic extensions were established. However, for semigroups of holomorphic (nonlinear) mappings, the investigation of the problem of analytic extension has only begun very recently. Moreover, using the numerical range, we are able to discover new criteria pertaining to a special class of semigroups of linear operators, namely, semigroups of composition operators. Thus the book concentrates not only on the numerical range itself, but also brings out its connections with various modern topics in infinite-dimensional holo-

xiv

Preface

morphy including geometric function theory in Banach spaces, as well as in linear and nonlinear operator theory. In conclusion, we would like to express our hope that this book might lead to further progress regarding the numerical range of holomorphic mappings and its manifold applications to several important areas of mathematical analysis. We also hope it attracts young researchers to these fields. We also take this opportunity to thank Professor Yuri Tomilov of IMPAN, Warsaw, Poland, for his useful comments concerning the first two chapters of our book and Ms. Dorothy Mazlum of Birkh¨ auser, Basel, Switzerland, for her patient and encouraging perseverance.

Mark Elin Simeon Reich David Shoikhet

Chapter 1

Semigroups of Linear Operators In this chapter we mostly use standard concepts of operator theory which can be found, for example, in [38, 54, 111, 116, 117] and [240].

1.1 Linear operators. Spectrum and resolvent Let X and Y be vector spaces over the field of complex numbers C. Let D be a linear subspace of X, that is, a subset of X which contains all linear combinations of its vectors. A mapping T : D → Y is called a linear operator with domain D = DT ⊂ X if T (αx + βy) = αT x + βT y for all α, β ∈ C and x, y ∈ D. From now on we assume that X and Y are topological vector spaces over the field C. A linear operator T : D → Y is said to be closed if its graph {[x, T x] : x ∈ D} ⊂ D × Y is closed in the product space X × Y . Equivalently, the operator T is closed if for any sequence {xn } ⊂ D, xn → x and T xn → y imply that x ∈ D and T x = y. Theorem 1.1.1 (Closed graph theorem). Let X and Y be Banach spaces. A linear operator T defined on all of X (that is, D = X) is closed if and only if it is continuous. In the case where X and Y are normed spaces, a linear operator T : X → Y is said to be bounded if its norm, defined by T  = sup {T xY : xX = 1}, is finite. It is well known that the boundedness of a linear operator T is equivalent to its continuity. By L(X, Y ) we denote the space of all bounded linear operators T : X → Y and we set L(X) = L(X, X). Let now X be a complex Banach space and let I be the identity mapping on X. Let T be a closed linear operator on X with domain D ⊂ X and values in X. We define the resolvent set ρ(T ) of T to be the set of all complex λ such that −1

R(λ, T ) := (λI − T )

© Springer Nature Switzerland AG 2019 M. Elin et al., Numerical Range of Holomorphic Mappings and Applications, https://doi.org/10.1007/978-3-030-05020-7_1

(1.1.1) 1

2

Chapter 1. Semigroups of Linear Operators

is a bounded linear operator on X. Then the operator-valued function R(λ, T ) is analytic in ρ(T ) and is called the resolvent of T . The complement σ(T ) of ρ(T ) in C is called the spectrum of T . It is well known that σ(T ) is a closed set. Let us first consider the spectral and resolvent characteristics of bounded linear operators. Proposition 1.1.1. For T ∈ L(X), the following assertions hold: (i) σ(T ) is a compact set in C;

 (ii) sup |λ| = lim n T n  ≤ T  ; λ∈σ(T )

n→∞

(iii) for |λ| > sup |λ|, the series λ∈σ(T )

∞

1 T n converges in L(X) and its sum n+1 λ n=0

equals R(λ, T ); (iv) for each  > 0, there is a norm  ·  on X, equivalent to the original norm on X, such that T  = sup T x < r(T ) + . x =1

Definition 1.1.1. For T ∈ L(X), the number r(T ) := sup |λ| = lim λ∈σ(T )

n→∞

n T n 

is called the spectral radius of the operator T . Remark 1.1.1. It immediately follows from assertion (iii) of Proposition 1.1.1 that the set {λ : |λ| > r(T )} is contained in the resolvent set ρ(T ) and that R(λ, T ) is analytic in this set. As a matter of fact, one can show that R(λ, T ) is analytic in the whole open set ρ(T ). In addition, the following resolvent identity holds: if λ and μ are in ρ(T ), then R(λ, T ) − R(μ, T ) = (μ − λ)R(λ, T )R(μ, T ). For a general closed operator T , the spectrum σ(T ) is not necessarily bounded (it may even be the whole plane in general), in contrast with the case of bounded linear operators. For any closed densely defined linear operator T , we denote by A(T ) the family of all analytic functions in a neighborhood Ω of σ(T ), considered a subset of C. Assume now that ∂Ω consists of a finite number of Jordan arcs and f ∈ A(T ) admits an analytic extension to Ω. Let also ∂Ω be positively oriented with respect to Ω. Then we define the linear operator I(T, f ) by

1 f (λ)R(λ, T )dλ. I(T, f ) := 2πi ∂Ω

1.2. Continuous semigroups and their generators

3

If T ∈ L(X), then one defines f (T ) := I(T, f ). Moreover, if σ(T ) ⊂ D, the open unit disk in C, and f (z) =

∞

cn z n is analytic

n=0

in D, then f (T ) =

∞

cn T n ,

n=0

where the right-hand series converges in L(X). More generally, for a closed densely defined linear operator T with ρ(T ) = ∅, consider the function ψα (μ) = μ1 + α, where α ∈ ρ(T ). Then one defines f (T ) := (f ◦ ψα )(−R(α, T )). It turns out that even if T is unbounded, then f (T ) = f (∞) + I(T, f ) (see, for example, [54]). The following operator calculus properties hold. Theorem 1.1.2. If f and g belong to A(T ), then (i) (f + g)(T ) = f (T ) + g(T ); (ii) (f · g)(T ) = f (T )g(T ); (iii) σ(f (T )) = f (σ(T ) ∪ {∞}). The last assertion is the so-called Spectral Mapping Theorem. In the case where T ∈ L(X), assertion (iii) can be simply written as σ(f (T )) = f (σ(T )).

1.2 Continuous semigroups and their generators Let X be a complex Banach space and let L(X) be the space of all bounded linear operators on X. Definition 1.2.1. A family S = {T (t) : t ≥ 0} ⊂ L(X) is called a one-parameter semigroup of linear operators if it has the following properties: (i) T (t + s) = T (t)T (s) for all t, s ≥ 0; (ii) T (0) = I. A semigroup S = {T (t) : t ≥ 0} ⊂ L(X) is said to be strongly continuous at the origin (or of class C0 ) if lim T (t)x = x

t→0+

for all x ∈ X.

(1.2.1)

4

Chapter 1. Semigroups of Linear Operators

Theorem 1.2.1. Let S = {T (t) : t ≥ 0} be a C0 -semigroup on X. Then (i) the operator-valued function T (·) : [0, ∞) → L(X) is continuous at each point t0 ∈ R + ; (ii) the limit 1 1 ω0 := lim log T (t) = inf log T (t) (1.2.2) t→∞ t t>0 t exists finitely and for each ω > ω0 , there is Mω such that for all t ≥ 0, T (t) ≤ Mω eωt .

(1.2.3)

The following definition contains some additional notions. Definition 1.2.2. (a) The semigroup S = {T (t) : t ≥ 0} of class C0 is called uniformly bounded if for all t ≥ 0, T (t) ≤ M < ∞. (b) If M = 1 in (a), then the semigroup is said to be a contraction semigroup. (c) If Mω = 1 in (1.2.3), then the semigroup S is called quasi-contractive. (d) The semigroup S is said to be uniformly continuous if T (t) − I → 0 as

t → 0+ .

Definition 1.2.3. Let S = {T (t) : t ≥ 0} be a semigroup of bounded linear operators on X of class C0 . The infinitesimal generator A of S is defined by Ax := lim

t→0+

1 [T (t) − I] x t

(1.2.4)

with domain DA consisting of all x ∈ X such that the limit in (1.2.4) exists. The following assertion describes the main properties of the infinitesimal generator A and its domain DA . Theorem 1.2.2. Let A be the infinitesimal generator of a one-parameter semigroup S = {T (t) : t ≥ 0} of class C0 . The following assertions hold. (i) DA is linear dense subspace of X; (ii) A is a linear closed operator on X; (iii) for each x ∈ DA and t ≥ 0, T (t)x ∈ DA . The trajectory {T (t)x : t ≥ 0} solves the Cauchy problem ⎧ ⎨ d T (t)x = AT (t)x, t ≥ 0, dt (1.2.5) ⎩ T (0)x = x, x ∈ DA .

1.2. Continuous semigroups and their generators

5

Remark 1.2.1. It can be shown that the operators A and T (t) commute in the sense that for each x ∈ DA and t ≥ 0, AT (t)x = T (t)Ax,

x ∈ DA .

Corollary 1.2.1. A semigroup S = {T (t) : t ≥ 0} of class C0 has a bounded infinitesimal generator A if and only if it is uniformly continuous. In this case, dT (t) x = AT (t)x = T (t)Ax dt

(1.2.6)

for all x ∈ X and the limit in (1.2.4) is uniform with respect the operator norm in X, that is,     1  lim  (T (t) − I) = 0. (1.2.7) A −   + t t→0 L(X) Moreover, T (t) = exp(tA) = I +

∞ 1 (tA)k , k!

t ∈ R+ .

(1.2.8)

k=1

It is known that if lim sup T (t) − I < 1 then, in fact, T (t) − I → 0, t→0+

and hence the semigroup is uniformly continuous, and has the form etA for some bounded operator A. Returning to the general case of semigroups of class C0 , we let Ah :=

1 [T (h) − I] , h

h ∈ R+ ,

(1.2.9)

so that Ax = lim Ah x, h→0+

x ∈ DA .

Since each Ah is a bounded linear operator on X, one can define the semigroup {Th (t) : t ≥ 0} using (1.2.8), that is, Th (t) := exp(tAh ).

(1.2.10)

We denote by s-lim the limit in the strong operator topology in L(X). Theorem 1.2.3. Let S = {T (t) : t ≥ 0} be a semigroup of class C0 generated by A, and let Ah and Th be defined by (1.2.9) and (1.2.10), respectively. Then for each t ≥ 0, T (t) = s- lim+ Th (t). (1.2.11) h→0

This limit is uniform on each bounded interval [0, t0 ]. Formula (1.2.11) is often called Hille’s first exponential formula.

6

Chapter 1. Semigroups of Linear Operators

Theorem 1.2.4. Let S = {T (t) : t ≥ 0} be a C0 -semigroup of bounded linear operators, and let the operator A with domain DA ⊂ X be its infinitesimal generator. If ω0 is defined by (1.2.2), then (i) the half-plane Πω0 := {λ ∈ C : Re λ > ω0 } lies in the resolvent set ρ(A) and

∞ R(λ, A)x =

e−λt T (t)xdt,

x ∈ X, λ ∈ ρ(A);

(1.2.12)

0

    (ii) for any ω > ω0 , there is a constant Mω = sup e−ωt T (t) such that for t≥0

each λ with Re λ > ω, [R(λ, A)]  ≤ Mω (Re λ − ω)−n . n

(1.2.13)

(iii) for each x ∈ X and λ ∈ C with Re λ > ω0 , 1 [R(λ, A)] x = (n − 1)!

∞

n

e−λt tn−1 T (t)xdt.

0

Theorem 1.2.5 (Hille–Yosida–Phillips Theorem). Let A be a closed linear operator with dense domain DA and let ω ∈ R be such that (ω, ∞) ⊂ ρ(A). Then A is an infinitesimal generator of a C0 -semigroup if and only if there is M < ∞ such that for each t > ω, n [R(t, A)]  ≤ M (t − ω)−n . (1.2.14) Moreover, the following representation formula holds:  n  n n T (t)x = lim R ,A x, x ∈ X. n→∞ t t

(1.2.15)

Remark 1.2.2. In particular, a densely defined closed linear operator A is the generator of a semigroup of contractions on X if and only if ρ(A) ⊃ (0, ∞) and R(t, A) ≤ 1 for all t > 0. Remark 1.2.3. If A is a generator of a quasi-contractive semigroup S = {T (t) : t ≥ 0}, then the operator A − ωI =: Aω is the generator of the semigroup of contractions Tω (t) = e−ωt T (t), where ω > ω0 and ω0 is defined in (1.2.2).

1.3 Numerical range of linear operators Let A ∈ L(H), where H is a Hilbert space. The numerical range of A is defined by V (A) := {(Ax, x) : x = 1, x ∈ H}.

(1.3.1)

1.3. Numerical range of linear operators

7

The number |V (A)| = sup {|λ| : λ ∈ V (A)} is called the numerical radius of A. Like the spectrum, the numerical range associates with A a set of complex numbers that reflects the properties of A. We now list several basic properties of the numerical range [21, 95]. Theorem 1.3.1. Let A ∈ L(H). Then (1) V (A) is convex, (2) A is Hermitian (that is, A∗ = A) if and only if V (A) is real; (3) σ(A) ⊂ V (A). Moreover, unlike the spectral radius, the numerical radius is an equivalent norm on L(H). Theorem 1.3.2. Let A ∈ L(H). Then |V (A)| ≤ A ≤ 2 |V (A)|. Moreover, if |V (A)| = A, then r(A) = A. Definition 1.3.1. Let X be a Banach space. An operator A ∈ L(X) is said to be power bounded if sup An  < ∞. (1.3.2) n∈N

For instance, Theorem 1.3.2 implies immediately that A is power bounded if there is K ∈ R such that |V (An )| < K for all n ∈ N. The following strong generalization of this fact was obtained in [85]. Theorem 1.3.3. Let A ∈ L(H). Suppose that V (An ) ⊂ S, n ∈ N, where S is a strip in the complex plane that is not parallel to the real axis. Then A is power bounded. Further, the numerical range has some mapping properties. For instance, assume that V (A) ⊂ D. If f ∈ Hol(D), then V (f (A)) ⊂ D; and if f ∈ Hol(D, C) with Re f (z) ≥ 0, then for each λ ∈ V (f (A)), we have Re(λ + f (0)) ≥ 0. The notion of the numerical range was successfully extended to operators on an arbitrary Banach space X by Lumer [155]. Let X ∗ denote the dual of the Banach space X and let x, x∗  denote the duality pairing of x∗ ∈ X ∗ and x ∈ X. For each x ∈ X, the set J(x), defined by   2 2 J(x) := x∗ ∈ X ∗ : x, x∗  = x = x∗  , (1.3.3) is not empty by virtue of the Hahn–Banach theorem and is a closed, convex and bounded subset of X ∗ . The mapping J is, in general, set-valued. It is single-valued if X is smooth. Moreover, in the case where X is a Hilbert space, the set J(x) consists of the single functional (y) = (y, x) by the Riesz representation theorem. Definition 1.3.2. For a linear operator A with dense domain DA , one defines the set V (A) := {Ax, x∗  : x ∈ DA , x = 1, x∗ ∈ J(x)} (1.3.4) to be the numerical range of A.

8

Chapter 1. Semigroups of Linear Operators

Clearly, the numerical range has the following properties: for all A, B ∈ L(X), (i) V (λA) = λV (A) for all λ ∈ C; (ii) V (A + B) ⊂ V (A) + V (B); (iii) V (A + λI) = V (A) + λ for all λ ∈ C. The following theorem is a well-known extension of Theorems 1.3.1 and 1.3.2. See [21] for this and many other properties of the numerical range. Recall that a bounded linear operator A is said to be Hermitian if its numerical range V (A) is a subset of the real line R. In a Hilbert space this is equivalent to its being self-adjoint. Theorem 1.3.4. Let A ∈ L(X). Then (1) (2) (3) (4) (5)

V (A) is connected; A is Hermitian if and only if eitA is a surjective isometry for all real t; σ(A) ⊂ V (A); |V (A)| ≤ A ≤ e |V (A)|; sup Re V (A) = lim+ (I + tA − 1)/t. t→0

Definition 1.3.3. A densely defined linear operator A is called dissipative if its numerical range lies in the closed left half-plane, that is, Re Ax, x∗  ≤ 0 for all x ∈ DA with x = 1 and x∗ ∈ J(x). Theorem 1.3.5. A linear operator A with domain DA ⊂ X is dissipative if and only if (tI − A)x ≥ t x for all x ∈ DA and t > 0. Theorem 1.3.6 (Lumer–Phillips Theorem [156]; see also Theorem 4.3 [175]). Let A be a densely defined linear operator with DA ⊂ X. The following assertions hold: (i) If A is dissipative and there exists λ0 > 0 such that (λ0 I − A)DA = X,

(1.3.5)

then A is the (infinitesimal) generator of a C0 -semigroup of contractions on X. (ii) If A is the generator of a C0 -semigroup of contractions on X, then (λI − A)DA = X for each λ > 0 and A is dissipative.

(1.3.6)

1.4. Analytic semigroups

9

Remark 1.3.1. In view of Theorem 1.3.6, for dissipative operators, conditions (1.3.5) and (1.3.6) are equivalent. Usually such operators are called m-dissipative. Thus the Lumer–Phillips Theorem can be formulated as follows (cf. [175]): A linear densely defined operator is the generator of a C0 -semigroup of contractions if and only if it is m-dissipative. In particular, it follows that every bounded linear dissipative operator is m-dissipative. Remark 1.3.2. If A is an m-dissipative operator on X, then ρ(A) ⊃ (0, ∞) and ω0 defined by (1.2.2) is at most zero. Thus it follows from Theorem 1.2.4 that, in fact, ρ(A) ⊃ Π+ = {Re λ > 0}; hence the spectrum σ(A) lies in the closure of the left half-plane. Consequently, condition (1.3.6) holds for all λ ∈ Π+ . In particular, if A is an m-dissipative operator with DA ⊂ X dense in X and the numerical range V (A) lies in a sector of the left half-plane, then there is α ∈ (0, π2 ) such that μA is still m-dissipative for each μ ∈ Π+ with |arg μ| ≤ α. So, the semigroup S = {T (t) : t ≥ 0} of contractions generated by A has an analytic continuation into this sector due to the exponential formula (1.2.15) and Remark 1.3.2 above. In the next section we present a general approach to the analytic extension of C0 -semigroups.

1.4 Analytic semigroups Although C0 -semigroup are originally defined only for real non-negative values of its parameter, it is natural to consider the problem of extending the domain of the parameter to a region in the complex plane that contains the non-negative real half-axis. It is clear that in order to preserve the semigroup structure, the domain in which the complex parameter should vary must be an additive semigroup of complex numbers. Hence, it is natural to assume that such domains

are sectors around the positive real axis. Throughout this section, given θ1 , θ2 ∈ 0, π2 , we set Λ(θ1 , θ2 ) := {ζ ∈ C : −θ1 < arg ζ < θ2 } and Λ(θ) := Λ(θ, θ) for short. Definition 1.4.1. Let {T (ζ) : ζ ∈ Λ} ⊂ L(X). Such a family is said to be an analytic semigroup in Λ if (i) ζ → T (ζ) is analytic in Λ; (ii) T (0) = I and lim T (ζ)x = x for every x ∈ X; Λ ζ→0

(iii) T (ζ1 + ζ2 ) = T (ζ1 )T (ζ2 ) whenever ζ1 , ζ2 ∈ Λ. If, in addition, T (·) is bounded in every closed subsector Λ1 ⊂ Λ ∪ {0}, then we call {T (ζ) : ζ ∈ Λ} a bounded analytic semigroup.

10

Chapter 1. Semigroups of Linear Operators

By this definition, the restriction of an analytic semigroup to the real axis is a C0 -semigroup. Multiplying the semigroup parameter by a unimodular complex constant, we see that we do not lose any generality if we only consider semigroups which are analytic in sectors of the form Λ(θ). We are interested in the possibility of extending a given C0 -semigroup to an analytic semigroup in some sector around the non-negative real axis. For a continuous semigroup of bounded linear operators on a complex Banach space, the problem of analytic extension with respect to the parameter goes back to the pioneering works [111] and [240]. In 1970, Kato [118] established a necessary and sufficient condition for a given C0 -semigroup to admit an analytic extension in terms of the spectrum of the semigroup elements T (t) for small t > 0. In particular, his result implies that if lim sup T (t) − I < 2, then the semigroup t→0+

has an analytic extension. Furthermore, according to Beurling’s theorem, a C0 semigroup {T (t) : t ≥ 0} has an analytic extension to a sector if and only if for some polynomial p, lim sup p(T (t)) < sup |p(z)|. t→0+

|z| 0 and n there is a constant C > 0 such t that the elements of the family AT (t) are uniformly continuous with C respect to n ∈ N and t ∈ (0, 1]; 

1 , (b) the semigroup S admits an analytic extension to the sector Λ arctan Ce  −ζ and the family e T (ζ) is uniformly continuous in a smaller sector. Corollary 1.4.1 (see Theorem 5.3 [175]). Let S = {T (t) : t ≥ 0} be a C0 -semigroup generated by an operator A. If lim sup tAT (t) < t→0+

1 , e

then A is a bounded operator (hence, S is uniformly continuous) and S can be extended analytically to the whole complex plane. Another criterion for the existence of an analytic extension is provided by the following theorem. Theorem 1.4.2 ([38]). Let S = {T (t) : t ≥ 0} be a C0 -semigroup of bounded linear operators on a Banach space X and let A be its infinitesimal generator with domain DA ⊂ X. The following statements are equivalent:

1.5. Ces` aro and Abel averages of linear operators

11

(a) T (t)x ∈ DA for all x ∈ X and t ∈ (0, 1], and there is a constant C > 0 such that tAT (t) ≤ C for all t ∈ [0, 1]; (b) the semigroup S admits

1  an analytic extension {T (ζ) : ζ ∈ Λ} to the sector =: Λ. {ζ : Re ζ > 0} ∩ Λ eC Another way of stating (b) is to say that T (ζ1 + ζ2 ) = T (ζ1 ) ◦ T (ζ2 ) for all ζ1 , ζ2 ∈ Λ, T (·) is analytic in Λ and T (ζ)x → x for each x ∈ X as ζ → 0 within a smaller sector | arg ζ| < ϑ(eC)−1 with ϑ < 1. To finish this section we note that analytic semigroups have important features; for instance, the spectral bound sup{Re λ : λ ∈ σ(A) for the semigroup generator equals the growth bound ω0 defined by (1.2.2), and for such semigroups the spectral mapping theorem etσ(A) = σ(T (t)) \ {0},

t ≥ 0,

holds (see, for example [71]).

1.5 Ces`aro and Abel averages of linear operators Let X be a complex Banach space. Let T be a bounded linear operator on X and let S = {T (t) : t ≥ 0} be a strongly continuous semigroup of bounded linear operators on X. Classical ergodic theory deals with the asymptotic behavior of the Ces` aro averages which are defined by Cn (T ) :=

 1

I + T + · · · + T n−1 n

(1.5.1)

for a single operator T , or by 1 Ct (S) := t

t

T (s)ds

(1.5.2)

0

for a continuous semigroup S = {T (t) : t ≥ 0}, respectively. The interest in such averages has its origins in statistical mechanics and probability theory. Definition 1.5.1. If the limit of Ces` aro averages lim Cn (T ) (respectively, lim Ct (S)) n→∞

t→∞

exists in the strong operator topology, then the operator T (respectively, the semigroup S) is called mean ergodic. In the case where the Ces` aro averages converge uniformly on bounded sets, that is, in the uniform operator topology, the operator T ∈ L(X) (respectively, the semigroup S ⊂ L(X)) is called uniformly mean ergodic. The study of such operators began in the 1930s and was initiated by von Neumann, Dunford, Riesz and others (see, for example, [141, 240]).

12

Chapter 1. Semigroups of Linear Operators

Definition 1.5.2. If an operator T (respectively, a semigroup S) is mean ergodic, then the operator P defined by P : x → lim Cn (T )x (respectively, P : x → n→∞

lim Ct (S)x) is a projection which is called the mean ergodic projection associated

t→∞

to T (respectively, to S). The Mean Ergodic Theorem of Yosida (see [240]) enables us to construct, for a class of linear operators T on X, a projection onto the kernel Ker(I − T ) of the operator I − T . Lemma 1.5.1. Let X be a Banach space and let T ∈ L(X) be power bounded. Then   (i) the set x ∈ X : lim Cn (T )x = 0 coincides with the closure Im(I − T ) of n→∞

the image (I − T )X of the operator I − T , where Cn (T ) is defined by (1.5.1); (ii) Im(I − T ) ∩ Ker(I − T ) = {0}. Theorem 1.5.1 (Mean Ergodic Theorem). Let X and T ∈ L(X) be as above. Assume that for some x ∈ X and for some subsequence {Cnk }∞ k=1 , the weak limit w- lim Cnk (T )x = x0 ∈ X k→∞

exists. Then T x0 = x0 and the limit lim Cn (T )x = x0 . n→∞

Corollary 1.5.1. Suppose that X is a reflexive Banach space and assume that T ∈ L(X) is power bounded. Then the operator T is mean ergodic. Moreover, the operator P : X → X defined by the equality P x = lim Cn (T )x n→∞

is a continuous linear projection of X onto Ker(I − T ). In addition, Ker(P ) = Im(I − T ); hence, X = Im(I − T ) ⊕ Ker(I − T ). The following result is a semigroup analog of Theorem 1.5.1. Theorem 1.5.2. Let X be a Banach space and let a strongly continuous semigroup S = {T (t) : t ≥ 0} be generated by an operator A. If T (t) ≤ M for all t ≥ 0, then the following assertions are equivalent. (a) S is mean ergodic. (b) The Ces` aro means C(r) = Cr (S) converge in the weak operator topology as r → ∞. (c) For each x ∈ X, there exists a monotone, unbounded sequence {rn } such that {C (rn ) x}∞ n=1 has a weak accumulation point in X. (d) For each x ∈ X, one has co{T (t)x : t ≥ 0} ∩ Fix S = ∅. In the further study of the Ces` aro and Abel averages the following condition (weaker than (1.3.2)) is very important: lim

n→∞

1 n T  = 0. n

(1.5.3)

1.5. Ces` aro and Abel averages of linear operators

13

We proceed with some results concerning the uniform convergence of the Ces` aro averages as well as the structure of the fixed point set of an operator T ∈ L(X). The following remarkable result is due to Dunford (see [53]). Theorem 1.5.3. Let X be a Banach space. An operator T ∈ L(X) is uniformly mean ergodic if and only if T satisfies (1.5.3) and the point 1 is at most a simple pole of the resolvent R(λ, T ) = (λI − T )−1 . Later on Lin [152] showed that under the assumption (1.5.3), uniform mean ergodicity is equivalent to the closedness of (I − T )X. As a matter of fact, in our setting this spectral condition in Theorem 1.5.3 can be replaced by the closedness of Im(I − T ), so one can arrive at a similar conclusion. Namely, if we just assume that sup Cn (T ) < ∞, then we have Im(I − T ) ∩ Ker(I − T ) = {0}.

(1.5.4)

We now turn our attention to a more recent result. Theorem 1.5.4 (Lyubich and Zem´anek [158]). Let T ∈ L(X) satisfy conditions (1.5.3) and (1.5.4). Then the following assertions are equivalent: (i) the number 1 is at most a simple pole of the resolvent R(λ, T ); (ii) Im(I − T ) is closed; (iii) Im(I − T ) ⊕ Ker(I − T ) = X. Thus, under the conditions of Dunford’s Theorem 1.5.3, we see that the operator P : X → X defined by P := lim Cn (T ) is a linear projection onto the n→∞

kernel Ker(I − T ) with Ker P = Im(I − T ), and hence Ker(I − T ) ⊕ Im(I − T ) = X. Since we will be mostly interested in the power boundedness condition (1.3.2), which is stronger than condition (1.5.3), we summarize the above information as follows. Corollary 1.5.2 (see [152]). Let T ∈ L(X) be power bounded and let the range Im(I − T ) be closed. Then the Ces` aro averages Cn (T ) converge uniformly to a projection P onto Ker(I − T ) and Ker(I − T ) ⊕ Im(I − T ) = X. Observe again that if T is uniformly mean ergodic (see Definition 1.5.1), then 1 is either a regular point of T , or 1 is an isolated point of its spectrum such that it is a simple pole of the resolvent R(λ, T ) = (λI − T )−1 . Moreover, in this case 1 must be an eigenvalue of T (see [158]). A stronger asymptotic property is the convergence of the powers {T n }. A spectral criterion for uniform convergence was established by Koliha [132]. His result can be reformulated as follows.

14

Chapter 1. Semigroups of Linear Operators

Theorem 1.5.5. The sequence {T n}∞ n=1 ⊂ L(X) converges uniformly if and only if T is power bounded, Im(I − T ) is closed and there is no spectral point of T on the unit circle of the complex plane except, perhaps, λ = 1, that is, σ(T ) ∩ {λ ∈ C : |λ| = 1, |λ| = 1} = ∅. Moreover, in this case the sequence {T n}∞ n=1 converges to a projection P onto Ker(I − T ). Note that the power boundedness condition (1.3.2) implies that σ(T ) ⊂ {λ : |λ| ≤ 1}. If σ(T ) lies inside the open unit disk, then I − T is invertible and lim T n x = 0 for all x ∈ X. This can easily be established by using an equivalent

n→∞

norm  · ∗ on X for which T ∗ < 1. Theorem 1.5.6. For a bounded strongly continuous semigroup S = {T (t) : t ≥ 0} generated by an operator A, the following conditions are equivalent: (a) S is uniformly mean ergodic; (b) the limit lim+ λR(λ, A) exists in the operator norm; λ→0

(c) Im A is closed in X; (d) 0 ∈ ρ(A) or 0 is a first-order pole of the resolvent of A. Some ergodic theorems deal with Abel averages, which are defined by Aα x := (1 − α)[I − αT ]−1 x

(1.5.5)

for a single operator T , or by λ x := λ A

∞

e−λs T (s)xds

(1.5.6)

0

for a continuous semigroup {T (t) : t ≥ 0} with suitable α and λ, respectively. More precisely,   if r(T ) is the spectral radius of T , then α in (1.5.5) can be chosen in 1 0, r(T ) , and if ω0 is defined by (1.2.2), then λ > ω0 in (1.5.6); cf. Theorem 1.2.4. The integral in (1.5.6) is a Bochner integral in the Banach space X. Note that the study of Abel averages makes sense in many situations. It goes back to Eberlein (1949) [57] and Hille (1945) [110]. Note that a densely defined closed linear operator T is dissipative if and only if, for all α ≥ 0, Aα  ≤ 1. (1.5.7) Uniform ergodic theorems for Abel averages (as well as for Ces` aro averages) were established by Lin [152, 153]. Combining several results from [153] and emphasizing the behavior of Abel averages, we now formulate the following assertions.

1.5. Ces` aro and Abel averages of linear operators

15

Theorem 1.5.7. Let T ∈ L(X) satisfy condition (1.5.3). Then the operators Aα defined by (1.5.5) are well defined for all α ∈ (0, 1) and the following properties are equivalent: (i) Im(I − T ) is a closed subset of X; (ii) the net {Aα } converges uniformly to a projection P of X onto Ker(I − T ) as α → 1− . Theorem 1.5.8. Let S = {T (t) : t ≥ 0} be a C0 -semigroup such that lim

t→∞

1 T (t) = 0 t

(1.5.8)

λ in (1.5.6) are well defined for all and let A be its infinitesimal generator. Then A λ > 0 and the following conditions are equivalent: (i) A has closed range;   λ converges uniformly as λ → 0+ ; (ii) the net A 1 is uniformly ergodic, that is, its Ces` (iii) the operator A aro averages converge uniformly.

1 N

N −1

n A 1

n=0

Remark 1.5.1. Observe that, actually, for all λ > 0, the limits (in the strong opN −1 n as N → ∞ exist and determine the same projection A erator topology) of N1 λ of X onto

n=0

Ker A =



Ker(I − T (t)),

t≥0

which coincides with

λ . P := lim A λ→0+

(1.5.9)

Remark 1.5.2. Returning to the discrete case, an analog of assertion (iii) of Theorem 1.5.8 can also be obtained by using some functional calculus   showing that for each 0 < α < 1, the operator Aα satisfies the condition  n1 Anα  → 0 as n → ∞, and (I −Aα )X is closed, so that according to Lin’s uniform ergodic Theorem 1.5.7, N −1 for each 0 < α < 1, the uniform limit of N1 Anα as N → ∞ exists and is a n=0

projection of X onto Ker(I − T ). This complements Theorem 1.5.7 in the spirit of Theorem 1.5.8. At the same time, under the hypotheses of Theorems 1.5.7 and 1.5.8, and by employing techniques used in [153], one can show that for α close enough to λ ) are 1− (respectively, λ close enough to 0+ ), the operators Aα (respectively, A uniformly power convergent.

16

Chapter 1. Semigroups of Linear Operators

As a matter of fact, if X is a complex Banach space, then the assumptions of Theorems 1.5.7 and 1.5.8 lead to the uniform power convergence of the operators λ for all 0 < α < 1 and λ > 0, respectively. The following assertion Aα and A can be deduced from the corresponding classical results of [152, 153] (see [134, Assertion 1.3]). Proposition 1.5.1. Let T be a bounded linear operator on a complex Banach space X which satisfies condition (1.5.3), and let Aα , 0 < α < 1, be its Abel averages. The following assertions are equivalent: (i) (I − T )X is closed; (ii) for each α ∈ (0, 1), the powers Anα are uniformly convergent (to a projection P : X → Ker(I − T )); (iii) there is α ∈ (0, 1) such that the powers Anα uniformly converge. The limit in (ii) and (iii) is given by the projection of X onto Ker(I − T ) along Im(I − T ). Proposition 1.5.2. Let X be a complex Banach space and S = {T (t) : t ≥ 0} be a C0 -semigroup on X, which satisfies condition (1.5.8), and let A be its infinitesimal generator. The following assertions are equivalent: (i) Im A is closed; (ii) for each λ > 0, λ x = λ A

∞

e−λs T (s)xds

(1.5.10)

0

is uniformly power convergent; λ is uniformly power convergent. (iii) there is λ > 0 such that A In this connection, see also the paper [205]. In fact, one can see that conditions (1.5.3) and (1.5.8), respectively, are not necessary for the uniform power convergence of Abel averages. In particular, by using the Lumer–Phillips Theorem 1.3.6, one can obtain the following assertion, which is of interest by itself, and can be applied to the study of the fixed point set of some nonlinear operators (see Chapter 5 below). Theorem 1.5.9. Let T be a bounded linear operator on X such that Re T x, x∗  ≤ 1,

where x = 1 and x∗ ∈ J(x).

(1.5.11)

Then a) Aα := (1 − α)[I − αT ]−1 is well defined for all α ∈ (0, 1) with Aα  ≤ 1. b) The following statements are equivalent: (i) (I − T )X is closed;

1.6. Abel averages: recent results

17

(ii) for each α ∈ (0, 1), the powers of Aα converge uniformly in X; (iii) for some α ∈ (0, 1), the powers of Aα converge uniformly on X. Moreover, if ReT x, x∗  < 1, where x = 1 and x∗ ∈ J(x), then lim Anα = n→∞

0; hence Ker(I − T ) = {0} and (I − T )X = X, that is, I − T is invertible.

The Ces` aro boundedness of T itself, in terms of the resolvent of T , was characterized in [169, Theorem 3.1] by the uniform boundedness of the partial Abel sums of the resolvent of T . Thus, the uniform boundedness of the partial sums, when combined with the closedness of Im(I − T ), imply uniform Abel convergence (that is, the existence of the uniform Abel limit), by the above-mentioned result from [88].

1.6 Abel averages: recent results In this section we are interested in general sufficient conditions which ensure that λ are uniformly power convergent, which are the operators Aα , respectively, A also necessary conditions, so that they give criteria which should cover Propositions 1.5.1 and 1.5.2 as well as Theorem 1.5.9. For the discrete case the problem becomes much more relevant if the operator T is unbounded. In this case condition (1.5.3) is not relevant. In this connection, we note that the common feature of the assumptions in Proposition 1.5.1 and Theorem 1.5.9 is that the spectrum σ(T ) of the operator T lies in the half-plane Π1 = {z ∈ C : Re z ≤ 1}. Finally, we observe that assumption (1.5.3) and condition (i) in Proposition 1.5.1 imply the condition Ker(I − T ) ⊕ Im(I − T ) = X.

(1.6.1)

See, for example, [158, 191]. We show that even for an unbounded operator T these conditions together are necessary and sufficient for the uniform power convergence of the Abel averages defined by Aα = (1 − α)[I − αT ]−1 , 0 < α < 1. (1.6.2) Moreover, in these terms a corresponding criterion for a strongly continuous semigroup becomes an elementary consequence of the following assertion. Theorem 1.6.1. Let T be a closed linear operator on a complex Banach space X. Then the Abel averages (1.6.2) are well defined on X for all 0 < α < 1 and are power convergent (to a projection P onto Ker(I − T ) along Im(I − T )) if and only if σ(T ) lies in the closed half-plane Π1 = {z ∈ C : Re z ≤ 1} and condition (1.6.1) holds.

18

Chapter 1. Semigroups of Linear Operators

Proof. Let T be a closed linear operator on X with the domain of definition DT (which is a dense linear subspace of X). Let ρ(T ) be the resolvent set of T and let σ(T ) be its spectrum. For λ ∈ ρ(T ), we denote by R(λ, T ) = (λI − T )−1 the resolvent of T . For a fixed λ ∈ ρ(T ), the operator R(λ, T ) is bounded and defines a one-to-one mapping of X onto DT . In addition, the following relations hold: (a) (λI − T )R(λ, T )x = x for all x ∈ X and (b) R(λ, T )(λI − T )x = x whenever x ∈ DT . See, for example, [54] and [240]. Assume now that for some α ∈ (0, 1), the number λ = For such an α, one can define the operator Aα by setting

1 α

belongs to ρ(T ).

Aα := (1 − α)(I − αT )−1 = (λ − 1)R(λ, T ). Then conditions (a) and (b) can be rewritten as follows: 

(a ) (I − αT )Aα x = (1 − α)x for all x ∈ X; (b ) Aα (I − αT )x = (1 − α)x whenever x ∈ DT . Let now x ∈ Ker(I − Aα x), that is, x = Aα x. Since the range R(Aα ) of Aα is contained in DT , it follows that x ∈ DT , and we have by (a ) that x−αT x = x−αx, that is, x = T x. So, Ker(I − Aα ) ⊂ Ker(I − T ). Conversely, let x ∈ Ker(I − T ) ⊂ DT . Then (b ) implies that Aα (x − αx) = (1 − α)x, that is, x = Ax. Thus, Ker(I − Aα ) = Ker(I − T ). Let now x ∈ Im(I − T ), that is, there is a point y ∈ DT , such that x = y − T y. Then we get by (b ), α(I − T )y = (I − Aα )(I − αT )y or x = (I − Aα )z, where z=

1 (I − αT )y. α

In other words, Im(I − T ) ⊂ Im(I − Aα ). Conversely, let x ∈ Im(I − Aα ), that is, x = y − Aα y, for some y ∈ X. Note α that the element z = 1−α Aα y ∈ DT . Then we have by (a ), 1 α (I − T )Aα y = (I − αT )Aα y − Aα y 1−α 1−α = y − Aα y = (I − Aα )y = x,

(I − T )z =

So, Im(I − T ) ⊇ Im(I − Aα ).

1.6. Abel averages: recent results

Thus we have shown that if for some α ∈ (0, 1), λ =

19 1 α

∈ ρ(T ), then

Ker(I − T ) = Ker(I − Aα )

(1.6.3)

Im(I − T ) = Im(I − Aα ).

(1.6.4)

and Given α ∈ (0, 1), consider the function f (z) = (1 − α)(1 − αz)−1 . This function is analytic in the domain Ωα = {z ∈ C : |z − λ| > λ−1}, λ = α1 , and maps Ωα univalently onto the open unit disk D in the complex plane C. Therefore, we have by the Spectral Mapping Theorem (assertion (iii) of Theorem 1.1.2), that the  spectrum σ(Aα ) lies in D if and only if the spectrum σ(T ) lies in Ωα . Since Ωα = Π1 = {z ∈ C : Re z ≤ 1}, this fact and relations (1.6.3) and (1.6.4) 0 0, λ := λR(λ, A) A is uniformly power convergent;

(1.6.5)

20

Chapter 1. Semigroups of Linear Operators

λ is uniformly power convergent and (iii) there is λ > 0 such that the operator A ρ(A) contains the whole right half-plane. Proof. Consider the operator T = I + A and for λ > 0, set α = we have the equality λ = λ(λI − A)−1 = (1 − α) A α



−1

∈ (0, 1). Then

−1 1−α I − (T − I) α

= (1 − α) [(1 − α)I − α(T − I)] = (1 − α) [I − αT ]

1 1+λ

−1

= Aα ,

(1.6.6)

which, when combined with Theorem 1.6.1, immediately yields our assertion.  Applying now Theorem VIII.1.11 from [54], we get the following assertion, which is also a generalization of Proposition 1.5.2. Corollary 1.6.1. Let S = {T (t) : t ≥ 0} be a strongly continuous semigroup of bounded linear operators on X and let A be its infinitesimal generator. Assume that log T (t) lim = 0. t→∞ t The following statements are equivalent: (i) Ker A ⊕ Im A = X; (ii) for each λ > 0, the sequence of operators

 ∞ n A λ

defined by

n=1

nλ = A

λn (n − 1)!

∞

 e−λt tn−1 T (t)dt = λn Rn (λ, A) ,

n ∈ N,

0

converges uniformly on X to a projection P onto the set {x ∈ X : T (t)x = x for all t ≥ 0};  ∞ n (iii) for some λ > 0, the sequence of operators A converges uniformly on λ n=1

X to a projection P onto the set {x ∈ X : T (t)x = x for all t ≥ 0}.

Chapter 2

Numerical Range In this chapter we introduce and study the main topic of our book: holomorphic mappings, their numerical range, growth estimates of it and related material. It is well known that in a finite-dimensional Banach space each mapping which is holomorphic in the unit ball B is bounded on each subset strictly inside B. This is no longer true in the general case. Relevant examples can be found in [40]. In general Banach spaces the problems of verifying whether a holomorphic mapping has unit radius of boundedness, as well as of studying its numerical ranges, arise in many aspects of infinite-dimensional holomorphy (see, for example, [78, 99, 191, 92]) and complex dynamical systems [9, 66, 191]. In particular, they play a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings [186, 187], the study of flow invariance and range conditions in nonlinear analysis [9, 185], and geometric function theory in finiteand infinite-dimensional Banach spaces [105, 92, 191]. This was specifically mentioned for the class of the so-called semi-complete vector fields (or infinitesimal generators) in the context of their applications to the study of starlike and spirallike mappings, and the Bloch (univalence) radii [103] for locally biholomorphic mappings.

2.1 Holomorphic mappings in Banach spaces Holomorphic mappings on abstract spaces were considered by Fr´echet around 1910 and subsequently studied by many authors; see [111, 49, 50]. We give two definitions of holomorphy. Let X and Y be complex Banach spaces, and let D be an open subset of X. Definition 2.1.1. A mapping f : D → Y is (strongly) holomorphic if for each x ∈ D, there exists a continuous complex-linear mapping A(x) : X → Y , denoted © Springer Nature Switzerland AG 2019 M. Elin et al., Numerical Range of Holomorphic Mappings and Applications, https://doi.org/10.1007/978-3-030-05020-7_2

21

22

Chapter 2. Numerical Range

by f  (x) (or Df (x)), such that f (x + y) − f (x) − f  (x)y = 0. y→0 y lim

There is an alternative weaker condition which reduces matters to the case of a complex-valued function of one complex variable. (We use λ to denote a complex variable.) Definition 2.1.2. A function f : D → Y is (weakly) holomorphic if it is locally bounded and if the mapping λ → (f (x + λy)) is holomorphic at λ = 0 for each x ∈ D, y ∈ X and linear functional ∈ Y ∗ . Clearly, the strong holomorphicity implies the weak one. It was shown by Dunford in 1938, using the uniform boundedness principle, that these two notions are, in fact, equivalent; see [111, Theorems 3.10.1 and 3.17.1]. We denote the set of all holomorphic mappings defined in D ⊂ X which take values in Ω ⊂ Y by Hol(D, Ω). In many cases, D will be the open ball of radius r and center x, that is, D = Br (x) = {y ∈ X : y − x < r}. We also write Hol(D) instead of Hol(D, D). Recall that a mapping F : X × · · · × X → Y is said to be multilinear (or, n

more precisely, n-linear) if it is linear in each one of its variable. Given any n-linear mapping F , one can define a symmetric mapping by Fs (x1 , . . . , xn ) =

 1

F xσ(1) , . . . , xσ(n) , n! σ

where the summation is taken over all permutations σ of (1, . . . , n). A mapping P : X → Y is said to be a homogeneous polynomial of degree n if P (x) = F ( x, . . . , x ),    n

where F : X × · · · × X → Y is a continuous n-linear mapping. Example 2.1.1. Each homogeneous polynomial P : X → Y of degree n is a holomorphic mapping on the whole space X and P  (x)y =

n−1 k=0

F ( x, . . . , x , y, x, . . . , x ),       k

n−k−1

x, y ∈ X.

2.1. Holomorphic mappings in Banach spaces

23

Moreover, if {Pn }∞ n=0 is a sequence of homogeneous polynomials, where Pn is of degree n, and we define Pn  := sup Pn (x), then the existence of positive x=1

constants M and r such that Pn  rn ≤ M for n = 0, 1, . . . , easily implies by the Weierstrass M -test that the series ∞ f (x) = Pn (x) n=0

converges to a holomorphic mapping in Br (0). In the other direction, the following theorem shows that every holomorphic mapping can be represented as the sum of an infinite series of homogeneous polynomials in a neighborhood of each of the points of its domain. Theorem 2.1.1 (Taylor’s theorem; see Theorem 2.6 in [191]). Let D ⊂ X be a domain and let f ∈ Hol(D, Y ) be bounded. Then for each x0 ∈ D, f (x0 + x) =

∞

x < dist(x0 , ∂D),

Pn [x0 ](x),

n=0

where Pn [x0 ], n = 0, 1, . . . , are homogeneous polynomials of order n. It is clear that P0 [x0 ] = f (x0 ) and P1 [x0 ] = f  (x0 ), the Fr´echet derivative of f at x0 . Theorem 2.1.2. In the settings of Theorem 2.1.1, let f (x) ≤ M for all x such that x − x0  < dist(x0 , ∂D). Then for each n = 0, 1, . . . , the following inequality holds: M Pn [x0 ] ≤ n . r Example 2.1.2. The holomorphic functional calculus associates classical holomorphic functions with holomorphic mappings on domains in spaces of operators. Specifically, let W be a complex Banach space and let X = L(W ) be the Banach space of all bounded linear operators on W . Suppose f is a holomorphic function in the disk {λ : |λ| < r}. Then f (λ) =

∞

for |λ| < r,

an λn

n=0

and hence f (A) =

∞

an An

n=0

is defined for those A ∈ X which have spectral radii r(A) less than r. It is not difficult to show that the set D = {A ∈ X : r(A) < r} is open and that f : D → X is holomorphic.

24

Chapter 2. Numerical Range

More generally, let Ω be a bounded open subset of the complex plane and let D = {A ∈ L(X) : σ(A) ⊂ Ω}. Define

1 f (A) = 2πi

f (λ)(λI − A)−1 dλ,

γ

where ζ consists of a finite number of positively oriented rectifiable Jordan curves in Ω whose interiors are a disjoint cover of σ(A). It is well known (see, for example, [54]) that this integral defines a holomorphic mapping f : D → X = L(W ). In contrast with the finite-dimensional case, in infinite-dimensional Banach spaces the radius of convergence of the Taylor series of a given mapping may be essentially different from its radius of boundedness. The following example shows that there are entire mappings which are unbounded in the unit ball. Example 2.1.3. We exhibit a mapping that is holomorphic everywhere in X = c0 , the space of all sequences {xk }∞ k=1 tending to zero, bounded on Br (0) for every r with 0 < r < 1, but unbounded on B = B1 (0). Define f : X → C by f (x) =

∞

xk k ,

for x = {xk }∞ k=1 .

k=1

  Then f is defined and holomorphic everywhere in X. Also, since xk k  ≤ xk for any x ∈ X, it follows that f (x) ≤

x , 1 − x

x ∈ B.

Now for each positive integer n define the point x(n) ∈ B1 (0) by 1 x(n)k = √ , 1 ≤ k ≤ n, n 2 x(n)k = 0, k > n. Then |f (x(n))| = f (x(n)) ≥

n k=1

[x(n)k ]n =

n . 2

Hence f is unbounded in B. This example shows that in the infinite-dimensional case, a mapping can be holomorphic in some ball (in the example, in the whole space), but to be unbounded in a smaller ball. This phenomenon leads us to the following notion. Definition 2.1.3. Let f ∈ Hol(B, X). One says that it has unit radius of boundedness if it is bounded on each ball Br = {x : x < r}, r < 1 ([78, 99]; see also [40]).

2.2. Spectrum and resolvent of holomorphic mappings

25

2.2 Spectrum and resolvent of holomorphic mappings We need below the following classical notions. Definition 2.2.1. Let D ⊂ X be a domain and let f ∈ Hol(D, X). (i) A mapping f is called univalent on D if it is injective; (ii) f is called biholomorphic on D if f (D) ⊂ X is a domain, and the inverse mapping f −1 exists and is holomorphic on f (D); (iii) f is called locally biholomorphic on D if for each point x ∈ D, there is a neighborhood U of x such that f |U is biholomorphic. In the finite-dimensional case, that is, if X = Cn , each univalent mapping on D is biholomorphic on D. However, in the infinite-dimensional case this result is not necessarily true. See [105] for a counterexample. Different versions of the Inverse Function Theorem can be found in [40, p. 94], [99, p. 1017], see also [103]. We formulate it as follows. Theorem 2.2.1. Let D, 0 ∈ D, be a domain in X, and let h ∈ Hol(D, X) with h(0) = 0. Then h is biholomorphic in a neighborhood of the origin if and only if

 the linear operator A = h (0) is invertible. In this case, h−1 (0) = A−1 . Definition 2.2.2. Let D, 0 ∈ D, be an open subset of X, and let h : D → X be a single-valued mapping. One says that the set ρ(h) ⊂ C of complex numbers is the resolvent set for h if it consists of those λ for which there is an open set Dλ ⊂ D with 0 ∈ Dλ , such that (λI − h)|Dλ is biholomorphic. The complement σ(h) := C \ ρ(h) is called the spectrum of h. In other words, the spectrum σ(h) consists of those λ ∈ C for which it is not possible to find an open subset Dλ and a neighborhood Uλ ⊂ (λI − h)(Dλ ) with Uλ  −h(0) such that (λI − h)−1 is a well-defined holomorphic mapping on Uλ with values in Dλ . By R(λ, h) we denote the operator (λI − h)−1 whenever it exists on an open domain Uλ ( −h(0)) and Uλ is called the domain of the resolvent R(λ, h). Proposition 2.2.1 ([99]). Let h be a holomorphic mapping in a neighborhood of the origin. The following relations hold: σ(h) = σ(h (0))

and, respectively,

ρ(h) = ρ(h (0)).

Proof. Denoting g = λI − h it is enough to show that 0 ∈ / σ(g) if and only if 0 ∈ / σ(g  (0)). Indeed, differentiation of the equalities g ◦ g −1 = I and g −1 ◦ g = I shows that 0 ∈ / σ(g) implies 0 ∈ / σ(g  (0)). The reverse implication is just a consequence of the inverse function theorem.  Let h ∈ Hol(D, X) and 0 ∈ D. Put h := sup h(x) when it is finite. x∈D

The following corollary is an immediate consequence of Proposition 2.2.1 and the Cauchy inequality for the first derivative (see Theorem 2.1.2).

26

Chapter 2. Numerical Range

Corollary 2.2.1. Let h ∈ Hol(B, X) satisfy the condition I − h < 1. Then 0 ∈ / σ(h), that is, h is a holomorphically invertible mapping in a neighborhood of the origin. Below we will see that properties of the resolvent set and the resolvent of a holomorphically invertible mapping, as well as the domain of its definition, can be described in terms of the numerical range of the mapping under consideration.

2.3 Numerical range As we have already mentioned in Section 1.3, Lumer [155] developed a theory of the numerical range for bounded linear operators on Banach spaces as a generalization of the classical theory in Hilbert spaces. The numerical range for nonlinear holomorphic mappings was introduced by Harris [99], who applied it to his previous results concerning a generalization and a continuous version of the Schwarz Lemma (see [97], Theorem 2; [98], Theorem 1). Let D be a bounded and convex domain in a complex Banach space X, 0 ∈ D, and let X ∗ denote the dual space of X. For x ∈ ∂D, the boundary of D, let Q(x) be the set of all linear functionals on X that are tangent to D at x, that is, Q(x) = { ∈ X ∗ : (x) = 1, Re (y) ≤ 1, y ∈ D}.

(2.3.1)

Definition 2.3.1 (cf. [99] and [103]). Let h : D → X have a continuous extension to the closure D of D. We define the set VD (h) := { (h(x)) : x ∈ ∂D, ∈ Q(x)} to be the numerical range of h with respect to D. The number |VD (h)| = sup{| (h(x))| : ∈ Q(x), x ∈ ∂D} is called the numerical radius of h with respect to D. It is natural to ask how to describe those mappings which attain their numerical radius in the sense that there are x0 ∈ ∂D and 0 ∈ Q(x0 ) such that |VD (h)| = | 0 (h(x0 ))|. It was proven in [7] that if a space X has the Radon– Nikod´ ym property, then for any mapping h ∈ Hol(B, X) which is bounded and continuous on B, for any m ∈ N and any  > 0, there is a homogeneous polynomial Pm of degree m such that Pm  <  and h + Pm attains its numerical radius. In addition, if X is a locally uniformly convex Banach space, then the set of numerical radius attaining elements in Hol(B, X) is dense (see [148]). In the opposite direction, an example of a Banach space X and a homogeneous polynomial P2 of degree 2 that cannot be approximated by numerical radius attaining holomorphic mappings was constructed in [7]. In view of the inequality | (h(x))| ≤  X ∗ h(x)X , one concludes that if h is bounded, then |VBR (h)| ≤ sup h(x), where BR = {x ∈ X : x < R}. The x=1

2.3. Numerical range

27

following result shows that the numerical radius may coincide with the norm for all homogeneous polynomials. Proposition 2.3.1 (see [89]). Let X = ∞ be the space of all bounded sequences endowed with the supremum norm. Then for any homogeneous polynomial P we have P  = |V (P )|. Let X be a Banach space. For ∈ X ∗ we mostly use the notation x,  in place of (x). Recall that for x ∈ X, we denote by J(x) the set of all x∗ ∈ X ∗ which satisfy Rex, x∗  = x2 = x∗ 2 . By the Hahn–Banach theorem such elements x∗ exist, but in general they are not unique. However, if X = H is a Hilbert space, then x∗ is unique and can be identified with x ∈ H = H∗ . From now on we just consider the case where D is a ball centered at the origin. This is because more general situations can be reduced to this special case by using the Minkowski functional (cf. [10] and [103]). If D = B is the open unit ball in X and x ∈ ∂B, then J(x) = Q(x), where Q is defined by (2.3.1). In this case we just write V (h) for the numerical range of h with respect to B. Clearly, as in the linear case, V (h) has the following algebraic properties: (i) V (h + g) ⊂ V (h) + V (g); and for any complex scalar λ (ii) V (λh) = λV (h); (iii) V (h + λI) = V (h) + λ. The following assertion has various applications related to the rigidity properties of holomorphic mappings. Proposition 2.3.2. Let h ∈ Hol(B, X) have a continuous extension to the closure of B with h(0) = 0, and let A = h (0) be its Fr´echet derivative at the origin. Then V (A) ⊂ coV (h). Proof. By the separation theorem for convex sets, it suffices to show that if α is any complex number such that Re αz ≤ 1 for all z ∈ V (h), then Re αz ≤ 1 for all z ∈ V (h (0)). Let x ∈ X with x = 1. Given x∗ ∈ J(x), define f (λ) = h(λx), x∗  /λ. Then f is holomorphic in the open unit disc D and continuous on D. So if |λ| = 1, then f (λ) is in V (h) since λx∗ ∈ J(x). Hence g(λ) := Re αf (λ) is a harmonic function in D with a continuous extension to D which satisfies g(λ) ≤ 1 when |λ| = 1. Hence, g(0) ≤ 1 by the maximum principle for harmonic functions. Thus Re α h (0), x∗  ≤ 1, as we wished to show.  For the case where a mapping h is not defined on the closure of the domain of its holomorphy, one extends Definition 2.3.1 as follows.

28

Chapter 2. Numerical Range

Definition 2.3.2. Let h ∈ Hol(BR , X). The number    |VBR (h)| = lim sup sup |λ| : λ ∈ VBR (h(s ·)) s→1−

is called the numerical radius of h with respect to BR . Clearly, if h ∈ Hol(BR , X) has a continuous extension to BR , then |VBR (h)| = sup {|λ| : λ ∈ VBR (h)} . For R = 1 we use the notations B and V (h) in place of B1 and V1 (h). The following result was obtained by Harris in [99]. Theorem 2.3.1. Let h ∈ Hol(B, X) and let Pm be the m-th term of the Taylor series expansion of h at the origin. The following relations hold: (i) |V (Pm )| ≤ |V (h)| ; (ii) Pm  ≤ km |V (h)|, where k0 = 1, k1 = e, and km = mm/m−1 for m ≥ 2; 2 (iii) sup h(x) ≤ (1−r) 2 |V (h)| for each r ∈ (0, 1). x=r

Proof. Assertions (i) and (iii) can be proved by reduction to the one-dimensional case and by using the classical Cauchy inequalities. The essential part of the theorem is assertion (ii). We quote here the proof given in [99]. So, we now proceed to prove the inequality Pm  ≤ km |V (Pm )| .

(2.3.2)

Since it is clear that (2.3.2) holds when m = 0, suppose that m ≥ 1 and let 0 < r < 1 be given. Since Pm  (0) = 0 it follows from the Inverse Function Theorem (Theorem 2.2.1) that there is δ > 0 such that (I + λPm )(B) covers Br and (I + λPm )−1 exists as a holomorphic mapping on Br for all |λ| < δ. If x ∈ B, x = 0, then choosing ∈ J(x/x), we have (I + λPm )(x) ≥ | (x + λPm (x))|     = x + λxm (Pm (x/x)) ≥ x (1 − δ |V (Pm )|) for all |λ| < δ. Hence   y ≥ (I + λPm )−1 (y) (1 − δ |V (Pm )|) for all y ∈ Br . Set β = 1 − δ|V (Pm )| and take a small δ if necessary so that β > 0. Then for all |λ| < δ, the mapping β(I + λPm )−1 maps Br into Br , so all of its iterates are defined and do the same. Let y ∈ Br and define  n fn (λ) := β(I + λPm )−1 (y),

2.3. Numerical range

29

where n indicates the nth iterate. Again by the Inverse Function Theorem, if fn is holomorphic in the disk |λ| < δ, then fn+1 is holomorphic in the same disk and   fn+1  (0) = β fn  (0) − Pm (fn (0)) . Since f0 is clearly holomorphic everywhere, it follows by induction that fn is holomorphic in the disk |λ| < δ and fn  (0) = −

n−1

β n+k(m−1) Pm (y)

(2.3.3)

k=0

for any positive integer n. But by Cauchy’s inequality, fn  (0) ≤ and consequently, Pm (y) ≤

r , δ

(2.3.4)

r(1 − β m−1 )  , 1 − β n(m−1)

δβ n

where we are disregarding for the time being the exceptional cases |V (Pm )| = 0 and m = 1. Since 1 − β m−1 ≤ (m − 1)(1 − β) = (m − 1)δ |V (Pm )| and y is an arbitrary point of Br , it follows that Pm  ≤

(m − 1)|V (Pm )|  . rm−1 β n 1 − β n(m−1)

(2.3.5)

Now for all n large enough, δ=

log m n(m − 1) |V (Pm )|

is small enough, hence by (2.3.5) and the fact that β n → m−1/(m−1) as n → ∞, we have km |V (Pm )| Pm  ≤ . rm−1 Letting r → 1− , we conclude that (2.3.2) holds provided |V (Pm )| = 0 and m ≥ 2. If |V (Pm )| = 0 or m = 1, it follows from (2.3.3) and (2.3.4) that Pm  ≤

1 . nδrm−1 β n

(2.3.6)

When |V (Pm )| = 0, we may take the limit in (2.3.6) as n → ∞ to get (2.3.2). When m = 1 and |V (Pm )| = 0, for all n large enough we may take δ=

1 ; (n + 1) |V (P1 )|

30

Chapter 2. Numerical Range

hence by (2.3.6),

 n+1 1 P1  ≤ 1 + |V (P1 )| n

and we then see that (2.3.2) holds by taking the limit as n → ∞ in the above inequality. This establishes (2.3.2) in all cases.  Assertion (iii) actually implies that if |V (h)| < ∞, then h has unit radius of boundedness. We will see below that to reach this conclusion it is enough to require only that the value N (h) is finite. Another immediate consequence of Theorem 2.3.1 is the following assertion. Corollary 2.3.1. If h ∈ Hol(B, X) and its numerical radius |V (h)| = 0, then h = 0 identically. For special classes of mappings with additional restrictions on the numerical range, the estimates of Theorem 2.3.1 can be improved. The next two theorems were proved in [89]. Theorem 2.3.2. Let g ∈ Hol(D, C) be univalent with g(0) = 1. Let h ∈ Hol(B, X) be normalized by h(0) = 0 and h (0) = I, and satisfy the following condition: 1 (h(x)) ∈ g(D) x

for all

x∈B

and all

∈ J(x).

Let Pm be the mth term of the Taylor series expansion of h at the origin. For any m ≥ 2, the following relations hold: (i) |V (Pm )| ≤ (m − 1)|g  (0)|; (ii) Pm  ≤ km (m − 1)|g  (0)|, where km = mm/(m−1) ; (iii) there is a constant M = M (r, g) > 0, which is independent of h, such that sup h(x) ≤ M (r, g) for each r ∈ (0, 1). x=r

In what follows we will use some notions and notations from geometric function theory. In particular, a univalent function g ∈ Hol(D, C) is called convex if its image g(D) is convex. One says that f ∈ Hol(D, C) is subordinate to g and writes f ≺ g if f (0) = g(0) and f (D) ⊂ g(D). For more details see, for example, [55, 84, 87, 92]. Theorem 2.3.3. Suppose that in the assumptions of Theorem 2.3.2 the function g is convex. For any m ≥ 2, the following relations hold: (i) |V (Pm )| ≤ |g  (0)|; (ii) Pm  ≤ km |g  (0)|, where km = mm/(m−1) ; ! r(1 − r log 2)  (iii) sup h(x) ≤ r 1 + 4|g (0)| for each r ∈ (0, 1). (1 − r)2 x=r

2.3. Numerical range

31

Proof. To prove assertion (i), fix y ∈ X with y = 1 and ∈ J(y). Define a function q ∈ Hol(D, C) by q(z) =

1 (h(zy)), z

ζ ∈ D \ {0}

(2.3.7)

and q(0) = 1. Then q(D) ⊂ g(D). Also, since q(0) = g(0) = 1, we see that q ≺ g. ∞  Let q(z) = 1+ qm z m be the Taylor expansion of q. Since g is a convex function, m=1

it follows that |qm | ≤ |g  (0)| for m ≥ 1. On the other hand, since h is holomorphic in B, by Taylor’s Theorem 2.1.1, there is r ∈ (0, 1] such that h can be represented by a series of homogeneous polynomials: h(x) = x +

∞

Pm (x),

x ∈ Br .

(2.3.8)

m=2

Now, since q(z) = 1 +

∞

qm z m = 1 +

m=1

∞

(Pm (y))z m−1 ,

|z| < r,

m=2

we deduce that qm−1 = (Pm (y)), m ≥ 2. Hence | (Pm (y))| ≤ |g  (0)|, m ≥ 2. Since y ∈ ∂B and ∈ J(y) are arbitrary, we deduce that |V (Pm )| ≤ |g  (0)| for m ≥ 2, and thus assertion (i) holds. Now the second relation follows from the first one and relation (ii) in Theorem 2.3.1. Next, we prove relation (iii). To this end, we remark that the right-hand side of (2.3.8) is uniformly bounded on each closed ball Br , r ∈ (0, 1), by relation (ii) ∞  and the fact that the series km z m has a unit radius of convergence. Thus the m=2

right-hand side of (2.3.8) converges uniformly on each closed ball Br , r ∈ (0, 1), to a holomorphic mapping on B. Consequently, equality (2.3.8) holds for all x ∈ B. Consequently, if x = r < 1, then h(x) ≤ r +

∞

Pm (x) ≤ r +

m=2

∞

Pm rm ≤ r + |g  (0)|

m=2

∞

km rm .

m=2

In view of the fact that ss/(s−1) ≤ 4(1 − log 2)(s − 2) + 4 for all s ≥ 2 (see [25]), we deduce that h(x) ≤ r + |g  (0)|

∞

[4(1 − log 2)(s − 2) + 4] rm

m=2

= r + 4r2 |g  (0)| · and the result follows.

1 − r log 2 (1 − r)2 

32

Chapter 2. Numerical Range

2.4 Real part estimates In this section we are interested in real part estimates for functions which are holomorphic in the open unit disk D of the complex plane C. Estimates of such type are frequently used in classical Geometric Function Theory and Harmonic Analysis, as well as in Analytic Number Theory. In our subsequent considerations they play a crucial role in establishing the growth estimates of the numerical range in complex Banach spaces. These estimates give us not only quantitative results, but also provide existence theorems in the fixed point theory for holomorphic mappings (see Chapter 3) in terms of their numerical ranges. For some historical remarks and additional details, we refer the reader to the monographs [84, 142, 179, 68] and [209]. In particular, sharp real part theorems are presented in the recent book of Kresin and Maz’ya [142], where the proofs are mostly based on various integral representations. Following the work [113] by Jensen (1919), we use another approach which is based on the classical Schwarz Lemma. This assertion is a fundamental result in Geometric Function Theory and in the study of fixed points in one and higher dimensions (see Chapter 3 below). Theorem 2.4.1 (The Schwarz Lemma). Suppose F ∈ Hol(D) and F (0) = 0. Then either (2.4.1) |F  (0)| < 1 and |F (z)| < |z|

(2.4.2)

for all z ∈ D \ {0}, or |F  (0)| = 1 and F is the rotation defined by F (z) = F  (0)z, z ∈ D. The Schwarz Lemma allows us to obtain estimates for different classes of holomorphic functions. We start with the following assertions which can be found in [9] and [160]; see also [209]. Lemma 2.4.1. Let p ∈ Hol(D, C) and assume that there is a non-negative real function ψ(r) on [0, 1) such that the following condition holds: Re [zp (z) + ψ(|z|)p(z)] ≥ 0,

z ∈ D.

Then Re p(z) ≥ 0 for all z ∈ D. Proof. Suppose that Re [zp (z) + ψ(|z|)p(z)] ≥ 0 for all z ∈ D and some function ψ : [0, 1) → R+ . ∂p Setting z = reiθ , we have zp (z) = r and hence ∂r     ∂p Re r (2.4.3) + ψ(r) Re p(z) ≥ 0, z = reiθ ∈ D. ∂r

2.4. Real part estimates

33

Assume that there exists a point z0 = r0 eiθ0 in D such that Re p(z0 ) < 0. Then (2.4.3) implies that Re p(0) ≥ 0. So, there is r1 ∈ [0, r0 ) such that Re p(r1 eiθ0 ) = 0 by continuity. Then one can find r2 ∈ (r1 , r0 ) such that Re p(r2 eiθ0 ) < 0

and

Re

∂p (r2 eiθ0 ) < 0. ∂r

But these inequalities contradict (2.4.3). Thus it follows that Re p(z) ≥ 0 everywhere, and we are done.  For a special choice of the function ψ, a converse assertion also holds. Lemma 2.4.2. Let p ∈ Hol(D, C) and assume that Re p(z) ≥ 0 for all z ∈ D. Then (1 − |z|2 )|p (z)| ≤ 2 Re p(z) and consequently, ! 1 + |z|2 Re zp (z) + p(z) ≥ 0, 1 − |z|2 

Proof. Fix a ∈ D and let pa (z) − 1 F (z) = , pa (z) + 1

 p where pa (z) =

z ∈ D.

a−z 1−az



− i Im p(a)

Re p(a)

.

Obviously, Re pa (z) ≥ 0 and pa (0) = 1. Therefore the function F satisfies the hypotheses of the Schwarz Lemma. The inequality |F  (0)| ≤ 1 means that |pa  (0)| ≤ 2. This fact implies, in its turn, the first asserted inequality. Consequently, Re(−zp (z)) ≤ |zp (z)| ≤

2|z| Re p(z) 1 + |z|2 ≤ Re p(z), 1 − |z|2 1 − |z|2 

which completes the proof. We now consider, in particular, functions of the Carath´eodory class P := {p ∈ Hol(D, C) : Re p(z) ≥ 0

for all z ∈ D, p(0) = 1} .

Lemma 2.4.3. Let p ∈ P. Then for all z ∈ D,   2  p(z) − 1 + |z|  ≤ 2|z| ,  1 − |z|2  1 − |z|2 Proof. Let F (z) = p(z)−1 p(z)+1 . By the Schwarz Lemma, |F (z)| ≤ |z|. A direct calculation shows that this inequality is equivalent to the required one. 

34

Chapter 2. Numerical Range

This fact can be generalized as follows. Proposition 2.4.1. Let p ∈ P satisfy | arg p(z)|
0, a ∈ R and let p ∈ Hol(D, C) satisfy p(0) = 1 and Re (p(z) + αzp (z)) ≥ a, Then

1 Re p(z) ≥ 0

z ∈ D.

! 1 − tα |z| (1 − a) dt + a ≥ (1 − a)κ(α) + a. 1 + tα |z|

Moreover, for every β ∈ [0, α], we have   β Re (p(z) + βzp (z)) > (1 − a) 1 − κ(α) + a, α

z ∈ D.

Corollary 2.4.5. Let α > 0, a ∈ R and let p ∈ Hol(D, C) satisfy p(0) = 1. Then (i) the condition Re (p(z) + αzp (z)) ≥ 0, implies that Re (p(z) + βzp (z)) >

z ∈ D,

 1−

β α

 κ(α)

for every β ∈ [0, α]; (ii) the condition 

 β 1− α κ(α)   Re (p(z) + αzp (z)) ≥ − 1 − 1 − αβ κ(α)

40

Chapter 2. Numerical Range

for some β ∈ [0, α] implies that Re (p(z) + βzp (z)) > 0; (iii) the condition Re (p(z) + αzp (z)) ≥ a   κ(α) with a ∈ − 1−κ(α) , 0 implies that Re (p(z) + βzp (z)) > 0 ! a + (1 − a)κ(α) for every β ∈ 0, α . (1 − a)κ(α) Note that assertion (i) with α0 = κ −1 (1/2) implies a generalization of a wellknown result due to Strohh¨acker which can be formulated as follows: If |p(z) + zp (z)| ≥ |zp (z)|, then Re p(z) ≥ 12 . 

Example 2.4.2. If Re (zp(z)) ≥ 0, then Re (p(z) + βzp (z)) ≥ (1 − β) (2 ln 2 − 1) for every β ∈ [0, 1]. To proceed, we denote by Pλ , λ ≥ 0, the class of functions analytic in the open unit disk D normalized by p(0) = 1 and satisfying the inequality   Re 2λ|z|2 p(z) + (1 − |z|2 ) (zp(z)) ≥ 0. (2.4.14) Proposition 2.4.2. Let λ ∈ (0, 1] and let the function p(= pλ ) be defined by p(z) = (1 − z)λ . Then p ∈ Pλ and p ∈ Pμ whenever μ < λ. Proof. Writing z ∈ D in the form z = 1 − reiφ with |φ| < we calculate

π 2

and r ∈ (0, 2 cos φ),



(2.4.15) 2μ|z|2 p(z) + (1 − |z|2 ) (zp(z))

 λ iφλ 2 = 2μ 1 + r − 2r cos φ r e + r(2 cos φ − r)  

 · rλ eiφλ − λ 1 − reiφ rλ−1 eiφ(λ−1)  

  = rλ 2μ 1 + r2 − 2r cos φ eiφλ + (2 cos φ − r) r(1 + λ)eiφλ − λeiφ(λ−1) . If μ < λ, we put φ = 0 in (2.4.15) to get 

2μ|z|2 p(z) + (1 − |z|2 ) (zp(z))

= rλ [2μ − 2λ + 2μr(r − 2) + (2 − r)r(1 + λ) + rλ] .

2.5. Holomorphically dissipative and accretive mappings

41

Clearly, this expression is negative when r is small enough, that is, p ∈ Pμ . If μ = λ, we get from (2.4.15),   (2.4.16) Re 2λ|z|2 p(z) + (1 − |z|2 ) (zp(z))  λ = r 2λ cos(φλ) + 2rλ(r − 2 cos φ) cos(φλ)  + (2 cos φ − r)r(1 + λ) cos(φλ) − (2 cos φ − r)λ cos (φ(λ − 1))  = rλ (2 cos φ − r)r(1 − λ) cos(φλ) + rλ cos (φ(λ − 1))  + 2λ (cos(φλ) − cos φ · cos (φ(λ − 1))) . Since obviously (2 cos φ − r)r(1 − λ) cos(φλ) ≥ 0 and rλ cos (φ(λ − 1)) ≥ 0, we have to examine whether the last summand is positive. Indeed, cos(φλ) − cos φ · cos (φ(λ − 1)) = cos(φλ) − cos2 φ · cos(φλ) − cos φ · sin(φλ) · sin φ = sin φ(sin φ · cos(φλ) − cos φ · sin(φλ)) = sin φ · (φ(1 − λ)) ≥ 0. Therefore p ∈ Pλ , as asserted.



2.5 Holomorphically dissipative and accretive mappings To motivate our approach below, we recall that a mapping h : H → H, where H is a Hilbert space with inner product (·, ·), is said to be monotone if for each pair x, y ∈ H, Re (h(x) − h(y), x − y) ≥ 0. This notion plays a crucial role in fixed point theory, optimization and in the theory of nonlinear semigroups generated by monotone mappings. It is not difficult to see that this condition is equivalent to (x + rh(x)) − (y + rh(y)) ≥ x − y,

0 0, are horodisks in D internally tangent to ∂D at ζ. In this situation we use the term ‘the horodisk function’ for d(z, ζ), z ∈ D, ζ ∈ ∂D, in place of ‘the horosphere function’. In addition, we use the following notion. Definition 3.2.2. Let B be the open unit ball in the Hilbert space H endowed with the inner product (·, ·). Let F be a continuous self-mapping of B. A point ζ ∈ ∂B is called a boundary contact point for F if F (ζ) := lim− F (rζ) ∈ ∂B. It is called r→1

a boundary regular contact point if the radial derivative ↑F  (ζ) := lim− r→1

1 − (F (rζ), F (ζ)) 1−r

exists finitely. We use the symbol ↑ F  (ζ) to distinguish the radial derivative (which is a complex number) from the Fr´echet derivative at a point x ∈ B, which is usually denoted by F  (x) (and is a linear operator on H). Note also that in the one-dimensional case, when F ∈ Hol(D) and ζ ∈ ∂D is a boundary regular contact point for F , we have ↑F  (ζ) = F (ζ)ζF  (ζ), where F  (ζ) is the angular derivative of F at ζ (see, for example, [47], [204] and [209]). So, if F ∈ Hol(D) and ζ ∈ ∂D is a boundary fixed point of F , then ↑F  (ζ) = F  (ζ).

3.3 Boundary fixed points and the horosphere function In this section we present and discuss some possible generalizations of the onedimensional results mentioned above to the horosphere function of the open unit ball B in a complex Hilbert space H. The first result we mention is a generalization of the classical Julia–Wolff–Carath´eodory Theorem (Theorem 3.1.3) to ρ-nonexpansive self-mappings of B.

72

Chapter 3. Fixed Points of Holomorphic Mappings

Theorem 3.3.1. Let F : B → B be ρ-nonexpansive. Assume that for some ζ ∈ ∂B, the radial limit lim− F (rζ) = η ∈ ∂B r→1

exists. Then the following conditions are equivalent: (i) there is K < ∞ such that d(F (x), η) ≤ K · d(x, ζ) for all x ∈ B; 1 − F (rζ) =: δ < ∞; (ii) lim inf 1−r r→1− 1 − (F (rζ), η) (iii) the radial derivative ↑F  (ζ) := lim− exists finitely. 1−r r→1 Moreover,

↑F  (ζ) = δ = inf K.

Proof. (i) =⇒ (ii). First we observe that the number K in (i) cannot be zero. Indeed, if it were, then it would follow that F (x) = η ∈ ∂B for all x ∈ B. This would contradict our assumption that F is a self-mapping of B. Setting now x = rζ in (i), we obtain |1 − (F (rζ), η) |2 (1 − r)2 1−r ≤K =K . 2 1 − F (rζ) 1 − r2 1+r

(3.3.1)

Furthermore, using (3.3.1) and some manipulations, we also obtain 1 − F (rζ) 1 − |(F (rζ), η)| ≤ 1−r 1−r 2 |1 − (F (rζ), η)| 1 + |(F (rζ), η)| ≤ 2 · 1−r 1 − |(F (rζ), η)|

0
0, are invariant under F . This point τ ∈ ∂B is called the sink point for F (see [81, 191]). Theorem 3.3.3 is a partial generalization of Theorem 3.1.4. Moreover, the sink point is a boundary regular fixed point of F and ↑F  (τ ) must be less than or equal to 1. However, not all of Theorem 3.1.4 admits a multi-dimensional generalization. Indeed, in contrast with the one-dimensional case, an example of Stachura [213] shows that in the infinite-dimensional case, the iterates of a holomorphic self-mapping of B need not converge to the sink point even if it is the unique boundary regular fixed point of F . However, if F is, for instance, a so-called firmly ρ-nonexpansive mapping which is fixed point free, then the iterates of F converge pointwise to its sink point (see Theorem 30.8 in [81]). The converse of Theorem 3.1.4 is also not true when the dimension is higher than one.   Example 3.3.1. Let H = C2 and B = (z1 , z2 ) ∈ C2 : |z1 |2 + |z2 |2 < 1 . The mapping F : B → B, defined by F (z1 , z2 ) := (z1 , −z2 ), is obviously a holomorphic self-mapping of B which fixes the origin and τ = (1, 0), while ↑F  (τ ) = 1. Finally, we present a sufficient condition for locally uniform convergence of the iterates of a holomorphic self-mapping F of B to an interior or a boundary fixed point.

76

Chapter 3. Fixed Points of Holomorphic Mappings

Although strong and weak convergence of iterates has been studied very intensively (see the survey [146]), little is known about locally uniform convergence of iterates in the infinite-dimensional case. Recall that a sequence {Fn }∞ n=1 of mappings on a bounded domain D is said to be locally uniformly convergent if it converges uniformly on each ball strictly inside D. For a general complex Banach space the Earle–Hamilton Theorem (see Section 3.5 below) implies that if F maps a bounded domain D in a complex Banach space strictly inside itself, that is, inf F (x) − y ≥  > 0,

x∈D y∈∂D

then F has a unique fixed point z ∈ D and the sequence {F n }∞ n=1 converges to z, locally uniformly in D. In the setting of the Hilbert ball it has recently been shown in [210, 64] that a similar conclusion holds even if the image F (B) is not strictly inside B, but instead is contained in a horosphere in B. Theorem 3.3.4. If F is a holomorphic self-mapping of the open unit ball B with F (B) ⊂ E(τ, 1/a) for some a > 0, then its iterates F n converge to a point ζ ∈ B locally uniformly. Moreover, if F is fixed point free, then ζ = τ . In the case where F is fixed point free, Theorem 3.3.4 was proved in [210]. For the general case it was obtained in [64].

3.4 Canonical representation of the fixed point set We start this section by recalling a remarkable result of Rudin [199] (see also [81, Theorem 23.2, page 120]): Theorem 3.4.1. If F : B → B has more than one fixed point in B, then its fixed point set F = FixB F is an affine manifold. In this section we study this assertion in more detail. The result presented here is only meaningful in the multi-dimensional case. Recall that every holomorphic self-mapping of D, which is not the identity, has at most one fixed point in D. This is a consequence of the Schwarz Lemma. This fact is no longer true for holomorphic self-mappings of B. Moreover, a result in [81] (Theorem 25.1 on page 125), when combined with Rudin’s Theorem 3.4.1, implies that if F ∈ Hol(B) has more than one fixed point in B, then its fixed point set F = FixB F is an affine manifold and each point τ ∈ F ∩ ∂B is a sink point for F with ↑F  (τ ) = 1. In the following theorem we show that, in a sense, the converse assertion also holds. To formulate it we need the following notations and observations.

3.4. Canonical representation of the fixed point set

77

For a point τ, τ  = 1, let P and Q be the two orthogonal projections defined by P x = (x, τ )τ and Q = I − P , respectively. If H1 = P H and H2 = QH, then each bounded linear operator A on H can be represented by the matrix operator block   A11 A12 A= , A21 A22 where Aij ∈ L(Hj , Hi ). Theorem 3.4.2. Let F ∈ Hol(B) with F (0) = 0 have a sink point τ ∈ ∂B such that ↑F  (τ ) = 1. If P and Q are the orthogonal projections defined above, then F = FixB F = {x ∈ B : x = λτ + u}, where λ ∈ C and u ∈ Ker(IH2 − QF  (0)Q) ∩ B ⊂ H2 . In other words, F = FixB F = Ker(I − A), where A is a block matrix of the form   IH1 0  A = F (0) = 0 A22 with A22 = QF  (0)Q. Proof. By A we denote the linear operator F  (0), the Fr´echet derivative of F at the origin. It follows from Theorem 3.3.2 that (Aτ, τ ) = A = 1. This implies that τ = Aτ is an eigenvector of A, which is different from the origin. Therefore, again by Rudin’s Theorem, F = FixB F = Ker(I − A) ∩ B ⊃ {x : x = λτ, λ ∈ C}. To prove the last assertion of the theorem we use “partial coordinates”: x1 = (x, τ ) and x2 = Qx. That is,we write x= x1 τ + x2 ∈ H and x2 = |x1 |2 + x2 2 . A11 A12 We claim that if A = , then A11 = IH1 and A12 = A21 = 0. A21 A22 Indeed, since Aτ = τ , it follows that AP x = A (x, τ ) τ = (x, τ ) τ = P z and P Ax = P AP x + P AQx = P x + P AQx. So, we have Ax = P AP x + P AQx + QAP x + QAQx = P x + P Ax2 + QP x + QAQx = x1 τ + (Ax2 , τ ) τ + QAxz2 . Hence A11 = P AP = IH1 and A21 = QAP = 0. In order to see that A12 = P AQ = 0, we write 2

2

Ax = |x1 + (Ax2 , τ )| + QAx2 |

2

= |x1 |2 + 2 Re x1 (Ax2 , τ ) + |(Ax2 , τ )|2 + QAx2 2 2

2

≤ |x1 | + x2 

for all x = (x1 , x2 ) ∈ H.

78

Chapter 3. Fixed Points of Holomorphic Mappings

Setting here x1 = n (Ax2 , τ ) , n = 1, 2, . . . , we get for a fixed x2 = 0, 2

(2n + 1)|(Ax2 , τ )|2 ≤ x2  . Since n is arbitrary, we obtain that (Ax2 , τ ) = 0. In other words, P AQ = 0, as claimed. So, P A = AP = P . 

3.5 Around the Earle–Hamilton fixed point theorem Higher-dimensional generalizations of the Poincar´e hyperbolic metric ρ on the open unit disk D ⊂ C are the well-known Carath´eodory and Kobayashi pseudometrics (see, for example, [130, 131, 49, 81] and [191]). Let D be an arbitrary domain in a complex Banach space X and let x, y ∈ D be any pair of points. Definition 3.5.1. The function CD defined by CD (x, y) := sup {ρ(f (x), f (y)) : f ∈ Hol(D, D)} is called the Carath´eodory pseudometric on D. Whereas the Carath´eodory pseudometric is defined by means of mappings from a domain D ⊂ X into the open unit disk D, the Kobayashi pseudometric deals with mappings from D into D. In order to define it, we first recall that the function δD defined by δD (x, y) := inf {ρ(z, w) : ∃f ∈ Hol(D, D) with x = f (z), y = f (w)} is called the Lempert function of D. This function does not always satisfy the triangle inequality; hence it cannot serve as a pseudometric. Definition 3.5.2. The Kobayashi pseudometric KD is defined by ⎧ ⎫ n ⎨ ⎬ KD (x, y) := inf δD (xj , xj+1 ) , ⎩ ⎭ j=0

where the infimum is taken over all n ∈ N and all chains (x0 , . . . , xn ) ⊂ D such that x0 = x and xn = y. More generally, Harris [100] introduced the so-called Schwarz–Pick systems of pseudometrics which contain the Carath´eodory and Kobayashi pseudometrics as the smallest and largest ones, respectively. Definition 3.5.3. A system which assigns a pseudometric to each domain in a normed linear space is called a Schwarz–Pick system if the following conditions hold:

3.5. Around the Earle–Hamilton fixed point theorem

79

(i) the pseudometric assigned to the open unit disc D is the Poincar´e hyperbolic metric; (ii) if ρ1 and ρ2 are the pseudometrics assigned to domains D1 and D2 , respectively, and F ∈ Hol(D1 , D2 ), then ρ2 (F (x), F (y)) ≤ ρ1 (x, y) for all x, y ∈ D1 . In particular, if D1 = D2 , each F ∈ Hol(D1 ) is nonexpansive with respect to a Schwarz–Pick system. In addition, the Carath´eodory and Kobayashi pseudometrics form the smallest and the largest Schwarz–Pick systems, respectively. Actually, it was shown by Dineen, Timoney and Vigu´e [51] (see also an earlier finite-dimensional result of Lempert [149]) that if D is a convex domain in a complex Banach space X, then all Schwarz–Pick systems of pseudometrics on D coincide. Moreover, if D is bounded, then this unique pseudometric is in fact a metric witch is called the hyperbolic metric on D. The hyperbolic metric ρ on a bounded convex domain D can be estimated by the norm and the size of D as follows (see [100, pages 353 and 355], [81, pages 89–90], [146, page 442] and [191, page 99]). Theorem 3.5.1. Let D be a bounded convex domain endowed with the hyperbolic metric ρ. For all x, y ∈ D, tanh−1 (x − y/diam(D)) ≤ ρ(x, y) and if x − y < dist(x, ∂D), then ρ(x, y) ≤ tanh−1 (x − y/ dist(x, ∂D)) . We now quote the celebrated Earle–Hamilton Theorem [56] (see also [81, 146, 209] and [191]). Theorem 3.5.2. Let D be a bounded domain in a complex Banach space X. If F ∈ Hol(D) maps D strictly inside itself, that is, inf

x∈D, y∈∂D

F (x) − y ≥ ε > 0,

then there is a metric ρ on D such that F is a strict contraction with respect to this metric: ρ(F (x), F (y)) ≤ kρ(x, y) for some k ∈ (0, 1). Hence F has a unique fixed point z ∈ D. Moreover, the sequence {F n }∞ n=1 converges to z locally uniformly in D. In this connection we also recall the following characterization of those subsets which lie strictly inside a bounded convex domain [100, Proposition 23]. See also [146, page 442] and [191, page 101]. Theorem 3.5.3. Let D be a bounded convex domain in a complex Banach space. A nonempty subset C ⊂ D lies strictly inside D if and only if it is ρ-bounded.

80

Chapter 3. Fixed Points of Holomorphic Mappings

For Cn the existence and uniqueness of the fixed point under our assumptions was earlier established by Herv´e (see [108] and [109]). A slightly more general formulation has been suggested by Harris [102]. Theorem 3.5.4. Let D be a nonempty domain in a complex Banach space X and let F ∈ Hol(D) be bounded. If F (D) lies strictly inside D, then F has a unique fixed point in D. Proof. We construct a metric ρ, called the Carath´eodory–Reiffen–Finsler (CRF) pseudometric, with respect to which F is a strict contraction. Define α(x, v) := sup{|g  (x)v| : g ∈ Hol(D, D)} for x ∈ D and v ∈ X, and set

1 L(γ) =

α(γ(t), γ  (t))dt

0

for γ in the set Γ of all curves in D with piecewise continuous derivative. Clearly α specifies a seminorm at each point of D. We view L(γ) as the length of the curve γ measured with respect to α. Define ρ(x, y) := inf{L(γ) : γ ∈ Γ, γ(0) = x, γ(1) = y} for x, y ∈ D. It is easy to verify that ρ is a pseudometric on D. Let x ∈ D and v ∈ X. By the chain rule, (g ◦ F ) (x)v = g  (F (x))F  (x)v for any g ∈ Hol(D, D). Hence, α(F (x), F  (x)v) ≤ α(x, v).

(3.5.1)

By integrating this and applying the chain rule, we obtain L(F ◦ γ) ≤ L(γ) for all γ ∈ Γ and thus the Schwarz–Pick inequality ρ(F (x), F (y)) ≤ ρ(x, y) holds for all x, y ∈ D. Now by hypothesis there exists an  > 0 such that B (F (x)) ⊂ D whenever x ∈ D. We may assume that D is bounded by replacing D by the subset 0 {B (F (x)) : x ∈ D} . Fix t with 0 < t < /δ, where δ denotes the diameter of F (D). Given x ∈ D, define F1 (y) := F (y) + t[F (y) − F (x)],

3.5. Around the Earle–Hamilton fixed point theorem

81

and note that F1 ∈ Hol(D, D). Given x ∈ D and v ∈ X, it follows from F1  (x)v = (1 + t)F  (x)v and (3.5.1), with F replaced by F1 , that α(F (x), F  (x)v) ≤

1 α(x, v). 1+t

Integrating this as before, we obtain ρ(F (x), F (y)) ≤

1 ρ(x, y) 1+t

for all x, y ∈ D. Now pick a point x0 ∈ D and let {xn } be the sequence of iterates given by xn = F n (x0 ). Then {xn } is a ρ-Cauchy sequence by the proof of the Banach contraction mapping theorem. Since X is complete in the norm metric, it suffices to show that there exists a constant m > 0 such that ρ(x, y) ≥ m x − y

(3.5.2)

for all x, y ∈ D. Since D is bounded, we may take m = 1/d, where d is the diameter of D. Given x ∈ D and v ∈ X, define g(y) := m (y − x), where ∈ X ∗ with   = 1. Then g ∈ Hol(D, D) is holomorphic and g  (x)v = m (v). Hence α(x, v) ≥ m v by the Hahn–Banach theorem. Integrating as before, we obtain (3.5.2).  The following theorem recently proved in [194] complements the Earle– Hamilton theorem by showing that most bounded holomorphic self-mappings of a strongly star-shaped domain in a complex Banach space map it strictly inside itself. Theorem 3.5.5. Let D be a strongly star-shaped domain in a Banach space X. The set of those F ∈ Hol(D) which map D strictly inside itself is an open and everywhere dense subset of the set of all bounded holomorphic self-mappings of D. Even in a general Banach space, the Earle–Hamilton theorem still applies in cases where the holomorphic function does not necessarily map its domain strictly inside itself. In fact, the following fixed-point theorem is a consequence of two applications of the Earle–Hamilton theorem. Theorem 3.5.6 (see [122, 186]). Let D be a nonempty bounded convex domain in a Banach space and let F ∈ Hol(D) have a uniformly continuous extension to D. If there exists an  > 0 such that F (x) − x ≥  whenever x ∈ ∂D, then F has a unique fixed point in D.

82

Chapter 3. Fixed Points of Holomorphic Mappings

Proof. Given 0 < t < 1 and x ∈ D, define ft (y) := (1 − t)x + tF (y) and let δ > 0 be such that Bδ (x) ⊂ D. Since D is convex, ft ∈ Hol(D). To show that ft (D) lies strictly inside D, take  = (1 − t)δ. Let y ∈ D and let w ∈ B (ft (y)). Then w − tF (y) z= ∈D 1−t since z ∈ Bδ (x), so w = (1 − t)z + tF (y) ∈ D. Hence B (ft (y)) ⊂ D for all y ∈ D. By Theorem 3.5.4, ft has a unique fixed point gt (x) in D. Since the CRFmetric is continuous, the proof of the contraction mapping theorem shows that the iterates of ft at a chosen point y0 ∈ D are holomorphic and locally uniformly Cauchy in x. Hence the limit function gt : D → D is holomorphic by [111, Theorem 3.18.1]. Now an x ∈ D is a fixed point for gt if and only if x is a fixed point for F . Thus, by Theorem 3.5.4, it suffices to show that gt (D) lies strictly inside D for some t > 0. Since F has a uniformly continuous extension to D, there exist  > 0 and δ > 0 such that F (x) − x ≥  whenever x ∈ D and d(x, ∂D) = inf {x − y : y ∈ ∂D} < δ. Since D is bounded, there is an M with x ≤ M for all x ∈ D. If x ∈ D, F (gt (x)) − gt (x) = (1 − t)[F (gt (x)) − x], so F (gt (x)) − gt (x) ≤ 2(1 − t)M. Choose t close enough to 1 so that 2(1 − t)M < . If d(gt (x), ∂D) < δ for some x ∈ D, then  ≤ F (gt (x)) − gt (x) , a contradiction. Thus, Bδ (gt (x)) ⊂ D for all x ∈ D, as required.



For example, the hypothesis that F (x) − x ≥  for all x ∈ ∂D is satisfied when D contains the origin and sup x∈∂D

F (x) < 1. x

For bounded convex domains another fixed point result which also is a generalization of the Earle–Hamilton Theorem is based on the notion of the numerical range of holomorphic mappings. Theorem 3.5.7 (cf. Theorem 3.7.2). Let D be a bounded convex domain in X which contains the origin and let F ∈ Hol(D, X) have the following property: the real part of the numerical range of F is less than 1. Then F has a unique fixed point in D.

3.6. Inexact orbits of holomorphic mappings

83

3.6 Inexact orbits of holomorphic mappings In this section we are going to show that if holomorphic self-mappings of the Hilbert ball (and certain self-mappings of other domains) have an inexact orbit which lies strictly inside the ball (respectively, the domain), then they have a fixed point. We begin by recalling the following known result (see [83, Theorem 11] and [81, Theorem 23.1]). Theorem 3.6.1. Let B be the open unit ball in a complex Hilbert space H, and let F ∈ Hol(B). If there is a point x0 ∈ B such that the sequence {F n (x0 )} of iterates lies strictly inside B, then F has a fixed point in B. In view of the recent interest in the influence of computational errors on the performance of iterative methods (see, for instance, [181], [193] and references therein), we quote the following result concerning inexact orbits of holomorphic self-mappings. Theorem 3.6.2. Let B be the open unit ball in a complex Hilbert space and let F ∈ Hol(B). Assume that a sequence {xn }∞ n=1 lies strictly inside B and satisfies lim xn+1 − F (xn ) = 0.

n→∞

Then there exists a point y0 ∈ B such that the sequence {F n (y0 )}∞ n=1 lies strictly inside B and the mapping F has a fixed point in B. It is clear that Theorem 3.6.2 follows from Theorem 3.6.1 and the following general result (see [195]). Theorem 3.6.3. Let D be a bounded convex domain in a complex Banach space X and let F ∈ Hol(D). Assume that a sequence {xn }∞ n=1 lies strictly inside D and satisfies lim xn+1 − F (xn ) = 0.

n→∞

Then there exists a point y0 ∈ D such that the sequence {F n (y0 )}∞ n=1 lies strictly inside D. Proof. There exist numbers M0 > sup{z1 − z2  : z1 , z2 ∈ D} and r∗ ∈ (0, 1) such that {x : xi − x < 2r∗ } ⊂ D,

i = 1, 2, . . . .

(3.6.1)

There also exists an integer i1 ≥ 1 such that for all integers i ≥ i1 , xi+1 − F (xi ) ≤ r∗ /4.

(3.6.2)

84

Chapter 3. Fixed Points of Holomorphic Mappings

By (3.6.1), (3.6.2) and Theorem 3.5.1, for every integer i ≥ i1 , there exists a real   number ξ ∈ 0, r∗−1 xi+1 − F (xi ) ⊂ [0, 1/4] such that

 ρ(xi+1 , F (xi )) ≤ tanh−1 xi+1 − F (xi )r∗−1 = (tanh−1 ) (ξ)xi+1 − F (xi )r∗−1 = (1 − ξ 2 )−1 r∗−1 xi+1 − F (xi ) ≤ 2r∗−1 xi+1 − F (xi ). Hence lim ρ(xn+1 , F (xn )) = 0.

(3.6.3)

n→∞

Now fix a point x0 ∈ D, let R := lim sup ρ(x0 , xn ), and consider the set n→∞

C := {y ∈ D : lim sup ρ(y, xn ) ≤ R}. n→∞

It is clear that this set is a nonempty ρ-bounded subset of D. If z ∈ C, then by (3.6.3), lim sup ρ(F (z), xn ) = lim sup ρ(F (z), xn+1 ) = lim sup ρ(F (z), F (xn )) n→∞

n→∞

n→∞

≤ lim sup ρ(z, xn ) ≤ R. n→∞

Hence C is invariant under F and therefore for each y0 ∈ C, the sequence {F n (y0 )}∞ n=1 lies strictly inside D by Theorem 3.5.3, as asserted. This completes the proof.  Now we consider the Banach space X = Hd , d ≥ 2, a finite power of a Hilbert space H. When X is equipped with the maximum norm, then its open unit ball is the finite power Bd of the Hilbert ball B. It is known [146, Theorem 7.3] that an analog of Theorem 3.6.1 holds for such powers. Combining this fact with Theorem 3.6.3, we obtain the following extension of Theorem 3.6.2. Theorem 3.6.4. Let Bd be a finite power of the Hilbert ball B and let F ∈ Hol(Bd ). d Assume that a sequence {xn }∞ n=1 lies strictly inside B and satisfies lim xn+1 − F (xn ) = 0.

n→∞

Then there exists a point y0 ∈ Bd such that the sequence {F n (y0 )}∞ n=1 lies strictly inside Bd and the mapping F has a fixed point in Bd . Remark 3.6.1. Note that since the proofs of Theorems 3.6.1, 3.6.3 and 3.6.4 have a metric character, these theorems continue to hold for all those self-mappings which are ρ-nonexpansive (even if they are not holomorphic). However, Example 11.5 in [146] shows that analogs of Theorems 3.6.1, 3.6.3 and 3.6.4 do not hold for the open unit ball B∞ of ∞ (H) even if H = C. This example exhibits a fixed point free holomorphic self-mapping of B∞ all the orbits of which are strictly inside B∞ .

3.7. The Bohl–Poincar´e–Krasnoselskii theorem

85

On the other hand, since an analog of Theorem 3.6.1 does hold for nonexpansive self-mappings of certain subsets of Banach spaces (cf., for example, [184, Theorem 1] and [81, Theorem 5.2]), the following analog of Theorems 3.6.3 and 3.6.4 can be established by using the idea of the proof of Theorem 3.6.3. Theorem 3.6.5. Let C be a boundedly weakly compact and convex subset of a (real or complex) Banach space X. Suppose that each weakly compact and convex subset of C has the fixed point property for nonexpansive mappings, and let the mapping T : C → C be nonexpansive. Assume that a bounded sequence {xn }∞ n=1 satisfies lim xn+1 − T (xn ) = 0.

n→∞

Then there exists a point y0 ∈ C such that the sequence {T n (y0 )}∞ n=1 is bounded and the mapping T has a fixed point in C.

3.7 The Bohl–Poincar´e–Krasnoselskii theorem For bounded convex domains a generalization of the Earle–Hamilton Theorem can be obtained by using one-sided estimates in terms of the numerical range. As we will see below this plays a crucial role in nonlinear semigroup theory for holomorphic mappings. One-sided estimates have been used many times to derive existence theorems for nonlinear operator equations and minimization problems. Recall, for example, the following theorem of Krasnoselskii ([136] and [138]). Theorem 3.7.1. Let B be the open unit ball of a real Hilbert space H with inner product (·, ·). Assume

that F : B → H is a continuous mapping on the closed unit ball B such that F B is compact. The following assertions hold. (i) If F satisfies the condition (F (x), x) ≤ 1, x ∈ ∂B, then F has a fixed point in B; (ii) If F satisfies the condition (F (x), x) < 1, x ∈ ∂B, then F has a unique fixed point in B. The first assertion can be proved by noting that (i) implies the Leray– Schauder boundary condition. The second one is a direct consequence of the Bohl– Poincar´e–Krasnoselskii null point theorem applied to the mapping I − F : If two completely continuous vector fields on B do not vanish on B and their directions are not antipodal, then they are homotopic. Quoting Krasnoselskii and Zabreiko [138] and Shinbrot [207], we note that despite its simplicity and perhaps because of it, Theorem 3.7.1 has many applications to the solvability of nonlinear equations. An extension of condition (i) to topological vector spaces, with applications, can be found in a paper by Fan [74].

86

Chapter 3. Fixed Points of Holomorphic Mappings

One-sided estimates of this type have systematically been used in many fields. We mention, for example, Galerkin’s approximation methods [221], the theory of equations with potential operators [226], monotone operator theory [32], and nonlinear integral and partial differential equations [34, 37, 12]. One of the main points in Theorem 3.7.1 is, of course, the compactness of the mapping F which allows us to use the methods of the theory of rotation of vector fields or degree theory (see [138]). Since we are interested in the class of holomorphic mappings, we note that in infinite-dimensional spaces this class is not contained in the class of compact mappings. Moreover, the intersection of these classes is quite small even for integral equations; see [221, 227] and [140]. Despite this lack of compactness, the fixed point theory for holomorphic mappings in Hilbert spaces and in some classes of Banach spaces is well developed. In the present section we will consider, inter alia, a complex Hilbert space H and mappings F : B → H which are holomorphic in B and uniformly continuous on B. Replacing condition (i) by Re (F (x) , x) ≤ 1,

x ∈ ∂B,

(3.7.1)

we will show below that this condition also ensures the existence of a fixed point of F in B. In addition, in the theory of holomorphic mappings, even in the finite-dimensional case, it is important to describe the structure of the analytic set FixB F and in particular to recognize when it is irreducible. As we will see in the sequel, condition (3.7.1) implies that FixB F has a very simple structure, namely, it is an affine submanifold of B. This is a generalization of Rudin’s theorem (see [200]). It turns out that such assertions are related to entirely different problems of nonlinear analysis, such as global solvability of autonomous Cauchy problems in Hilbert or Banach spaces. More precisely, we have in mind holomorphic vector fields which are dissipative in B. As we will show below, such mappings generate continuous flows on a domain (see Chapter 4). In turn, it is clear that the property of holomorphic mappings f in B, which are uniformly continuous on B, of being dissipative, is equivalent to the representation f = F − I where F satisfies condition (3.7.1). Finally, turning to the finite-dimensional case, we note that such assertions may be considered comparison-type theorems resembling Rouch´e’s theorem for two vector fields. We remark in passing that Rouch´e’s theorem is also a consequence of the Bohl–Poincar´e–Krasnoselskii theorem. Therefore it is of interest to establish theorems involving more general onesided estimates, which will contain the Bohl–Poincar´e–Krasnoselskii theorem, and which may be applied to a wider class of domains. This will be done in the last two theorems of the section. We now need the following notion.

3.7. The Bohl–Poincar´e–Krasnoselskii theorem

87

Definition 3.7.1. Let B be the open unit ball in a complex Banach space X. A mapping F : B → X is called a pseudo-contraction on B if    N (F ) = lim sup sup Re λ : λ ∈ VB (Fs ) ≤ 1, s→1−

where Fs (x) = F (sx), and VB (Fs ) is the numerical range of Fs . It is called a strict pseudo-contraction if N (F ) < 1. It is clear that each holomorphic self-mapping of B is a pseudo-contraction. Historically, the notion of a pseudo-contraction is defined by using the socalled non-linear resolvent. We will see in Chapter 5 that these definitions are equivalent. We continue with the following extension of the Earle–Hamilton Theorem. Theorem 3.7.2. Let F : B → X be a holomorphic mapping on B which is a strict pseudo-contraction on B. Then F has a unique fixed point in B. Proof. Since Fs is uniformly continuous on B it suffices to prove our assertion for this subclass of pseudo-contractive mappings. So, let F be uniformly continuous on B and pseudo-contractive on B. Then by Theorem 2.5.1 and our hypothesis, N (F − I) = lim

t→0+

I + t(F − I) − 1 < 0. t

Thus, for ε sufficiently small and t ∈ (0, ε), we have I + t(F − I) < 1 − tε. This means that (I + t(F − I)) (B) lies inside the ball (1 − tε)B. Hence, by the Earle–Hamilton Theorem, the mapping I + t (F − I) has a unique fixed point in B, which is obviously the unique fixed point of F.  The following result, which partially solves the so-called coefficient problem, is a consequence of Theorem 3.5.7. Corollary 3.7.1. Let D = B be the open unit ball in a complex Banach space X and let F ∈ Hol(B) be a holomorphic self-mapping of B. Assume that LF :=

sup u=1, u∗ ∈J(u)

Re F  (0)u, u∗  < 1 − 4F (0).

(3.7.2)

Then F has a unique fixed point x0 in B with x0  ≤ r0 , where r0 is the fixed point in the interval (0, 1) of the scalar mapping φ(r) = F (0)(1 + r)2 + rLF . Remark 3.7.1. Inequality (3.7.2) reminds us of the well-known one-dimensional Schwarz inequality |F  (0)| ≤ 1 − |F (0)|2 . However, even in the one-dimensional case the last inequality does not imply the existence of an interior fixed point of F.

88

Chapter 3. Fixed Points of Holomorphic Mappings

In the case of a Banach space, it is natural to replace condition (3.7.1) by the condition N (F ) ≤ 1. However, the following simple example shows that there is no complex analogue of assertion (i) in Theorem 3.7.1 without additional requirements. Example 3.7.1. Let X = c0 be the space of all complex sequences x = (x1 , x2 , . . ., xn , . . .) such that xn → 0 as n → ∞. Let F (x) = (a, x2 , x3 , . . .), where a ∈ C and |a| ≤ 1. This mapping is a pseudo-contraction (even a contraction) on the open unit ball of c0 , but has no fixed point in c0 whenever a = 0. More can be said when B is the open unit ball in a complex Hilbert space H. Namely, the following assertion is a generalization of Theorem 3.2.1. Theorem 3.7.3. Let B be the open unit ball in a complex Hilbert space H, and let F ∈ Hol(B, H) be a pseudo-contraction on B. If F has a continuous extension to B, then F has a fixed point in B. Proof. For a fixed t ∈ (0, 1) and y ∈ B, we define the mapping Ft : B → H by Ft (x) = (1 − t) y + tF (x) , that is, for all t ∈ (0, 1) and y ∈ B, the following identity holds: G(t, y) = (1 − t) y + tF (G (t, y)) . Clearly, Ft is a strict pseudo-contraction on B. So by Theorem 3.7.2, Ft has a unique fixed point x = G(t, y) ∈ B. In addition, since F has unit radius of boundedness, it can be shown by using the Implicit Function Theorem that the mapping G(t, ·) is a holomorphic self-mapping of B. Assume now that F has no fixed point in B. We claim that for each t ∈ (0, 1), the mapping G(t, ·) has no fixed point in B as well. Indeed, otherwise if y0 = G(t, y0 ) ∈ B for some t ∈ (0, 1), then by the uniqueness of the fixed point of Ft we get y0 = Ft (y0 ) = (1 − t)y + tF (y0 ) . So, y0 = F (y0 ) which is a contradiction. On the other hand, it follows again from the Earle–Hamilton Theorem  that for any fixed x ∈ B, t ∈ (0, 1) and each n ≥ 1, the mapping n1 x + 1 − n1 G(t, ·) must have a fixed point zn ∈ B. However, it follows from Theorem 27.3 in [81] and from our assumption that F , hence G(t, ·), have no fixed point in B, that the sequence {zn }n≥1 implicitly defined by the equation:   1 1 zn = x + 1 − G (t, zn ) n n must strongly converge to a point z ∈ ∂B. In turn, the sequence yn := G (t, zn ) = n 1 n−1 zn + n−1 x, n ≥ 2, converges to the point z ∈ ∂B as well. At the same time, for the points yn we have by definition the identity yn := G (t, zn ) = (1 − t) zn + tF (G (t, zn )) = (1 − t) zn + tF (yn ) ,

3.7. The Bohl–Poincar´e–Krasnoselskii theorem

89

which immediately implies that F (yn ) also converges to z and so F (z) = z by continuity. The proof is complete.  Similarly, one can prove the following generalization of Rudin’s Fixed Point Theorem (see [200]). Theorem 3.7.4. Let B be the open unit ball in a complex Hilbert space H, and let F ∈ Hol(B, H) be a pseudo-contraction on B. Then the fixed point set of F in B is an affine manifold. In particular, if F (0) = 0 and A = F  (0), then FixB F = FixB A = Ker(I − A)|B . The following generalization of the Cartan Uniqueness Theorem is an immediate consequence of this theorem. Corollary 3.7.2. Let B be the open unit ball in a complex Hilbert space H and let F ∈ Hol(B, H) be a pseudo-contraction on B. If F (0) = 0 and F  (0) = I, then F = I identically on B. Finally, we deduce a fixed point theorem in terms of the numerical range for the finite-dimensional case. Let X = Cn with the Euclidean norm and the inner product (·, ·). Proposition 3.7.1 (see Shih [206]). Let D ⊂ Cn be a bounded domain containing the origin. If F ∈ Hol(D, Cn ), then the inequality (F (x), x) < 1 for x ∈ D implies that F has a unique fixed point in D. Actually, if D is the open unit ball in Cn , one can prove the following assertion; see [9]. Proposition 3.7.2. Let B be the open unit ball in Cn , and let F ∈ Hol(B, Cn ) satisfy the following condition: The numerical range of the mapping f = I − F does not contain zero and does not separate zero and infinity. Then F has a unique fixed point in B. Example 3.7.2. Let G map the unit disc D ⊂ C conformally onto a domain Q ⊂ C such that 0 ∈ / Q and C \ Q contains a curve {z(t) = 1 − 1/λ(t)}, where λ(t), 0 ≤ t ≤ 1, is a curve joining 0 and 1. Then the mapping F (z1 , z2 ) = (z1 G(z1 ), z2 G(z1 )) satisfies the conditions of Proposition 3.7.2 on B, the open unit ball in C2 , and hence has a unique fixed point in B. At the same time, (F, z) = G(z1 ) on ∂B, and if Q intersects R− = {z ∈ C : Re z < 0, Im z = 0}, then Re(F, z) is not always less than 1 on ∂B. Thus we cannot use the Bohl–Poincar´e–Krasnoselskii Theorem as well as Theorems 3.7.2 and 3.7.3 in this case.

90

Chapter 3. Fixed Points of Holomorphic Mappings

3.8 Fixed points of pseudo-contractive holomorphic mappings Observe that it follows from the Earle–Hamilton Theorem 3.5.2 that each holomorphic mapping on B is pseudo-contractive if and only if for each t ∈ [0, 1) and y ∈ B, the equation x = tF (x) + (1 − t)y (3.8.1) has a unique solution x = Φt (y) ∈ B which is holomorphic in y ∈ B. This solution Φt : B → B is nonexpansive with respect to the hyperbolic (Kobayashi) metric ρ on B: ρ(Φt (y1 ), Φt (y2 )) ≤ qρ(y1 , y2 ) (3.8.2) for some q ∈ [0, 1] and all y1 , y2 in B (cf. [36, 129, 170, 171]). Moreover, a mapping F ∈ Hol(B, X) is strictly pseudo-contractive on B if condition (3.8.2) holds for all y1 , y2 ∈ B with some q ∈ [0, 1). Even in the one-dimensional case, the class of pseudo-contractive mappings is much wider than the class of holomorphic self-mappings (see Example 3.8.1 below). It even contains mappings which are unbounded on B. Theorem 3.8.1. Let B be the open unit ball in X, and let F ∈ Hol(B, X). Assume that (3.8.3) a = sup Re F  (0)x, x∗  < 1 x∈∂B x∗ ∈J(x)

and for each x ∈ B and x∗ ∈ J(x), Re F  (x)x, x∗  + 2x2 F (0) ≤ (1 + k)x2 , where k ≤ κ := (1 − a)

2 log 2 − 1 . 2(1 − log 2)

(3.8.4)

(3.8.5)

Then F is pseudo-contractive. Moreover, if the inequality in (3.8.5) is sharp, then F is strictly pseudo-contractive and hence has a unique fixed point. Remark 3.8.1. If F (0) = 0, then the origin is the unique fixed point even when k = κ = (1 − a)

2 log 2 − 1 . 2(1 − log 2)

Indeed, since a < 1, it follows that the linear operator A = I − F  (0) is invertible. Therefore the origin is an isolated fixed point of F . On the other hand, the fixed point set of F in D is a holomorphic retract, hence a connected analytic submanifold of B. Thus it must contain the unique fixed point.

3.8. Fixed points of pseudo-contractive holomorphic mappings

91

In the one-dimensional case, where X = C and B = D is the open unit disk in C, the parameter a is just Re F  (0) and condition (3.8.4) can be rewritten as Re F  (z) < 1 + k − 2|F (0)|,

(3.8.6)

where k satisfies (3.8.5). Example 3.8.1. Consider the function F ∈ Hol(D, C) defined by F (z) = 3(z − log(1 + z)). This function as well as its real part are unbounded on D. At the same time, F (0) = 0 and     1 3 1−z F  (z) = 3 1 − = 1− . 1+z 2 1+z Therefore we have a = Re F  (0) = 0 and Re F  (z) ≤

3 2 log 2 − 1 < . 2 2(1 − log 2)

Thus, Theorem 3.8.1 implies that F is strictly pseudo-contractive. Hence it has a pseudo-attractive fixed point τ ∈ D in the sense that for each t ∈ (0, 1), the limit lim Φn (t, z) = τ, n→∞

where Φ (= Φ(t, z)) is the solution of equation (3.8.1). Remark 3.8.2. We will see below in Theorem 4.6.2 that actually the condition Re F  (0)x, x∗  ≤ 1 (cf. (3.8.3)), which means that the numerical range of the operator A = F  (0) lies in the half-plane Π = {z ∈ C : Re z ≤ 1}, is necessary for F ∈ Hol(B, X) to be pseudo-contractive. Now we prove Theorem 3.8.1. Proof. Fix u ∈ ∂B and u∗ ∈ J(u) ⊂ X ∗ , and consider the function g ∈ Hol(D, C) defined by g(z) (= gu (z)) = z − F (zu), u∗ + F (0), u∗  − F (0), u∗  z 2 .

(3.8.7)

Since g(0) = 0, one can write g(z) = zp(z) with p ∈ Hol(D, C) and Re p(0) = Re g  (0) = 1 − Re F  (0)u, u∗  ≥ 1 − a > 0.

(3.8.8)

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Chapter 3. Fixed Points of Holomorphic Mappings

Also, setting x = zu, x = |z| = 0, we have Re (p(z) + zp (z)) = Re g  (z)   = 1 − Re F  (zu)u, u∗ − 2zF (0), u∗  ≥ 1 − Re F  (x)x, x∗ 

1 − 2F (0) x2

 1

Re F  (x)x, x∗  + 2x2 F (0) 2 x 1 − 2 log 2 . ≥ 1 − (1 + k) = −k ≥ (1 − a) 2(1 − log 2)

=1−

Setting now β = −k in Theorem 2.4.3, we have Re p(z) ≥ ε(β) = −2k(1 − log 2) + Re p(0)(2 log 2 − 1) ≥ 0 whenever −

(3.8.9)

Re p(0) · (2 log 2 − 1) ≤ β ≤ Re p(0), 2(1 − log 2)

or, which is one and the same, − Re p(0) ≤ k ≤ κ. In turn, it follows from (3.8.8) and (3.8.9) that for all z ∈ D, Re g(z)z = |z|2 Re p(z) ≥ |z|2 ε(−k) ≥ 0. On the other hand, by (3.8.7), we realize that Re g(z)z = |z|2 − Re F (zu), (zu)∗ + Re F (0), u∗  z − Re F (0), u∗ z|z|2 = x2 − Re F (x), x∗  + Re F (0), x∗  · (1 − x2 ). This implies that Re F (x), x∗  ≤ x2 + (1 − x2 )F (0)x − ε(−k)x2 = x2 (1 − ε(−k)) + (1 − x2 )F (0)x.

(3.8.10)

Now fix any y ∈ B and t ∈ [0, 1), and consider the equation x = tF (x) + (1 − t)y. Define the mapping Ψ ∈ Hol(B × B, X) by Ψ(x, y) = x − (tF (x) + (1 − t)y) and set x = r ∈ [0, 1). We have by (3.8.10) that Re Ψ(x, y), x∗  ≥ r2 − (t Re F (x), x∗  + (1 − t)ry) ≥ r2 − (tr2 ν + (1 − r2 )F (0)r + (1 − t)ry) =: rϕ(r),

(3.8.11)

3.8. Fixed points of pseudo-contractive holomorphic mappings

93

where ν := 1 − ε(−k) < 1 and ϕ(r) = r − trν − (1 − r2 )F (0) − (1 − t)y = r(1 − tν) − (1 − r2 )F (0) − (1 − t)y. Since ϕ(1) = 1 − tν − (1 − t)y = (1 − t)(1 − y) + tε(−k) > 0,

(3.8.12)

we obtain that for every fixed y ∈ B and t ∈ [0, 1), there is r0 ∈ [0, 1) and δ > 0 such that inf Re Ψ(x, y), x∗  > δ. (3.8.13) x=r0

Now it follows from a version of the Implicit Function Theorem given in [186] (see also [191]) that equation Ψ(x, y) = 0, hence (3.8.11), has a unique solution x = Φt (y) with x < r0 (= r0 (y, t)) < 1 which holomorphically depends on y ∈ B. Thus F is pseudo-contractive on B. Finally, if k < κ, hence ε(−k) > 0, it follows from (3.8.12) that inequality (3.8.13) holds for all y ∈ B, hence r0 can be chosen independently of y and t ∈ (0, 1]. Since Φt (y) < r0 < 1, y ∈ B, it follows from the Earle–Hamilton Theorem (Theorem 3.5.2) that ρ(Φt (y1 ), Φt (y2 ) ≤ qρ(y1 , y2 ),

y1 , y2 ∈ B,

(3.8.14)

for some 0 ≤ q < 1, that is, F is strictly pseudo-contractive. The theorem is proved.  As we have already mentioned, if q = 1, that is, F is pseudo-contractive, but not strictly pseudo-contractive, the existence of a fixed point does not necessarily follow even when F is a self-mapping of B having a continuous extension to B. In addition, even when the fixed point set of F in B is not empty, it is not clear whether the net {Φt (y)}t∈[0,1) is pointwise (strongly) convergent to a fixed point of F when t tends to 1− (compare with equation (3.8.1)). For a self-mapping of the Hilbert ball the answer is affirmative (see [81] and references therein). We show that this fact holds in general as well. To formulate this theorem we need the following property of the solution Φt ∈ Hol(B). Set t = 12 and denote for simplicity Φ := Φ 21 that is, Φ ∈ Hol(B) satisfies the equation 1 Φ = (F ◦ Φ + I). (3.8.15) 2 Since Φ is a self-mapping of B, for each s ∈ [0, 1), the equation Gs = sΦ ◦ Gs + (1 − s)I has a solution Gs ∈ Hol(B).

(3.8.16)

94

Chapter 3. Fixed Points of Holomorphic Mappings

Lemma 3.8.1. For all t ∈

1

2,1

 , the following identity holds Φt = Φ ◦ Gs ,

where s = 2 −

1 t

∈ [0, 1).

Proof. Since for s = 0, the assertion is trivial, we proceed with s ∈ (0, 1). Rewrite equations (3.8.15) and (3.8.16) in the form

and

I − Φ = (I − F ) ◦ Φ

(3.8.17)

! s I+ (I − Φ) ◦ Gs = I. 1−s

(3.8.18)

Substituting (3.8.17) into (3.8.18), we get the equality Gs +

s (I − F ) ◦ Φ ◦ Gs = I. 1−s

Denote Φ ◦ Gs = Ψs ∈ Hol(B) and write the last equality as Gs +

s (I − F ) ◦ Ψs = I. 1−s

(3.8.19)

On the other hand, by (3.8.16), we have Gs = sΨs + (1 − s)I. Comparing this relation with (3.8.19), we get sΨs +

s (I − F ) ◦ Ψs = sI, 1−s

or, equivalently, (1 − s)Ψs + (I − F ) ◦ Ψs = (1 − s)I, 1 1−s Ψs = F ◦ Ψs + I 2−s 2−s and 1



Ψs = tF ◦ Ψs + (1 − t)I,

(3.8.20)

1 where t = 2−s ∈ 2 , 1 whenever s ∈ [0, 1). Finally, since (3.8.20) has a unique solution which by definition is Φt , we have Φt = Ψs := Φ ◦ Gs and we are done. 

Theorem 3.8.2. Let B be the open unit ball in a complex Hilbert space H. Suppose that F ∈ Hol(B, H) is pseudo-contractive, that is, the equation Φt = tF ◦ Φt + (1 − t)I,

t ∈ [0, 1),

has a holomorphic solution Φt ∈ Hol(B) for each t ∈ [0, 1). Then for each x ∈ B, the curve Φt (x) converges to a point x  ∈ B.

3.8. Fixed points of pseudo-contractive holomorphic mappings

95

Proof. Denote by {Gs }s∈[0,10 the net defined by (3.8.16) and fix x ∈ B. Then {Gs (x)} is an approximating curve in B. This curve converges to a point a ∈ B as s → 1− (see [81]). Also we have by (3.8.16), Gs (x) − (Φ ◦ Gs )(x) = (1 − s)(I − Φ ◦ Gs )(x) ≤ (1 − s) [Φ ◦ Gs (x) + x] ≤ 2(1 − s), which tends to zero as s → 1− . By Lemma 3.8.1, this fact completes the proof. 

Chapter 4

Semigroups of Holomorphic Mappings In this chapter we consider certain autonomous dynamical systems acting on the open unit ball of a complex Banach space. Our interest in such systems is based on the fact that if a dynamical system is differentiable with respect to time, then its derivative is a holomorphically dissipative mapping. Furthermore, different estimates on the numerical range lead to rather detailed information on the asymptotic behavior of the system. We pay special attention to stationary points of dynamical systems and to so-called flow invariance conditions. In the case where there is a unique stationary point, we establish rates of convergence of a semigroup of holomorphic mappings to this point. If the stationary point set is empty, the situation is more delicate. In the case of a general Banach space, the asymptotic behavior of semigroups with this property is still unknown. Nevertheless, in the case of a Hilbert space, some ‘boundary analogs’ of the previous results can be obtained. To this end, we consider modifications of the numerical range for the special case where a domain D  0 is replaced by ellipsoids which are contained in the closed unit ball of a Hilbert space.

4.1 Generated semigroups Let D be a domain in a complex Banach space X. Recall that the set Hol(D) consists of all holomorphic self-mappings of D. Note that this set is a semigroup with respect to the composition operation. Definition 4.1.1. A family S = {Ft }t≥0 ⊂ Hol(D) is called a one-parameter semigroup on D if the following properties hold: (i) Ft+s = Ft ◦ Fs for all t, s ≥ 0; (ii) F0 = I, where Ix = x for each x ∈ D. Such a semigroup is called continuous at the point t = 0 if lim Ft = I, t→0+

where the limit is taken with respect to the strong topology of X. If this limit is © Springer Nature Switzerland AG 2019 M. Elin et al., Numerical Range of Holomorphic Mappings and Applications, https://doi.org/10.1007/978-3-030-05020-7_4

97

98

Chapter 4. Semigroups of Holomorphic Mappings

uniform on each subset strictly inside D (bounded away from the boundary of D), then one says that the semigroup is locally uniformly continuous on D. Given a family S = {Ft }t≥0 ⊂ Hol(D) satisfying both (i) and (ii) for all real s and t, one easily sees that each Ft is a biholomorphic mapping (automorphism) of D the inverse of which is F−t . In this case S has actually the structure of a continuous one-parameter group of automorphisms of D. The set of all the automorphisms of D is usually denoted by Aut(D). From the composition rule it follows immediately that if a semigroup is continuous at t = 0, then it is continuous at each point s ≥ 0, that is, Ft+s → Fs pointwise in D as t → 0+ . Moreover, it is known that if D is a bounded convex domain in X and S is a locally uniformly continuous semigroup on D, then it is also differentiable in t ∈ R+ := [0, ∞) (see Berkson and Porta [19] for the onedimensional case, Abate [5], and Reich and Shoikhet [186, 191] for Banach spaces). However, there are one-parameter semigroups which are not uniformly continuous on D, hence are not differentiable at each point of D. Definition 4.1.2. Let S = {Ft }t≥0 ⊂ Hol(D) be a one-parameter semigroup on D. If the limit 1 f = lim [Ft − I] (4.1.1) t→0+ t exists, the mapping f is called the (infinitesimal) generator of the semigroup S. Thus, if a semigroup S = {Ft }t≥0 ⊂ Hol(D) is locally uniformly continuous on D, it is generated by a mapping f ∈ G(D) (see below). Moreover, in this case, the mapping v(t, x) = Ft (x) is the solution to the Cauchy problem ⎧ ⎨ ∂v(t, x) = f (v(t, x)) (4.1.2) ∂t ⎩ v(0, x) = x. Note that since every holomorphic mapping f on a domain D is locally Lipschitzian, it follows that for every f ∈ Hol(D, X) the Cauchy problem (4.1.2) has a unique continuous solution v(·, x) defined on the interval [0, T ], where T depends on the initial value x ∈ D. Definition 4.1.3. Let D be a domain in X and let f ∈ Hol(D, X). The mapping f is said to be a semi-complete vector field on D if the Cauchy problem (4.1.2) has a solution {v(t, x) : t ≥ 0} ⊂ D which is well defined on R+ for each initial value x ∈ D. Note that since f ∈ Hol(D, X) is locally bounded, the solution to (4.1.2) is unique and the family {Ft }t≥0 , where Ft (x) := v(t, x), is a one-parameter semigroup of holomorphic self-mappings of D. In the case where this semigroup consists of automorphisms of D, it can be extended to a one-parameter group and the Cauchy problem (4.1.2) has a unique solution {v(t) : t ∈ R} ⊂ D defined on all of R = (−∞, ∞) for each initial value x ∈ D. The converse is also true. In such

4.1. Generated semigroups

99

a situation the mapping f is said to be a complete vector field (see [13, 49, 225]). Complete vector fields also play a crucial role in the study of bounded symmetric domains in Banach spaces and their algebraic representations. Now we observe that if the solution Ft = v(t, ·) of (4.1.2) is known, then f ∈ Hol(D, X) can be recovered as the strong limit (4.1.1). As a matter of fact, for a hyperbolic domain ([187, p. 233] and [191, p. 100]) the converse is also true: If a locally uniformly continuous semigroup S = {Ft }t≥0 ⊂ Hol(D) is generated by f ∈ Hol(D, C), then v(t, x) = Ft (x) is the solution of the Cauchy problem (4.1.2) for all t ≥ 0 and x ∈ D. In other words, f ∈ Hol(D, X) is the generator of a semigroup on D if and only if it is a semi-complete vector field (see, for example, [186]). We let G(D) denote the set of all semigroup generators (semicomplete vector fields) on D. If D is a convex domain, the set G(D) is a real cone in Hol(D, X), while the set of all group generators on D is a real Banach algebra. The set of group generators on D (complete vector fields) is usually denoted by aut(D) (see [186]). Returning to the limit (4.1.1), we note that it exists if and only if Ft converges  strictly inside D, that is, to I as t → 0+ uniformly on each subset D lim sup Ft (x) − x = 0.

t→0+ x∈D 

In other words, a semigroup of holomorphic self-mappings of D is right differentiable at t = 0+ if and only if it is locally uniformly right continuous at t = 0+ (see [19, 5, 190]). Note also that in this case f ∈ Hol(D, X) defined by (4.1.1) is bounded on each subset strictly inside D. Finally, we observe that if {Gt : t ∈ [0, a]} ⊂ Hol(D), a > 0, is an arbitrary family of holomorphic mappings such that the strong limit f (x) = lim+ t→0

Gt (x) − x t

exists locally uniformly on D and f is bounded on each subset strictly inside D, then f is a semi-complete vector field on D [190]. In addition, the solution v(t, x) of the Cauchy problem (4.1.2) can be obtained by the product formula   v(t, ·) = lim G nt n→∞

n

,

(4.1.3)

where the limit in (4.1.3) is taken with respect to the norm of X, uniformly on each subset strictly inside D. (Recall that [F ]n denotes the n-fold iterate of a mapping F .) A particular, but very important case is where such a family can be obtained by using the so-called nonlinear resolvents Jr = (I − rf )−1 , r > 0.

(4.1.4)

100

Chapter 4. Semigroups of Holomorphic Mappings

Definition 4.1.4. We will say that a mapping f ∈ Hol(D, X) satisfies the range condition (RC) if for each r > 0, (I − rf )(D) ⊃ D and the mapping Jr in (4.1.4) is a well-defined holomorphic self-mapping of D. The following nonlinear version of the classical Lumer–Phillips Theorem 1.3.6 can be obtained by using Theorem 2.5.7 and results in [186, 190, 103]. Theorem 4.1.1. Let D be a bounded convex domain in X and f ∈ Hol(D, X). Then f is a semi-complete vector field on D if and only if it satisfies the range condition (RC). Moreover, in this case the solution v(t, x), t ≥ 0, x ∈ D, of the Cauchy problem (4.1.2) can be obtained by the exponential formula −n  t (x), (4.1.5) v(t, x) = lim I − f n→∞ n where the limit in (4.1.5) is uniform on each subset strictly inside D.

4.2 Stationary points of semigroups Let D be a domain in X and let f be a semi-complete vector field on D. Suppose that S = {Ft }t≥0 is the semigroup of holomorphic mappings generated by f . The uniqueness of the solution to the Cauchy problem (4.1.2) implies that the null point set of f in D coincides with the common fixed point set of S, that is,  NullD f = FixD S := FixD Ft . t≥0

In the theory of evolution equations this set is usually called the stationary point set of the semigroup. The topics in which we are interested here are: The structure of NullD f , existence and uniqueness theorems and the attractivity of a stationary point of S (that is, convergence theorems). Definition 4.2.1. We say that a point a0 ∈ NullD f is quasi-regular if A = f  (a0 ) satisfies the following condition: Ker A ⊕ Im A = X. If, in particular, A is an invertible linear operator (that is, Ker A = {0}), then we will say that the point a0 is a regular null point of f . Proposition 4.2.1 ([126]). Let D be a bounded convex domain in X and let f be a semi-complete vector field on D with a nonempty null point set NullD f . Suppose that one of the following hypotheses holds: (i) X is reflexive; (ii) there is a quasi-regular null point a0 ∈ NullD f .

4.2. Stationary points of semigroups

101

Then NullD f is a holomorphic retract of D. Moreover, it is a complex analytic submanifold of D which is tangent to Ker A = Ker f  (a0 ). Consequently, if a0 is a regular point of NullD f , then this point is unique, that is, NullD f = {a0 }. A particular, but important case compatible with (i) is the case where X = H is a complex Hilbert space. In this case NullD f is an affine submanifold of D which coincides with Ker A ⊕ {a0 } (see [126]). Definition 4.2.2 (see [78, 112]). Let f be a semi-complete vector field on a domain D in X with NullD f = ∅. A point a0 ∈ NullD f is said to be locally uniformly attractive if the semigroup S generated by f converges to a0 in the topology of locally uniform convergence over D. Proposition 4.2.2 (cf. [186, 124]). Let D be a hyperbolic domain in X and let f be a semi-complete vector field on D with NullD f = ∅. Then 1) for each a0 ∈ NullD f , the linear operator A = f  (a0 ) generates the semigroup {T (t) : t ≥ 0} of uniformly bounded linear operators defined by T (t) = [Ft ] (a0 ), where S = {Ft }t≥0 is the semigroup generated by f ; 2) for each a0 ∈ NullD f , the spectrum σ(A) of the linear operator A = f  (a0 ) lies in the left half-plane; 3) if f is bounded on each subset strictly inside D, then a0 is an attractive point of S if and only if σ(A) lies strictly inside the left half-plane, that is, there is ε > 0 such that Re λ ≤ −ε < 0 for all λ ∈ σ(A). Note that in the latter case a0 is, of course, a regular null point. In such a situation it is also called strictly regular. Propositions 4.2.1 and 4.2.2 follow immediately from the so-called Stabilization Theorem. Theorem 4.2.1. Let S = {Ft }t≥0 ⊂ Hol(D) be a uniformly continuous semigroup generated by f ∈ Hol(D, X). Suppose that NullD f = ∅ and that one of the following conditions holds: (i) X is reflexive; (ii) Ker f  (a) ⊕ Im f  (a) = X for some a ∈ NullD f . Then there exists δ > 0 such that for all t ∈ (0, δ), FixD Ft = NullD f. Proof. Let ft := 1t (I − Ft ) such that FixD Ft = NullD ft . Since both NullD ft and NullD f are connected complex submanifolds of D and NullD f ⊂ NullD ft , it suffices to show that their  tangent spaces coincide. A direct calculation shows that (ft ) (a) = 1t I − etA for any a ∈ NullD f , where A = f  (a). Thus, our aim is to show that for some positive δ and for all t ∈ (0, δ), Fix etA = Ker A.

102

Chapter 4. Semigroups of Holomorphic Mappings

When X is reflexive, we first note that the semigroup {(Ft ) (a) = etA } is uniformly bounded by the Cauchy inequalities. Let P denote the projection of X onto Ker A obtained from the mean ergodic theorem.

1 t sA Now let B(t) = e ds. There is δ > 0 such that B(t) is invertible for t 0 all t ∈ (0, δ). For such t, let Pt be the mean ergodic projection onto Fix etA (see Definition 1.5.2 and Corollary 1.5.1). A computation shows that for all natural m, ⎞ ⎛ m 1 B(mt) = ⎝ e(j−1)tA ⎠ B(t). m j=1 Letting m → ∞, we see that P = Pt B(t) = B(t)Pt . Since B(t) is invertible, Ker P = Ker Pt and then Fix P = Fix Pt . Therefore, Ker A = Fix P = Fix Pt = Fix etA . When hypothesis (ii) holds, there is ε > 0 such that Az ≥ εz

(4.2.1)

for all z ∈ Im A. Let x = y + z, where y ∈ Ker A 7 and z ∈ Im A, belong to Fix etA . 6∞ tn−2 Then etA z = z and hence Az = −t An z. n! n=2

ε Let 0 < t < eA −1−A . If z = 0, then it follows that Az < εz, which contradicts (4.2.1). Thus z = 0 and x = y ∈ Ker A. 

Another approach to proving Propositions 4.2.1 and 4.2.2 is based on the resolvent method (see [191]) and on using Lemma 8.2 in that book.

4.3 Flow invariance conditions Let D be a convex subset of a Banach space X and let f : D → X be a continuous mapping on D, the closure of D. Then the following tangency condition of flow invariance lim+ dist(x + rf (x), D)/r = 0, x ∈ D, (4.3.1) r→0

is a necessary condition for the solvability of the evolution equation dv = f (v), dt

v(0) = x ∈ D.

(4.3.2)

This condition was systematically used to study the classes of monotone and accretive operators because of their connections with the theory of semigroups of nonexpansive mappings. For instance, a result of Martin [162] shows that if D is a convex subset of X and f : D → X is a continuous dissipative mapping on D,

4.3. Flow invariance conditions

103

then (4.3.1) is also sufficient for the existence of a solution to the Cauchy problem (4.3.2). Observe also that it was shown in [183] that in this case condition (4.3.1) is equivalent to sup Re f (x), x∗  ≤ 0, x ∈ ∂D, (4.3.3) for each support functional x∗ of D at x ∈ ∂D (that is, x∗ ∈ X ∗ and Re x, x∗  ≥ Re y, x∗  for all y ∈ D). Clearly, if D ⊂ X is a convex domain, and a mapping f ∈ Hol(D, X) is continuous on D and dissipative on D, then condition (4.3.3) holds, and so f is semi-complete. As a matter of fact, we will see below that even if f is not necessarily continuous on D, then its dissipativity already implies the semi-completeness of f . For the class of holomorphic mappings an analog of Martin’s theorem was established in [186]. Theorem 4.3.1. Let D be a bounded convex domain in X. If f ∈ Hol(D, X) has a uniformly continuous extension to D, then it is semi-complete if and only if it satisfies the boundary flow invariance condition (4.3.1) (or (4.3.3)). To illustrate some applications of this condition we consider a question regarding the solvability of autonomous differential equations of order n. Example 4.3.1. Let X be the n-dimensional complex space Cn = {(z1 , z2 , . . . , zn ) : zj ∈ C} equipped with the p -norm 6 z =

n

71/p |zk |

p

,

k=1

and let D = B be the open unit ball in X. Suppose that g : B → C is a holomorphic function on B which has a continuous extension to B. Defining f = (f1 , . . . , fn ) : B → X in (4.3.2) by the formulae fi (z1 , z2 , . . . , zn ) = zi+1 , 1 ≤ i ≤ n − 1, fn (z1 , z2 , . . . , zn ) = g(z1 , . . . , zn ), and using the standard method of rewriting an nth order differential equation as a first-order system of n equations, we get that the boundary condition Re g(z)zn + |zn |2−p Re

n−1 k=1

zk+1

|zk |p ≤ 0, zk

is fulfilled if and only if the equation   x(n) = g x, x , . . . , x(n−1)

z ∈ ∂B,

104

Chapter 4. Semigroups of Holomorphic Mappings

with the initial data x(0) = z1 , x (0) = z2 , . . . , x(n−1) (0) = zn , has a unique solution x = x(t, z1 , . . . , zn ), defined for all t ≥ 0 and (z1 , . . . , zn ) ∈ D, which satisfies the estimate  p 1/p    x(t)p,T = max |x(t)|p + |x (t)|p + · · · + x(n−1) (t) 0. Example 4.3.2 (Li´enard’s Equation [14], [115]). Let us consider the second-order differential equation x ¨ + p(x)x˙ + q(x) = 0 (LE) with the initial data x(0) = z1 ,

x(0) ˙ = z2 ,

where p and q are holomorphic functions on the closed unit disk of the complex plane, and |z1 |2 + |z2 |2 < 1. In this situation the boundary condition of Example 4.3.1 becomes   Re p(z1 )z2 + q(z1 ) − z1 z¯2 ≥ 0. It is easy to see that the latter inequality is equivalent to the conditions Re p(x) ≥ 0

and q(x) = x,

|x| < 1.

The important particular case of Li´enard’s Equation is obtained by setting p(x) = ε(x2 − 1) and q(x) = x: x¨ + ε(x2 − 1)x˙ + x = 0.

(V dP E)

This well-known Van der Pol’s Equation describes nonharmonic oscillators. It has a unique solution x = x(t, z1 , z2 ), defined for all t ≥ 0 and |z1 |2 + |z2 |2 < 1, which satisfies ˙ = z2 , x(0) = z1 , x(0) and 2 |x(t)|2 + |x(t)| ˙ < 1,

t ≥ 0,

if and only if ε ≤ 0. Example 4.3.3. The Lorenz equations x˙ = a(y − x) y˙ = x − y − xz z˙ = xy − cz

4.3. Flow invariance conditions

105

arose in a model for convective motion in the atmosphere [115]. Here we show that for each pair of real parameters (a, c), there is an initial point in the unit ball of C3 such that the solution of the system does not remain in the ball for all time. This is to be expected on physical grounds. Indeed, setting

 f (x, y, z) := a(y − x), x − y − xz, xy − cz , we see that Re f (x, y, z), (x, y, z) = Re{(a¯ x − y¯)(y − x) + z y¯(−x + x ¯) − c|z|2 }. Now letting x = y be purely imaginary and z real, we obtain Re f (x, y, z), (x, y, z) = −2z|x|2 − c|z|2 which is positive when x = 0, z < 0, and |z| is small enough. Thus boundary flow invariance conditions are seen to be quite useful. At the same time, there are many examples of semi-complete vector fields defined on a domain D which have no continuous extension to D. Hence, it is natural to raise the following question: What conditions on a mapping f that is not necessarily uniformly continuous on D imply that f is a semi-complete vector field? Since we will mainly concentrate our discussions on domains which are biholomorphically equivalent to a ball, we assume in the sequel that D = B is the open unit ball of a complex Banach space X. Our approach to the search for different (but equivalent) characterizations of the class of semi-complete vector fields on the open unit ball B is based on the following theorem. Theorem 4.3.2. Let B be the open unit ball in a complex Banach space X. Then f ∈ Hol(B, X) is holomorphically dissipative if and only if there is a real continuous function α : [0, 1) → R such that Re f (x), x∗  ≤ α(x)x

(4.3.4)

for all x ∈ B and some x∗ ∈ J(x), and the equation s − rα(s) = μ

(4.3.5)

has a unique solution s(μ) ∈ [0, 1) for all μ ∈ [0, 1) and for all r > 0. Proof. Assume that f is dissipative. By Theorem 2.5.4, it satisfies the inequality Ref (x), x∗  ≤ α(x)x, where α(s) = f (0)(1−s). Obviously, equation (4.3.5) has a unique solution.

106

Chapter 4. Semigroups of Holomorphic Mappings

Assume now that for a continuous function α equation (4.3.5) has a unique solution for all μ ∈ [0, 1) and r > 0. First, this implies that lim inf α(s) ≤ 0. Indeed, otherwise, there is  ∈ (0, 1) s→1−

such that α(s) ≥ 2ε whenever 1 −  < s < 1. Let A < 2

min

α(s) and A < 0.

s∈[0,1−]

−A  Choosing μ = − ∈ [0, 1) and r = −A > 0, we see that (4.3.5) has no −A solution. Second, this implies that lim sup α(s) ≤ 0. Otherwise, lim sup α(s) = s→1−

s→1−

lim inf α(s) and there are positive numbers A and δ, two sequences {sn } and s→1−

{sn } tending to 1− such that for all n ∈ N, sn < sn < sn+1 ,

α(sn ) = A,

α(sn ) = A − δ.

Choosing now μ = 0 and r = A1 , we see that equation (4.3.5) has infinitely many solutions. Hence, inequality (4.3.4) implies that f is holomorphically dissipative.  The next theorem asserts that each holomorphically dissipative mapping is a semi-complete vector field. We will see in the sequel that the converse assertion is also true. Theorem 4.3.3. Let a mapping f satisfy the equivalent conditions of Theorem 4.3.2. Then f is a semi-complete vector field on B and the semigroup {Ft }t≥0 generated by f admits the following estimate: Ft (x) ≤ β(t, x),

x ∈ B,

where β(t, s) is the solution of the Cauchy problem ⎧ ⎨ ∂β(t, s) = α(β(t, s)) ∂t . ⎩ β(0, s) = s ∈ [0, 1) Proof. Let α : [0, 1) → R be a continuous function such that inequality (4.3.4) holds and equation (4.3.5) has a unique solution s(μ) ∈ [0, 1) for all μ ∈ [0, 1) and for all r > 0. (By Theorem 4.3.2 such a function exists. It can be chosen as α(s) = f (0)(1 − s).) Fix y ∈ B and r > 0, and consider the equation s − rα(s) = y. Let s0 = s(y) be its unique solution in [0, 1). It was shown in the proof of Theorem 4.3.2 that lim sup α(s) ≤ 0. So, for s close enough to 1, s − rα(s) is s→1−

bigger than y. Thus the continuity of α implies that ε := s1 − rα(s1 ) − y > 0 for every s1 ∈ (s0 , 1).

4.4. Semi-complete vector fields on bounded symmetric domains

107

Taking x ∈ B such that x = s1 > s0 , we have by (4.3.4) for such x and some x∗ ∈ J(x), Re x − rf (x) − y, x∗  ≥ s21 − rα(s1 )s1 − ys1 = s1 ε. Now it follows from Theorem 3 in [10] that the equation x − rf (x) = y has a unique solution x = x(y) such that x(y) ≤ s1 . Since s1 ∈ (s0 , 1) is arbitrary, we must have x(y) ≤ s0 . In terms of nonlinear resolvents the latter inequality can be rewritten as Jλ (y) ≤ (1 − rα)−1 y. Now we obtain our assertion by using Theorem 4.1.1 and the exponential formula given there. 

4.4 Semi-complete vector fields on bounded symmetric domains First we study complete vector fields. To do this we need the following definition. Definition 4.4.1. A domain D in X is called symmetric if for each a ∈ D, there exists Fa ∈ Aut(D) such that Fa2 = ID and a is an isolated fixed point of Fa . For the case where D is a bounded symmetric domain, the class aut(D) of all complete vector fields on D has been well described by using an algebraic approach (see, for example, [120, 225, 48, 13, 43]). By Kaup’s theorem [120], every bounded symmetric domain can be realized as the open unit ball of a so-called JB ∗ -triple system. The simplest example of a bounded symmetric domain is the open unit disk D in the complex plane C. In this case, if f ∈ Hol(D, C) is a complete vector field (that is, f ∈ aut(D)) which has a continuous extension to D, then the boundary flow invariance condition (4.3.1) implies that Re f (z)¯ z = 0,

z ∈ ∂D.

(4.4.1)

It is not difficult to check that a function f satisfying (4.4.1) must be a polynomial of degree at most two, that is, f (z) = a + bz + cz 2 . Moreover, the coefficients a, b, c satisfy the relations a = −c,

Re b = 0.

So, f is not only continuous on D; it is, in fact, holomorphic on all of C. This also holds for the general case. The following assertion can be found in [225].

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Chapter 4. Semigroups of Holomorphic Mappings

Proposition 4.4.1. Let B be the open unit ball in a complex Banach space X. Then aut(B) is a real Banach Lie algebra and each f ∈ aut(B) is a polynomial of degree at most 2. Moreover, if P = {f ∈ aut(B) : f  (0) = 0} and K = {f ∈ aut(B) : f (0) = 0}, then aut(B) can be decomposed into the direct sum aut(B) = P ⊕ K. Note that K is actually the closed subalgebra of aut(B) consisting of the linear conservative operators A : X → X, that is, for all x ∈ X, and x∗ ∈ J(x), Re Ax, x∗  = 0 (see [225, 49]). The set P is the closed subspace consisting of the so-called “transvections”: P = {f ∈ aut(B) : f (x) = a − Wa (x)}, where a is an element of X and Wa is a homogeneous polynomial of degree two such that Wia = −iWa . In fact, the existence and uniqueness of transvections for each a ∈ X is a necessary and sufficient condition for the unit ball of X to be symmetric. Proposition 4.4.2. Let B be the open unit ball in a complex Banach space X. The following are equivalent: (i) B is a bounded symmetric domain in X; (ii) for each a ∈ X, there is a unique homogeneous polynomial Wa of degree two such that Wia = −iWa and the mapping f ∈ Hol(X, X) defined by f (x) = a − Wa (x) is a complete vector field on B; (iii) B is a homogeneous (or transitive) domain, that is, for each pair x and y in B, there is F ∈ Aut(B) such that F (x) = y. Returning to our main question concerning the formulation of an interior flow invariance condition for a semi-complete vector field, we first consider the case where the open unit ball B of X is a homogeneous (or bounded symmetric) domain. As we mentioned above, aut(B) is a real Banach Lie algebra, while G(B) (the set of all semi-complete vector fields) is only a real cone. However, these facts immediately lead to the following representation (see [123]).

4.4. Semi-complete vector fields on bounded symmetric domains

109

Proposition 4.4.3. Let X be a complex Banach space such that its open unit ball B is homogeneous. Then the cone G of semi-complete vector fields on B admits the decomposition G(B) = P ⊕ N0 , where P is the real closed subspace of Hol(B) consisting of transvections and N0 is the subcone of G such that for each f ∈ N0 we have f (0) = 0. In other words, f ∈ G admits the unique representation f = f1 + f2 ,

(4.4.2)

f1 (x) = f (0) − Wf (0) (x)

(4.4.3)

where is complete and f2 ∈ G is semi-complete with f2 (0) = 0.

(4.4.4)

Note (see Theorem 4.4.1 below) that the cone N0 can be described as follows:  N0 = f ∈ Hol(B, X) : f (0) = 0 and  (4.4.5) Re f (x), x∗  ≤ 0, x ∈ B, x∗ ∈ J(x) . (Compare this with Definition 2.5.2 of holomorphically dissipative mappings and the class N (B) studied in Section 2.7.) In the particular case where X = C is the complex plane and B = D is the open unit disk, (4.4.2), when combined with (4.4.3)–(4.4.5), becomes f (z) = f (0) − f (0)z 2 − zp(z),

(4.4.6)

where p(z) ∈ Hol(D, C) with Re p(z) ≥ 0,

z ∈ D.

(4.4.7)

As a consequence of Theorem 4.3.3, we also get that if f ∈ Hol(D, C) has the representation (4.4.6) with (4.4.7), then it is a semi-complete vector field on D. Now by using the method of reduction to the one-dimensional case, one can characterize semi-complete vector fields in the general situation (cf. [9]). Theorem 4.4.1. Let B be the open unit ball in an arbitrary complex Banach space X and let f ∈ Hol(B, X). Then f is semi-complete on B if and only if it is holomorphically dissipative. Therefore, in this case, the following estimates hold: 8 9 1 + x  Re f (0)x + (1 − x2 )f (0), x∗ 1 − x 8 9 1 − x  ≤ Re f (x), x∗  ≤ Re f (0)x + (1 − x2 )f (0), x∗ . 1 + x

110

Chapter 4. Semigroups of Holomorphic Mappings

Corollary 4.4.1. Let f ∈ Hol(B, X) be holomorphically dissipative. Then the linear operator A = f  (0) is dissipative on X. The operator A is conservative if and only if f is a complete vector field. Remark 4.4.1. Theorems 4.4.1 and 4.3.3 also show that the simple inequality Re f (x), x∗  ≤ m(1 − x2 ), where x∗ ∈ J(x) and m ≥ 0, is a necessary and sufficient condition for f to be semi-complete.

4.5 Rates of convergence of semigroups Let D be a domain in a Banach space X and let G(D) be the family of all semicomplete vector fields on D. Definition 4.5.1. A mapping f ∈ G(D) is said to be a strongly semi-complete vector field if it has a unique null point in D which is a locally uniformly attractive fixed point for the semigroup generated by f . We have seen in Proposition 4.2.1 that if D is a bounded domain, then f ∈ G(D) with f (τ ) = 0, τ ∈ D, is strongly semi-complete if and only if Re σ(f  (τ )) ≤ −ε < 0. In this section we give several sufficient conditions for f to be strongly semicomplete on the open unit ball B of X and obtain rates of convergence for the semigroups generated by such mappings. It can be easily seen that strong dissipativity (see (4.5.10) below) implies strong semi-completeness. The following theorem gives quantitative characteristics of this fact. To formulate it, recall that for a bounded convex domain D in X, all metrics assigned to it by a Schwarz–Pick system coincide (see Section 3.5). This unique metric is called the hyperbolic metric on D. Theorem 4.5.1. Let B be the open unit ball in X and let f ∈ Hol(B, X) satisfy the following condition: Re f (x), x∗  ≤ α(x) · x,

x ∈ B, x∗ ∈ J(x),

(4.5.1)

where α is a real continuous function on [0, 1] such that α(1) = −ω < 0. Then (i) f is strongly semi-complete;

(4.5.2)

4.5. Rates of convergence of semigroups

111

(ii) for each pair x and y in B, the semigroup S = {Ft }t≥0 generated by f satisfies the following estimate: ω

ρ(Ft (x), Ft (y)) ≤ e− 2 t ρ(x, y),

(4.5.3)

where ρ is the hyperbolic metric on B. In particular, if τ ∈ B is the null point of f , then for all x ∈ B, ω

ρ(Ft (x), τ ) ≤ e− 2 t ρ(x, τ ).

(4.5.4)

Theorem 4.5.1 is different from Theorem 4.3.3 because we impose different conditions on the function α. Moreover, as we have already mentioned, strong semi-completeness is a local property of vector fields. So, a strongly semi-complete vector field is not necessarily strongly dissipative. Proof. Consider for each n = 1, 2, . . . , the mappings fn ∈ Hol(B, X) defined by fn (x) := x −

t g(x) − y, n

x ∈ B,

where t ≥ 0 and y ∈ B. Let Br be the open ball centered at the origin of radius r ∈ [0, 1). For all x ∈ ∂Br = {x ∈ X : x = r} and for all x∗ ∈ J(x), we have by (4.5.1), t Re f (x), x∗  − Re y, x∗  n   t t 2 ≥ r − rα(r) − ry = r r + α(r) − y . n n

Re fn (x), x∗  = x2 −

Since α(1) < 0, it follows that for n big enough the equation ϕn (r) := r −

t α(r) = 1 n

(4.5.5)

has a solution rn ∈ [0, 1). Indeed, ϕn (0) = −(t/n) α(0) < 1 for n > t|α(0)| and ϕn (1) = 1 + tω n > 1. Inequality (4.5.1) implies in its turn that for such n and rn , and for all x with x = rn and x∗ ∈ J(x), the following inequality holds: Re fn (x), x∗  ≥ rn (1 − y). Since fn is bounded on Brn , it follows from [10] that the equation t f (x) − y = 0 n

−1 has a unique solution x = J nt (y) := I − nt f (y) ∈ Brn for each y ∈ B. In other words, the resolvent mapping J nt maps B into Brn . fn (x) = x −

112

Chapter 4. Semigroups of Holomorphic Mappings

It now follows from the Earle–Hamilton Fixed Point Theorem 3.5.2 that J nt has a unique fixed point τ in B. This point is also a null point of f . In addition, repeating the proof of Theorem 3.5.4, we obtain the estimate   1 ρ J nt (x), J nt (y) ≤ ρ(x, y) (4.5.6) t 1 − n · α(r2n ) for each pair of points x and y in B. Since α(r) is continuous on the interval [0, 1], it follows from (4.5.5) that rn → 1 and hence α(rn ) → −ω as n → ∞. Therefore, by using (4.5.6) and the exponential formula (4.1.5), we get by induction estimates (4.5.3) and (4.5.4). The proof is complete.  Example 4.5.1. Let D = D be the open unit disk in the complex plane C and let f ∈ Hol(D, C) be defined by f (z) =: a − az 2 − bz

1 − cz , 1 + cz

where a, b ∈ C, Re b > 0 and 0 ≤ c < 1. If we take α(s) = |a|(1 − s2 ) − Re b · s

1 − cs , 1 + cs

then we get Re f (z)z ≤ α(|z|)|z| and α(1) = − Re b on D.

1−c < 0. Hence f (z) is a strongly semi-complete vector field 1+c

Example 4.5.2. In the theory of autonomous systems the following system is often considered: $ x˙ 1 = x2 + x1 ϕ(x1 , x2 ) x˙ 2 = −x1 + x2 ϕ(x1 , x2 ) We assume that the function ϕ is holomorphic in the unit ball   B = x ∈ C2 : |x1 |2 + |x2 |2 < 1 . It is clear that for any point x = (x1 , x2 ) ∈ B, the support functional x∗ is defined by y, x∗  = y1 x1 + y2 x2 . Hence, for the mapping f (x) = (x2 + x1 ϕ(x), −x1 + x2 ϕ(x)), we have f (x), x∗  = (|x1 |2 + |x2 |2 )ϕ(x). Thus we have to examine the following three cases:

4.5. Rates of convergence of semigroups

113

1) There exists a point x0 = (x01 , x02 ) ∈ B such that Re ϕ(x0 ) > 0; β ∈ R,

2) ϕ(x) = iβ, 3) Re ϕ(x) < 0.

In the first case the mapping f is not a semi-complete vector field. In the third case f is a strongly semi-complete vector field. In the second case f is a group generator. The third case applies to the often used function ϕ(x) = −(1 + x21 + x22 ) (see, for example, [115, p. 327]). Thus the solution of the system $ x˙ 1 = x2 − x1 (1 + x21 + x22 ), x˙ 2 = −x1 − x2 (1 + x21 + x22 ) is well defined for all t ≥ 0 and for all initial values in B, and converges globally on B to the origin. Example 4.5.3. Now we will return to the differential equation from Example 4.3.1 in Section 4.3. It is clear that if we set α(s) := sup{Φ(x1 , . . . , xn1 ) : |x1 |p + · · · + |xn |p = sp }, where ⎡

⎤ n−1 p xj+1 |xj |p g(x , . . . , x )|x | 1 n n ⎦, Φ(x1 , . . . , xn ) = Re ⎣ + xn x j j=1 then inequality (4.5.1) holds. Consequently, if lim sup {Φ(x1 , . . . , xn ) : |x1 |p + · · · + |xn |p = sp } < 0, s→1−

then by Theorem 4.5.1, the solution to the Cauchy problem in this example converges to the point (x0 , 0, . . . , 0) which is the unique null-point of the generator f . Note that if f ∈ Hol(B, X) is known to be a semi-complete vector field on B, then condition (4.5.2) can be replaced by a slightly more general condition, namely, α( ) < 0 for some ∈ (0, 1], (4.5.2 ) which will still ensure assertion (i) of Theorem 4.5.1. This implies the following very simple and interesting sufficient condition. Recall that a bounded operator A ∈ L(X) is said to be strongly dissipative if Re Ax, x∗  ≤ −kx2 for some k > 0 and all x ∈ X, x∗ ∈ J(x).

(4.5.7)

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Chapter 4. Semigroups of Holomorphic Mappings

Corollary 4.5.1. Let f ∈ G(B) and suppose that the linear operator A = f  (0) is strongly dissipative, that is, it satisfies (4.5.7) for some k > 0. If k > 4f (0),

(4.5.8)

then f is a strongly semi-complete vector field. Note that if A = f  (0) is strongly dissipative and f (0) = 0, then condition (4.5.8) is fulfilled automatically. Hence the origin is an attractive fixed point of the semigroup generated by f . Actually, this fact also follows from more general considerations, and in this case one can obtain an exponential rate of convergence. Proposition 4.5.1. Let f ∈ G(B) be such that f (0) = 0 and A = f  (0) is strongly dissipative with Re Ax, x∗  ≤ −kx2 . Suppose that {Ft }t≥0 is the semigroup generated by f . Then for all x ∈ B and for all t ≥ 0 the following estimates hold: 1−x

(i) Ft (x) ≤ xe−k 1+x t ; (ii)

Ft (x)

(1 − Ft (x))2

≤ e−kt

x . (1 − x)2

Estimate (i) is due to Gurganus [94], while (ii) was obtained by Poreda [180]. The condition f (0) = 0 is essential in their considerations as well as in our approach above. Note that both estimates (i) and (ii) are not uniform with respect to x ∈ B. Therefore we need the following notion. One says that ε > 0 is an exponential squeezing coefficient for a semigroup {Ft }t≥0 if for all x ∈ B, Ft (x) ≤ e−εt x. (4.5.9) The following assertion and Theorem 2.7.3 show that for semigroups generated by mappings from each filtration class Kα (B) an exponential squeezing coefficient

1 1 − tα exists and equals κ(α) = dt. 1 + tα 0

Theorem 4.5.2 (cf. Remark 2.7.1). A semigroup S = {Ft }t≥0 has an exponential squeezing coefficient ε if and only if its generator f ∈ N (B) satisfies Ref (x), x∗  ≤ −εx2 .

(4.5.10)

If a mapping f satisfies (4.5.10) with some ε > 0, one says that f is strongly dissipative on B. Proof. Suppose that S has an exponential squeezing coefficient ε. Fix u ∈ ∂B, u∗ ∈ J(u) and for every t ≥ 0, define gt ∈ Hol(D) by gt (ζ) = Ft (ζu), u∗ . By (4.5.9), we have 2 |gt (ζ)| ≤ exp(−2εt)|ζ|2 .

4.6. Semigroups and pseudo-contractive holomorphic mappings

115

Since both sides of this inequality coincide at t = 0, we can differentiate it at t = 0+ and then obtain    ∂gt (ζ) 2 Re gt (ζ)  ≤ −2ε|ζ|2 . (4.5.11) ∂t t=0+ Taking into account that ∂gt (ζ) = ∂t

8

∂Ft (ζu) ∗ ,u ∂t

9 =

f (Ft (ζu)), (ζu)∗  , ζ

(4.5.12)

and setting x = ζu with obviously x = |ζ|, we get immediately from (4.5.12) and (4.5.11) that inequality (4.5.10) holds. Conversely, suppose that f is strongly dissipative and satisfies (4.5.10). Then our assertion follows immediately from Theorem 4.3.3. The proof is complete.  Remark 4.5.1. It can be shown that the existence of an exponential squeezing coefficient in inequality (4.5.9) as well as the strongly dissipativity of the mapping f in (4.5.10) are equivalent to the following estimate of the nonlinear resolvent:   (I − λf )−1 (x) ≤

x , 1 + λε

λ > 0.

Thus, Theorem 4.5.2 can be proved by using the nonlinear exponential formula given in Theorem 4.1.1. Now as a consequence of Theorems 2.7.1 and 4.5.2 we get the following result. Corollary 4.5.2. Let f ∈ Hol(B, X), f (0) = 0, f  (0) = −I, satisfy the generalized Noshiro–Warschawski condition: Re f  (x)x, x∗  ≤ 0, x ∈ B. Then f generates a semigroup S = {Ft }t≥0 which has an exponential squeezing coefficient ε = 2 log 2 − 1. In the case of a Hilbert space sharper estimates have been obtained in [65]. In Section 4.7 we consider modifications of the numerical range for the special case where a domain D  0 is replaced by ellipsoids which lie in the closed unit ball of a Hilbert space. Then we prove a general result concerning the asymptotic behavior of one-parameter semigroups in Hilbert ball. In particular, we show below how more general estimates can be obtained when f has an arbitrary null point which is strictly regular.

4.6 Semigroups and pseudo-contractive holomorphic mappings In this section we consider the connection between semigroup generators and pseudo-contractive holomorphic mappings.

116

Chapter 4. Semigroups of Holomorphic Mappings

As we have mentioned above, for a bounded convex domain, the notions of a semi-complete vector field and an infinitesimal generator are one and the same. Moreover, one can establish the following assertion which we formulate for the case where D = B is the open unit ball in a complex Banach space X. Proposition 4.6.1. Let f ∈ Hol(B, X). Then f is a semi-complete vector field if and only if the mapping F = I + f is pseudo-contractive. Proof. If F is pseudo-contractive, then we set Ft = Φt , where Φt is the solution of the equation Φt (x) = tF (Φt (x)) + (1 − t)x. (4.6.1) We have Ft (x) − x 1 = [tF (Ft (x)) + (1 − t)x − x] = F (Ft (x)) − x. t t

(4.6.2)

Since Ft (x) continuously depends on t, we get that there is y ∈ B such that lim Ft (x) = y

t→0+

and y = x by (4.6.1). Thus by (4.6.2) we see that lim

t→0+

Ft (x) − x = f (x) := F (x) − x. t

Conversely, let f = F − I be a semi-complete vector field. Then the equation x − r(F (x) − x) = y

(4.6.3)

has a unique solution x = x(r, y) for each pair r ≥ 0 and y ∈ B, holomorphically t depending on y ∈ B. Setting here r = 1−t , t ∈ [0, 1), one transforms equation (4.6.3) to the form x = tF (x) + (1 − t)y, 

and we are done.

As we have mentioned above, the common fixed point set of a locally uniformly continuous semigroup coincides with the null point set of its infinitesimal generator. Therefore Theorem 3.8.1 and Proposition 4.6.1 imply immediately the following result which complements Theorem 4.5.2 and Corollary 4.5.2. Corollary 4.6.1. Let f ∈ Hol(B, X) satisfy the conditions a=

sup Ref  (0)x, x∗  < 0 x∈∂B x∗ ∈J(x)

and

2x2 f (0) − Re f  (x)x, x∗  ≤ kx2

4.6. Semigroups and pseudo-contractive holomorphic mappings

117

2 log 2 − 1 . Then f is a semi-complete vector field. 2(1 − log 2) Moreover, if k < κ, then the semigroup S = {Ft }t≥0 generated by f has a unique common fixed point x in B. In addition, if f (0) = 0, then the following global uniform estimate holds: Ft (x) ≤ e−bt x, for some k ≤ κ := (1 + a)

where b = −2(1 − log 2)k + (2 log 2 − 1)(1 − a) > 0. In its turn, Corollary 4.6.1 enables us to prove a new rigidity theorem. To explain it, we observe the following clear fact: if a real function F is continuous in [0, 1], preserves the endpoints of this segment, is smooth inside it and satisfies F  (x) ≤ 1, then it must be the identity mapping. We show that for functions analytic in the open unit disk in C, the last inequality can be weakened in some sense. Theorem 4.6.1. Let F ∈ Hol(D, C) satisfy F (0) = 0, Re F  (0) ≤ 1 and Re F  (z) ≤ 1 +

2 log 2 − 1 (1 − Re F  (0)), 2(1 − log 2)

z ∈ D.

Assume that either one of the following conditions holds: (i) the radial derivative F  (1) = lim− r→1 or (ii) F  (0) = 1.

F (r) − 1 exists with Re F  (1) ≤ 1; r−1

Then F (z) ≡ z. Proof. Define the functions fn by fn (z) := F (z) − n+1 n z. These functions satisfy the conditions fn = 0, Re fn (0) < 0 and   1 2 log 2 − 1 2 log 2 − 1 Re fn (z) ≤ − +1 − Re fn (0) n 2(1 − log 2) 2(1 − log 2) 2 log 2 − 1 0, we have:       1 Ht ◦ Hs (x, y) = Ht Hs (x, y) = Ht Fs (x), (JFs (x)) n+1 Gs (y)    1 1 = Ft (Fs (x)), (JFt (Fs (x))) n+1 Gt (JFs (x)) n+1 Gs (y)   1 1 = Ft (Fs (x)), (JFt (Fs (x))) n+1 (JFs (x)) n+1 Gt (Gs (y))   1 = (Ft ◦ Fs )(x), (JFt ◦Fs (x)) n+1 Gt+s (y) = Ht+s (x, y). This calculation completes the proof.



Theorem 4.6.4. If F is a pseudo-contractive holomorphic mapping on Bn and G is a linear pseudo-contractive operator on Y , then the mapping H which is holomorphic in the open unit ball B of the space Cn × Y and defined by   trace F  (x) − n H(x, y) = F (x), Gy + y n+1 is pseudo-contractive on B. Moreover, if F is strictly pseudo-contractive, then H has a unique fixed point in B.

120

Chapter 4. Semigroups of Holomorphic Mappings

Proof. Since F and G are pseudo-contractive, the mappings f = IX + F and g = IY +G are semi-complete vector fields by Proposition 4.6.1. Denote by {Ft }t≥0 the semigroup generated by f and by {Gt }t≥0 the (linear) semigroup generated by g. Then by Theorem 4.6.3, the family {Ht }t≥0 defined by   1 Ht (x, y) = Ft (x), (JFt (x)) n+1 Gt (y) forms a one-parameter continuous semigroup on the ball B ⊂ Cn × Y . To find its generator h ∈ Hol(B, Cn × Y ), we just differentiate Ht (x, y) at t = 0 using regular rules of differentiation of determinants:   ∂  det Ft (x) = trace f  (x). ∂t t=0 Hence, h(x, y) =

    ∂ trace f  (x) Ht (x, y) = f (x), y + g(y) . ∂t n+1 t=0

Using Proposition 4.6.1 again, we conclude that the mapping H := ICn ×Y + h is pseudo-contractive on B, which proves the first assertion of the theorem. If F is strictly pseudo-contractive, then it has a unique fixed point x0 ∈ Bn . Then there is q ∈ (0, 1) such that Re F  (x0 )x, x∗  < q

(4.6.7) ∂F (x )

j 0 for all x ∈ ∂Bn and x∗ ∈ J(x). In particular, this implies that Re ∂x < q for j every j = 1, . . . , n. Obviously, the point (x0 , 0) is a fixed point of H. We have to show that it is the only fixed point of H. Consider the linear operator A = H  (x0 , 0). A straightforward calculation shows that it acts as follows:   trace F  (x0 ) − n  A(x, y) = F (x0 )x, Gy + y , n+1

and hence, for any point (x, y) on the unit sphere of the space Cn × Y with x = 0, we have trace F  (x0 ) − n A(x, y), (x, y)∗  = F  (x0 )x, x∗  + Gy, y ∗  + y2 n+1 8 9 x x∗ trace F  (x0 ) − n  ≤ F (x0 ) , (1 − x2 ). x2 + (1 − x2 ) + x x n+1 Therefore (4.6.7) implies that Re A(x, y), (x, y)∗  < qx2 +

1 + nq 1 + nq (1 − x2 ) ≤ < 1. n+1 n+1

Thus the fixed point (x0 , 0) is isolated and hence unique. The proof is complete. 

4.7. Semigroups on the Hilbert ball

121

Remark 4.6.1. Various extension operators for one-parameter semigroups were studied in [58]. Using a similar scheme one can use such operators to obtain new extension operators for pseudo-contractive mappings. For instance, using the op  β  h(x) erator h(x) → h(x), x y with β > 0 introduced in [91], one shows that if F is pseudo-contractive on the open unit disk D in C and F (0) = 0, then the mapping H ∈ Hol(D, C × Y ) defined by     F (x) H(x, y) := F (x), 1 − β + β y x is pseudo-contractive. Moreover, it has a unique fixed point provided F is strictly pseudo-contractive. Example 4.6.1. We have already seen in Example 3.8.1 that the function F ∈ Hol(D, C) defined by F (z) := 3(z − log(1 + z)) is a strictly pseudo-contractive mapping. Let Y be an arbitrary Banach space and β be a positive number. Then by Theorem 4.6.4 and Remark 4.6.1 the mappings H1 , H2 , which are holomorphic in the open unit ball   B := (z, y) ∈ C × Y : |z|2 + y2 < 1 and defined by  H1 (z, y) := and

1 + 4z 3(z − log(1 + z)), y 2(1 + z)



    log(1 + z) H2 (z, y) := 3(z − log(1 + z)), 1 + 2β − 3β y , z

are pseudo-contractive on B and have a unique fixed point.

4.7 Semigroups on the Hilbert ball Let now X = H be a complex Hilbert space. In this case the open unit ball B can be equipped with the non-Euclidean “distance” d(x, τ ) from x ∈ B to τ ∈ B defined by (3.2.7). Then the sets E(τ, s) = {x ∈ B : d(x, τ ) < s} , s > 0, τ ∈ B,

(4.7.1)

are ellipsoids in B (see [81] for details). If τ ∈ ∂B, the boundary of B, then E(τ, s) ∩ ∂B = {τ }. If τ ∈ B, then the ellipsoids E(τ, s) are strictly inside B and

122

Chapter 4. Semigroups of Holomorphic Mappings

they are, in fact, open balls with respect to the hyperbolic metric ρ (see Section 3.2) centered at τ in the metric space (B, ρ), that is, : s E(τ, s) = {x ∈ B : ρ(x, τ ) < R}, R = arctanh−1 s + 1 − τ 2 (compare (3.2.3)–(3.2.4) and (3.2.7)). For fixed τ ∈ B and x ∈ ∂E(τ, s), x = τ , consider the non-zero vector 6 7 1 1 1 † x = x− τ . (4.7.2) 1 − σ (x, τ ) 1 − x2 1 − τ, x It can be easily verified that x† is a support functional of the smooth convex set E(τ, s) at x. (This functional is normalized by the condition

 lim x − τ, x† = 1 x→τ

whenever τ ∈ B.) Hence, for a mapping f ∈ Hol(B, H), the inequality

 Re f (x), x† ≤ 0

(4.7.3)

(cf. (4.3.2)) is necessary for f to be the generator of a continuous semigroup for which the sets E(τ, s) are invariant, that is, (4.7.3) is the flow-invariance condition for the sets E(τ, s). In our situation, when f ∈ G(B), this is exactly the case if τ ∈ B is a null point of f , since (4.7.4) ρ (Ft (x), τ ) = ρ (Ft (x), Ft (τ )) ≤ ρ(x, τ ). In fact, inequality (4.7.4) shows that if condition (4.7.3) holds for some τ ∈ B and all x ∈ B, then τ must be a stationary point of S = {Ft }t≥0 , hence a null point of f . If f has no null point, then it can be shown exactly as in Theorem 3.1 in [8] that there is a unique boundary point τ ∈ ∂B such that (4.7.3) holds. This point τ is the sink point for the semigroup generated by f . By these reasons, the sets   

(4.7.5) f (x), x† , x ∈ B ∩ ∂E(τ, s) can be considered a modification of the numerical range of a mapping f ∈ Hol(B,H) where the ellipsoids E(τ, s) are taken in place of a domain D  0. In particular, this enables us to classify the asymptotic behavior of semigroups in the Hilbert ball. For this purpose, we consider a condition finer than (4.7.3). More precisely, given a point τ ∈ B and f ∈ G(B), let ω  be the real non-negative function on (0, ∞) defined by

 ω  (s) := sup 2 Re f (x), x† , s > 0, (4.7.6) d(x,τ )=s

where x† is defined by (4.7.2).

4.7. Semigroups on the Hilbert ball

1 ω

123

Let M(0, ∞) denote the class of all negative functions ω on (0, ∞) such that is Riemann integrable on each closed interval [a, b] ⊂ (0, ∞) and

ds is divergent. ω(s)s 0+

Note that for each ω ∈ M(0, ∞), the function Ω defined by d(x,τ

)

Ω(s) := −

dλ ω(λ)λ

(4.7.7)

s

is a strictly decreasing positive function on (0, d(x, τ )] and maps this interval onto [0, ∞). We denote its inverse function by V : [0, ∞) → (0, d(x, τ )]. Theorem 4.7.1. Let f ∈ G(B) and let S = {Ft }t≥0 be the semigroup generated by f . Given a point τ ∈ B and a function ω ∈ M(0, ∞), the following conditions are equivalent: (i) For all s ∈ (0, ∞),

ω  (s) ≤ ω(s)

where ω  (s) is defined by (4.7.6); (ii) for any differentiable function W on [0, ∞) such that V (t) ≤ W (t), V (0) = W (0) and V  (0) = W  (0), we have d(Ft (x), τ ) ≤ W (t),

x ∈ B, t ≥ 0,

where V = Ω−1 and Ω is defined by (4.7.7). In particular, d(Ft (x), τ ) ≤ V (t); hence τ is a globally attractive point for S. Proof. Consider the function Ψ : R+ × B → R+ defined by Ψ(t, x) = d(Ft (x), τ ). By direct calculations we have 

 ∂Ψ(t, x)  = 2Ψ(0, x) Re f (x), x† .  ∂t t=0+

(4.7.8)

(4.7.9)

Let us first assume that condition (ii) holds. Since Ψ(0, x) = d(x, τ ) = V (0), we get by (4.7.9) and (ii) that   

∂Ψ(t, x)  dW (t)  ≤ 2Ψ(0, x) Re f (x), x† = ∂t t=0+ dt t=0+  dV (t)  1 = =  = d(x, τ )ω (d(x, τ )) . dt t=0+ Ω (d(x, τ )) Varying x ∈ ∂E(τ, s) = {x ∈ B : d(x, τ ) = s}, we see that the latter inequality immediately implies (i).

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Conversely, let condition (i) hold. It follows from (4.7.8) and the semigroup property that for all x ∈ B and s, t ≥ 0, Ψ(s + t, x) = Ψ (s, Ft (x)) . Hence by (4.7.9) and the continuity of f , Ψ is differentiable at each t ≥ 0 and we deduce from (i) and (4.7.9) that ∂Ψ(t, x) ≤ Ψ(t, x)ω  (Ψ(t, x)) ≤ Ψ(t, x)ω (Ψ(t, x)) . ∂t Separating variables we get

d(Ft (x),τ ) d(x,τ )

dΨ = Ω (d (Ft (x), τ )) ≥ t. ω(Ψ)Ψ

This is equivalent to condition (ii). Theorem 4.7.1 is proved.



Example 4.7.1. Let τ ∈ ∂B. Assume that the semigroup generator f can be represented in the form f (x) = p(x) (x (·, τ ) − I) (x − τ ), where p ∈ Hol(B, C) satisfies Re p(x) > a > 0. Then

 p(x)x − τ 2 p(x)x − τ 2 f (x), x† = − = −d(x, τ ) · . 1 − x2 |1 − (x, τ )2 So, one can  choose ω(s)  = −2as in condition (i) of Theorem 4.7.1. In this case 1 1 1 Ω(s) = 2a d(x,τ ) − s , and then d(Ft (x), τ ) ≤ V (t) =

d(x, τ ) 1 + 2atd(x, τ )

for all x ∈ B and t ≥ 0. It turns out that in the case where τ is an interior point of the unit ball B, using the support functionals x† defined by (4.7.2) and the sets (4.7.5) as an analog of the numerical range, we can characterize the class Nτ which consists of all the semi-complete vector fields on B which vanish at a given point τ ∈ B. Observe that E(τ, s) can be rewritten in the form ' ( : s E(τ, s) = x ∈ B : Mτ (x) ≤ , (4.7.10) s + 1 − τ 2 where Mτ is the M¨ obius involution of B defined by (3.2.1). Thus Mτ maps the ellipsoid E(τ, s), s > ; 0, onto the open ball Br = {x ∈ B : x < r} centered at s the origin, where r = s+1−τ 2 < 1.

4.7. Semigroups on the Hilbert ball

125

Let now f ∈ Nτ , τ ∈ B, and let {Ft }t≥0 be the semigroup of holomorphic self-mappings of B generated by f . Then for all t ≥ 0 we have Ft (τ ) = τ , hence ρ(Ft (x), τ ) ≤ ρ(x, τ ) for all x ∈ B. It follows from (4.7.1) that for each s > 0 the ellipsoid E(τ, s) is Ft -invariant for all t ≥ 0. In other words, f ∈ G(E(τ, s)), the class of all semi-complete vector fields on E(τ, s). Consider now the semigroup {Gt }t≥0 defined by Gt = Mτ ◦ Ft ◦ Mτ . Clearly, its generator g is defined by g = lim+ t→0

1 [Gt − I] = (Mτ ) (Mτ ) [f ◦ Mτ ] ; t

hence g vanishes at the origin, that is, g ∈ N0 . But as we already know the class N0 can be described by (4.4.5), that is, Re (g(x), x) ≤ 0. Thus, we conclude that

 Re (Mτ ) (Mτ (x)) [f ◦ Mτ (x)] , x ≤ 0. Calculating the adjoint operator to (Mτ ) (Mτ ), we get by direct calculations the following assertion. Theorem 4.7.2. A mapping f ∈ Hol(B, H) belongs to Nτ = {g ∈ G(B) : g(τ ) = 0} for some τ ∈ B if and only if f (τ ) = 0 and the following inequality holds:

 Re f (x), x† ≤ 0, x ∈ B. (4.7.11) An additional important problem which arises when τ ∈ B is to establish a characterization of the cone G = G(B) and its subcones Nτ which could show when f ∈ G is a strongly semi-complete vector field. Moreover, we will also establish a one-to-one correspondence between our characterization of strongly semi-complete vector fields and the rates of convergence of their generated flows. For this purpose, it is more convenient to use another non-Euclidean distance, namely, < d(x, τ ) . Mτ (x) = d(x, τ ) + 1 − τ 2 We already saw in (4.7.10) that the ellipsoids E(τ, s) can be expressed by Mτ (x). Let α : [0, 1] → R be a continuous function on the interval [0, 1] such that for some δ > 0 and for each r ∈ [0, δ), the function s − rα(s) is increasing on [0, 1]. We also assume that α satisfies the following range condition: for each r ∈ [0, δ) and for all p ∈ [0, 1], the equation s − rα(s) = p has a unique solution s = s(r, p) ∈ [0, 1]. This solution is an increasing function of p ∈ [0, 1] for each fixed r ∈ [0, δ). Also, for each t ≥ 0, there exists the limit   1 [n] t, p , β(t, p) = lim s n→∞ n

126

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where by s[n] (r, p) we denote the n-fold iterate of s(r, ·), that is, s[0] (r, p) = p, s[n] (r, p) = s[n−1] (r, s(r, p)), n = 1, 2, . . .. This limit is the solution of the Cauchy problem ⎧ ⎨ ∂β(t, p) = α(β(t, p)) (4.7.12) ∂t ⎩ β(0, p) = p. Let now f be a holomorphic mapping on B, the open unit ball of the Hilbert space H. Since f is continuously differentiable, hence locally Lipschitzian, the Cauchy problem ⎧ ⎨ ∂u(t, x) = f (u(t, x)) ∂t (4.7.13) ⎩ u(0, x) = x has a unique local solution u = u(t, x) which is real-analytic in t in some neighborhood of zero and holomorphic in a neighborhood Ux of x. We intend to compare this solution with the function β(t, p) defined by (4.7.12) and to recognize when u = u(t, x) can be extended to a global solution of (4.7.13) defined on R+ × B. More precisely, we compare the function β (t, M−τ (x)) with the function mτ : [0, δ) × Ux defined by mτ (t, x) := Mτ (u(t, x)) . Theorem 4.7.3. Let f ∈ Hol(B, H) and let α, β, u and mτ be the functions defined above. Then the following assertions are equivalent: (i) for some τ ∈ B,

mτ (t, x) ≤ β (t, Mτ (x))

whenever u(t, x) is defined; (ii) for some τ ∈ B,  ∂mτ (t, x)   + ≤ α(Mτ (x)); ∂t t=0 (iii) for some τ ∈ B, the mapping f belongs to Nτ and

 Re f (x), x† ≤

α(Mτ (x)) , Mτ (x)σ(τ, x)

where x† is defined in (4.7.2). When these assertions hold, u(t, x) has a unique extension to all of R+ × B and the estimate (i) is true globally. The point τ in (i)–(iii) is one and the same. Proof. Note that mτ (0, x) = β(0, Mτ (x)), so (i) =⇒ (ii) is trivial. Furthermore, by direct calculations we obtain 

 ∂mτ (t, x)  †  + = Mτ (x)σ(x, τ ) Re f (x), x . ∂t t=0

This concludes the proof of the implication (ii) =⇒ (iii).

4.7. Semigroups on the Hilbert ball

127

Thus it remains to be shown that (iii) =⇒ (i). First, we note that (iii) and Theorem 4.7.2 imply that f ∈ Nτ , hence u(t, x) is well defined and belongs to B for all (t, x) ∈ R+ × B. As we have already seen above, the mapping g = (Mτ ) (Mτ ) [f ◦ Mτ ] belongs to N0 . Since the explicit expression for the linear operator A (= A(x)) := Mτ (Mτ (x)) is  1 − (x, τ )  2 (I − P ) , A= x (·, τ ) − P − 1 − τ  τ τ 1 − τ 2 we get A∗ =

 1 − (τ, x)  2 (I − P ) τ (·, x) − P − 1 − τ  τ τ 1 − τ 2

and then (g(x), x) = (Af (Mτ (x)) , x) = (f (Mτ (x)) , A∗ x)   |1 − (x, τ ) |2 1 − x2 = f (Mτ (x)), Mτ (x) − τ . 1 − τ 2 1 − (τ, x) Also, if z = Mτ (x), then 1 − (x, τ ) =

(4.7.14)

1 − τ 2 and 1 − x2 = σ(τ, z). 1 − (z, τ )

Hence by (4.7.14) we obtain   1 − τ 2 1 − z2 f (z), z − τ |1 − (z, τ ) |2 1 − (τ, z)   τ z − = σ(τ, z) f (z), 1 − z2 1 − (τ, z)

 = σ(τ, z)Mτ (z)2 f (z), z † .

(g(x), x) =

Thus (iii) implies that Re (g(x), x) ≤ α(x)x. Now, if v(t, x) is the solution of the Cauchy problem ⎧ ⎨ ∂v(t, x) = ϕ(v(t, x)) ∂t ⎩ v(0, x) = x, then it follows from Theorem 4.3.3 that v(t, x) ≤ β(t, x). But v(t, Mτ (x)) = Mτ (u(t, x)) and this concludes the proof.



Remark 4.7.1. If β(t, p) → 0 as t → ∞ for a fixed s ∈ [0, 1), then condition (i) establishes also a rate of convergence of the semigroup Ft = u(t, ·) to its stationary point τ ∈ B. Hence this point is unique in B, and f is a strongly semi-complete vector field.

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Chapter 4. Semigroups of Holomorphic Mappings

We saw in this section that the sets (4.7.5) taken as an analogue of the numerical range and condition (4.7.3) taken as an analogue of holomorphic dissipativity play a crucial role in the study of semigroups of holomorphic self-mappings on the Hilbert ball. In particular, the upper bounds of the numerical range of a mapping f provide the rate of convergence of the semigroup generated by f . The influence of the lower bounds on the asymptotic behavior of semigroups was studied in [66].

Chapter 5

Ergodic Theory of Holomorphic Mappings Ergodic approximations naturally associated with a given holomorphic mapping essentially determine the asymptotic behavior of the nonlinear mapping in a way similar to how the “big bang” seems to determine future developments. Most important among such approximations are those associated to fixed points of the given holomorphic mapping, stationary points of semigroups of holomorphic mappings, and null points of semigroup generators. In this chapter, which is based on [192] and [134], we pay attention to the Fr´echet derivatives at the fixed points of a given holomorphic mapping and study the asymptotic behavior and ergodic properties of the given mapping around its fixed points.

5.1 General remarks The following theorem is a significant result of a general character. Theorem 5.1.1 ([165] and [208]; see also [190] and [191], pp. 154-155). Let D be a bounded and convex domain in a reflexive Banach space X, and let F be a holomorphic self-mapping of D. If the fixed point set F = FixD F is not empty, then it is a holomorphic retract of D. If X is not necessarily reflexive, then the same conclusion holds under the additional local assumption that for a point a ∈ F , Ker(I − F  (a)) ⊕ Im(I − F  (a)) = X,

(5.1.1)

where I is the identity operator on X (see, for example, [165, 186]). First of all, note that Im(I − F  (a)) in (5.1.1) is automatically closed by a theorem of Kato; see, e.g., [219, Theorem IV.5.10]. © Springer Nature Switzerland AG 2019 M. Elin et al., Numerical Range of Holomorphic Mappings and Applications, https://doi.org/10.1007/978-3-030-05020-7_5

129

130

Chapter 5. Ergodic Theory of Holomorphic Mappings

Condition (5.1.1) is actually equivalent to the fact that the point 1 is at most a simple pole of the resolvent of the linear operator F  (a). It appears in the uniform ergodic theorem for linear operators; see, for example, Theorems 1.5.4 and 1.6.1. Since F (a) = a, it follows that (F n ) (a) = (F  (a))n . Therefore the Cauchy inequalities (see, for example, [191], Proposition 2.3, p. 57) show that the power boundedness of F (see Proposition 5.2.1 below) yields the same property for F  (a). Hence, in view of Corollary 1.5.2, the reflexivity assumption in Theorem 5.1.1 can be replaced not only by (5.1.1) but just by the closedness of Im(I − F  (a)). Definition 5.1.1 (cf. [191], p. 144). A point a ∈ F = FixD F is said to be a quasiregular point of F if the following condition holds: Ker(I − F  (a)) ⊕ Im(I − F  (a)) = X. If, in addition, Im(I − F  (a)) = X, that is, I − F is invertible on a neighborhood of a, then a is said to be a regular point of F . We also mention that for the finite-dimensional case, a fixed point a of a holomorphic mapping F is quasi-regular whenever F  (a) is power bounded (or, merely, Ces`aro bounded); see, e.g., [241, Theorems 7 and 8]. On the other hand, even for a linear mapping F = A, a fixed point need not be quasi-regular (e.g., if the eigenvalue 1 of A is not a simple pole of the resolvent of A). Note that by Cartan’s theorem [39], a holomorphic retract of D is a complex analytic submanifold of D; see also [191, p. 146]. Although fixed point sets of holomorphic self-mappings have been studied very intensively, little is known in the case of unbounded domains or mappings which have fixed points, but do not necessarily map a domain into itself (that is, they are not self-mappings). An additional essential restriction in the results mentioned above is the convexity of D. Indeed, an example given in [165] points out that the holomorphic self-mapping F of the annulus {z ∈ C : 12 < |z| < 2} defined by F (z) = z1 has two isolated fixed points, that is, F = FixD F is not a retract. However, a result in [164] or Proposition 5.2.1 below show that each of the two isolated fixed points has a bounded convex invariant neighborhood; hence they are local retracts by Theorem 5.1.1. Therefore it is natural to study the local structure of the fixed point set of a holomorphic mapping under weaker restrictions. In this chapter we investigate the local structure of the fixed point set of a holomorphic mapping defined on a (not necessarily bounded or convex) domain in a complex Banach space by using ergodic theory and the nonlinear concepts of numerical range and the nonlinear spectrum introduced and studied by Harris; see Chapter 2. We also generalize Theorem 5.1.1 to holomorphic mappings not necessarily preserving the domain D (see Theorem 5.4.4 below and Remark 5.4.3).

5.2. Power bounded holomorphic mappings

131

Below we study, in turn, further deep relations between the asymptotic behavior (of the iterates or of their Ces`aro averages) of a holomorphic mapping and of its linear part, the Fr´echet derivative at a fixed point. On the way, we observe (in Theorem 5.4.2) that the numerical range of the Fr´echet derivative at the origin is contained in the closed convex hull of the numerical range of the holomorphic mapping, thus improving the spectral inclusion established in Proposition 2.3.2.

5.2 Power bounded holomorphic mappings In this section we prove a simple assertion regarding the local structure of a locally power bounded holomorphic mapping. It is a natural analogue of a corresponding property of power bounded linear operators (cf. Section 1.5). Proposition 5.2.1. Let D be a domain in a Banach space X and let 0 ∈ D. Let F : D → D be a holomorphic mapping on D such that (i) F (0) = 0 and (ii) the iterates F n = F ◦ F n−1 , n = 1, 2, . . . , F 0 = I, of F are uniformly bounded on some open ball B containing 0 and contained in D, that is, sup F n (x) < M < ∞ x∈B

for all n = 1, 2, . . .. Then there is a bounded domain U  0 such that F (U ) ⊂ U. In particular, if X is finite-dimensional, then U ⊂ B can be chosen to be convex. Thus, in this case, FixU F is a holomorphic retract of U . Proof. Let BR = {x ∈ X : x < R} ⊂ D and define the function ρ : BR → R+ by ρ(x) := sup {F n (x)} . n≥0 1 Note that the mappings Gn defined by Gn (y) := M F n (Ry) map the open unit ball of X into itself. Hence, setting x = Ry, y < 1, we get from the generalized Schwarz Lemma (see, for example, [49, 191]) that for all x ∈ BR ,

x ≤ ρ(x) ≤

M x . R

Now consider the set Ω := {x ∈ B : ρ(x) < R} . We claim that (a) Ω ⊂ B, (b) Ω is open, and (c) F (Ω) ⊂ Ω.

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Chapter 5. Ergodic Theory of Holomorphic Mappings

Indeed, the first claim is obvious. To prove (b), note first that we can choose BR strictly inside D, so that there is a ball B  ⊂ D such that 0 < dist(BR , ∂B  ) < ∞. Then it follows from the Cauchy inequalities (see, for example, [49] and [191]) that M (F n ) (x) ≤ =: k. dist(BR , ∂B  ) Hence F n is uniformly Lipschitzian on B, that is, F n (x) − F n (y) ≤ k x − y for all x, y ∈ BR and n = 1, 2, . . .. Therefore, if x ∈ Ω, that is, ρ(x) ≤ R − ε for some ε > 0, then for all y ∈ B such that ε x − y < , 2k we have ρ(y) = sup {F n (y) − F n (x) + F n (x)} n≥0

≤ ρ(x) + k x − y < ρ(x) +

ε < R. 2

Thus Ω is open. Finally, it is clear that F (Ω) ⊂ Ω because   ρ(F (x)) = sup {F n (F (x))} = sup{F n+1 (x)} ≤ ρ(x). n≥0

n≥0

Now, since Ω is bounded and 0 = F (0) ∈ Ω, it follows from a result of Mazet [164] that if X is finite-dimensional, then there is a convex open neighborhood U ⊂ Ω of the point x = 0 which is F -invariant, that is, F (U ) ⊂ U .  Example 5.2.1. Let · be the Chebyshev norm on X = Ck+1 , that is, x =

max |xj | ,

1≤j≤k+1

x = (x1 , . . . , xk , xk+1 ),

k ≥ 2.

Let D = X and BR = {x ∈ X : x < R} be the ball of radius R (polydisk) in X. Consider the holomorphic mapping F : X → X, defined by ' ( F (x) = x1 + · · · + xk , 0, . . . , 0, xk+1 2 .    k−1

It is clear that BR is not F -invariant for any R > 0. At the same time, for R = 1 and the open unit ball B, we have sup F n (x) = k < ∞. x∈B

5.3. Ergodicity and fixed points

133

Hence the set Ω = {x ∈ B : ρ(x) < 1}, where ρ(x) = sup {F n (x)} = max {|x1 + · · · + xk | , |x1 | , |x2 | , . . . , |xk+1 |} , n≥0

is F -invariant. It is clear that Ω itself is convex and obviously bounded as a subset of B. The following example also illustrates Proposition 5.2.1, but in an opposite direction. Example 5.2.2. Let F : C2 → C2 be given by F = (F1 , F2 ),F1 (x1 , x2 ) = x1 (1−x2 ), F2 (x1 , x2 ) = x2 . It is clear that F (0) = 0 and FixC2 F = x ∈ C2 : x1 · x2 = 0 is not an analytic manifold in a neighborhood of the origin. So, F cannot be power bounded in any ball around the origin. Indeed, a direct calculation shows that F n (x1 , x2 ) = (x1 (1 − x2 )n , x2 ) . Thus, for example, for each x = (x1 , x2 ) such that x1 = 0 and x2 < 0, F n (x) → ∞ as n → ∞. Remark 5.2.1. Unfortunately, the compactness argument used in the proof of the result of Mazet [164] does not allow us to extend completely (the proof of) Proposition 5.2.1 to the infinite-dimensional case. Nevertheless, using an ergodic approach (see Section 5.3), one can obtain the final conclusion of Proposition 5.2.1 for general Banach spaces under an additional condition of quasi-regularity, which is automatically satisfied in the finite-dimensional case. Namely, the following assertion is a direct consequence of Theorem 5.3.1 below. Proposition 5.2.2 (cf. [191], p. 141). Let D be a domain in a Banach space X and let 0 ∈ D. Let F : D → D be a holomorphic mapping on D such that (i) F (0) = 0 and this fixed point is quasi-regular; (ii) the iterates F n = F ◦ F n−1 , n = 1, 2, . . . , F 0 = I, of F are uniformly bounded on some open ball B containing 0 and included in D, that is, sup F n (x) ≤ M < ∞ for all n = 1, 2, . . . . x∈B

Then there is a bounded convex neighborhood U of the origin such that FixU F is a holomorphic retract of U .

5.3 Ergodicity and fixed points Observe that even for a bounded convex domain D and F ∈ Hol(D), the locally uniform convergence of the iterates F n , n = 1, 2, . . . , to a retraction ρ ∈ Hol(D) onto F = FixD F = ∅ is equivalent to the fact that there is a quasi-regular point a ∈ F such that the spectrum of F  (a) is contained in D ∪ {1}, where D is the open unit disk in C; see [230] and [191, p. 140], cf. [157].

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Chapter 5. Ergodic Theory of Holomorphic Mappings

If the latter spectral condition is not satisfied, then one can try to find another Φ ∈ Hol(D), with the same fixed point set F , the iterates of which do converge to a retraction onto F . The method used in [165], for example, is as follows. It is shown that there exists an N large enough such that for each fixed n > N , the mapping Mn (F ) defined by Mn (F )(x) :=

 1

I + F + · · · + F n−1 (x), n

is power convergent to a holomorphic retraction; hence Φ can be chosen to be any of these Mn (F ), n > N . In the finite-dimensional case, another construction of Φ was proposed by Vigu´e [234] as follows. Due to Montel’s theorem, there is a subsequence of Mn (F ) converging to a holomorphic mapping Φ the iterates of which, in turn, converge to the desired retraction. In fact, independently of the dimension, in the above situation, we can choose just Φ = Mn (F ) with any fixed n ≥ 2, see [189], since Φ (a) obviously satisfies the above spectral condition. Nevertheless, it would be interesting to characterize when the whole sequence {Mn (F )}∞ n=1 converges in some sense. For instance, for linear operators uniform ergodicity is characterized by Dunford’s well-known Theorem 1.5.3; see also Theorems 1.5.4 and 1.5.7. The problem becomes much more complicated when the domain D is not convex or not bounded. In this section, we are going to study this situation. Definition 5.3.1. Let D be a domain in X. A mapping F ∈ Hol(D, X) is said to be locally uniformly ergodic (UE) at a point a ∈ D, if its iterates F n , n = 1, 2, . . . , are well defined in an open connected neighborhood U of the point a, and their Ces` aro averages 1 Mn (F )(x) := (I + F + · · · + F n−1 )(x) n converge locally uniformly in U , that is, uniformly on each open ball contained in U , with respect to the norm of X. If follows immediately from the equation n+1 1 Mn+1 (F ) − Mn (F ) = F n n n that condition (UE) implies that 1 F n (x) → 0, n uniformly in a neighborhood of the point a. We call a mapping F satisfying this condition locally uniformly pre-ergodic (UPE). The point where this property is crucial is formula (5.3.5) below. In the infinite-dimensional case, condition (UPE) is weaker than (UE), even for linear mappings.

5.3. Ergodicity and fixed points

135

Example 5.3.1 (Hille [110]). Consider the Volterra operator V , defined by

s x(t)dt, 0 ≤ s ≤ 1,

(V x)(s) := 0

on X = L1 (0, 1), and set A = I − V . It is shown in [110] that An L1 (0,1) ≈ n1/4 . Hence

An  = 0. n→∞ n Thus the operator A is uniformly pre-ergodic. In addition, A is pointwise ergodic, also by [110]; hence its Ces` aro averages ∞ {Mn (A)}n=1 are uniformly bounded by the Banach–Steinhaus theorem. However, A is not uniformly ergodic since 1 is not a simple pole of the resolvent of the operator A (see, e.g., Theorem 1.5.4). lim

Note that this example also shows that, in general, condition (UPE) does not imply the power boundedness of the operator in question. Moreover, Example 4.6 and Lemma 2.1 (iii) in [220] show that (UPE) does not, in general, imply even Ces`aro boundedness. On the space X = L2 (0, 1), the operator A above is power bounded (see, e.g., [220], p. 213), but still not uniformly ergodic, for the same reason: 1 is not a simple pole of its resolvent. If F (0) = 0, then it can be shown that local uniform ergodicity of F implies that the origin is a quasi-regular fixed point of F . Indeed, if A = F  (0), then it follows from the Cauchy inequalities that lim Mn (A) =: P

n→∞

exists uniformly in X (that is, in the operator norm topology of L(X)) and P is the projection of X onto Ker(I − A) along Im(I − A). Moreover, it follows from Dunford’s Theorem 1.5.3 and Lyubich and Zem´ anek’s Theorem 1.5.4 that a bounded linear operator A is uniformly ergodic (UE) if and only if the following two conditions, taken together, lim

n→∞

1 An  = 0 and n

Ker(I − A) ⊕ Im(I − A) = X

hold. At the same time, uniform ergodicity does not imply, in general, either the power boundedness or the existence of a ball around the origin which is invariant under F .

136

Chapter 5. Ergodic Theory of Holomorphic Mappings

Example 5.3.2. The operator −A = V − I on L1 (0, 1), where V is the Volterra operator from Example 5.3.1, is uniformly ergodic, but not power bounded. So, the natural question is whether locally uniform ergodicity, or even preergodicity, implies that the germ of the fixed point set F at the origin is still a complex analytic manifold. The following theorem, motivated by Theorem 1.5.4 (see also [158]), gives an affirmative answer to this question. Theorem 5.3.1. Let D be a domain in X and let F belong to Hol(D, X) with F = FixD F = ∅. Suppose that a ∈ F is quasi-regular and the following condition holds: (UPE) There is a ball B  D (strictly inside D), B  a, such that the iterates F n of F are well defined on B, and sup F n (x) lim

n→∞

x∈B

n

= 0.

Then there exists a bounded convex neighborhood U of the point a such that F ∩U is a holomorphic retract of U . Moreover, it is a complex analytic submanifold of U tangent to Ker(I − F  (a)). In particular, a ∈ F is an isolated point of F if and only if it is regular. Proof. Without loss of generality assume that a = 0, and let P be the projection of X onto Ker(I −F  (0)) along Im(I −F  (0)). Note that P commutes with A = F  (0), that is, P A = AP = P . Since P is a bounded linear operator, the norm  · 1 on X defined by x1 := max {P x, (I − P )x} is equivalent to the original norm  ·  of X and therefore one can find R > 0 such  R := {x ∈ X : x1 < R} is contained in B. Then the equation that the ball B x = F (x)  R is equivalent to the system in B u = P F (u + v) v = (I − P )F (u + v),

(5.3.1) = (5.3.2)

where we denote u = P x and v = (I − P )x. Since each holomorphic mapping is locally bounded and the operator (I − P )(I − F  (0)) is invertible on Im(I − F  (0)), it follows from the Implicit Function Theorem (see, for example, [40] and [191]) that there is r ∈ (0, R) such that for each u, u < r, the second equation of (5.3.2) has a unique holomorphic solution v := f (u) defined on {x ∈ X : x1 < r} ∩ Ker(I − F  (0))

5.3. Ergodicity and fixed points

137

with values in Im(I − F  (0)) and such that f (0) = 0. We will now show that actually f (u) ≤ cu2 (5.3.3) for some c > 0 and r small enough. Indeed, since F (0) = 0, we can represent F in the form F (x) = F  (0)x + G(x), where G(x) contains the terms of order greater than or equal to 2 in the Taylor expansion of F at the origin. Consequently, the second equation of (5.3.2) has the form v = (I − P )F  (0)v + (I − P )G(u + v), which is equivalent to (I − P )(I − F  (0))v = (I − P )G(u + v), or, v = Γ(I − P )G(u + v), where Γ : Im(I − F  (0)) → Im(I − F  (0)) is the inverse operator for (I − P )(I − F  (0)) on Im(I − F  (0)). Therefore, f (u) satisfies the equality f (u) = Γ(I − P )G(u + f (u)), which means that the Taylor representation of f at the origin begins with a term of order m ≥ 2. By the Schwarz Lemma we have (5.3.3). Now we will show that for each u ∈ Ker(I − F  (0)) with small enough norm, the following identity holds: P F (u + f (u)) ≡ u.

(5.3.4)

To see this, fix u near 0 and consider the functions ϕn (λ) := P F n (λu + f (λu)), where λ ∈ D, the closed unit disk of the complex plane C. Since λu + f (λu)1 < R by (5.3.3), these functions are well defined and map D holomorphically into Ker(I − F  (0)). In addition, we have ϕ1 (λ) = P F (λu + f (λu)) = P F  (0)(λu + f (λu)) + P G(λu + f (λu)) = λu + λm1 hm1 (u) + · · · ,

m1 ≥ 2,

because f (λu) ∈ Im(I − F  (0)) and hence P F  (0)f (λu) ≡ 0.

138

Chapter 5. Ergodic Theory of Holomorphic Mappings

Suppose that hm1 (u) = 0. Then we have by induction ϕn (λ) = λu + n · λm1 hm1 (u) + · · · .

(5.3.5)

But it follows from our (UPE) assumption and the Cauchy inequalities that hm1 (u) → 0

as n → ∞.

This implies that hm1 (u) = 0. So, ϕ1 (λ) ≡ λu for all λ ∈ D. Since u is arbitrary, setting λ = 1 we get (5.3.4). Comparing (5.3.4) with the first equation of (5.3.2), we obtain that x = u + f (u)

(5.3.6)

is a solution of equation (5.3.1), or which is one and the same, that the mapping  R (with R sufficiently small) by ρ defined on U := B ρ(x) := P x + f (P x)  R onto B  R ∩ F . In addition, formula (5.3.6) gives a parametrizais a retraction of B  R tangent to Ker(I − F  (0)) because f  (0) = 0 tion of this set as a submanifold of B by (5.3.3).  Corollary 5.3.1. Let F ∈ Hol(D, D) have a fixed point in D. If F is locally uniformly ergodic at a, then for some bounded convex neighborhood U of a, the set F = FixU F is a holomorphic retract of U . Corollary 5.3.2. Let D be a hyperbolic domain in X and let F ∈ Hol(D) have a quasi-regular fixed point. Then there is a bounded convex neighborhood U of this point such that F ∩ U is a holomorphic retract of U . Proof. Since D is hyperbolic, the Kobayashi pseudometric KD is locally bounded, that is, there is r > 0 such that sup {y : y ∈ D,

KD (a, y) < r} < ∞.

In other words, there is a KD -ball in D of center a, which is a bounded domain in X. The Schwarz–Pick inequality KD (F (a), F (y)) ≤ KD (a, y) and the condition a = F (a) imply that this ball is invariant under F , and (UPE) follows for each ball B which is inside this KD -ball. Hence we can apply either Theorem 5.3.1 or Proposition 5.2.1 in the finite-dimensional case. 

5.3. Ergodicity and fixed points

139

Remark 5.3.1. In fact, for the finite-dimensional case, Corollary 5.3.2 can be obtained in another way, which also gives a direct method for constructing a local retraction on F (cf. [189]). Indeed, it follows from a result of Mazet [164] that there is a KD -ball U centered at a, which is bounded and convex and F -invariant. Therefore the mapping Φ = 12 (I + F ) belongs to Hol(U ). It is clear that FixU Φ = F ∩ U and Ker(I − Φ (a)) ⊕ Im(I − Φ (a)) = X. In addition, note that the spectrum σ(F  (a)) lies in D. But the spectrum σ(Φ (a)) lies in D ∪ {1} and it follows from the Vesentini theorem [230] that Φ is power convergent on U in the topology of locally uniform convergence over U . It is clear that the limit of the sequence {Φn }∞ n=1 is a retraction onto F ∩ U . 

On the other hand, the following example shows that, in general, condition (UPE) does not imply even power boundedness of the mapping Φ = 12 (I + F ). Therefore, Theorem 5.3.1 cannot be proved by using the approach described in Remark 5.3.1. Example 5.3.3. Let V be the Volterra operator on the space L1 (0, 1), as in Example 5.3.1. Then the operator   I −V −V T := 0 I −V on the space L1 (0, 1) ⊕ L1 (0, 1) satisfies T n  → 0; n see [220], Example 4.6. But the operator  I +T I − 12 V = 0 2

− 12 V I − 12 V



is not even Ces`aro bounded (in particular, not power bounded), since the operator I − 12 V is not power bounded on L1 (0, 1) by [222, Theorem 2]. See also [220, Lemma 2.1]. These remarks lead us to the following notion. Definition 5.3.2. Let F ∈ Hol(D) with F = FixD F = ∅. A point a ∈ F is said to be strictly quasi-regular (strictly regular) if condition (5.1.1) holds and σ(F  (a)) ⊂ D ∪ {1} (σ(F  (a)) ⊂ D, respectively). So, similarly to the above discussion we have the following assertion. If D is a hyperbolic domain and F ∈ Hol(D) with F = FixD F = ∅, then a ∈ F is a strictly quasi-regular fixed point if and only if there is a neighborhood U of the point a such that the sequence {F n }∞ n=1 converges to a retraction onto F ∩ U , locally uniformly on U .

140

Chapter 5. Ergodic Theory of Holomorphic Mappings

This implies in turn that for all n = 1, 2, . . . , FixU F n = FixU F.

(5.3.7)

Actually, this fact also holds in a more general situation. Corollary 5.3.3. Let the conditions of Theorem 5.3.1 be satisfied. Then equality √ (5.3.7) holds for all n except possibly for those n for which n 1 ∈ σ(F  (a)). Proof. It is clear that for each n = 1, 2, . . . , FixU F ⊂ FixU F n . √ Since F (a) = a and n 1 ∈ / σ(F  (a)), it follows from the Spectral Mapping Theorem and the chain rule that Ker(I − (F n ) (a)) = Ker(I − F  (a)) and Im(I − (F n ) (a)) = Im(I − F  (a)). This means that a is a quasi-regular fixed point for such F n , too. Hence it follows from Theorem 5.3.1 that FixU F n is a complex analytic submanifold of U tangent to Ker(I − F  (a)). This means that FixU F n = FixU F , as asserted.  Theorem 5.3.2 (Implicit Function Theorem). Let {Fλ } be a family of holomorphic self-mappings of a domain D in X, depending holomorphically on λ ∈ Ω ⊂ C, that is, for each x ∈ D the mapping F (λ, x) := Fλ (x) is holomorphic in λ ∈ Ω, which satisfies the condition 1 sup F n (x) → 0 as n → ∞, n x∈B λ

(5.3.8)

λ∈Ω

where B is a ball in D. Suppose that, for some λ0 ∈ Ω, the mapping Fλ0 has a fixed point x0 ∈ B such that Im(I − A) is closed (or, equivalently, x0 is quasi-regular by Theorem 1.5.4), where A = (Fλ0 ) (x0 ). Then, there exists a neighborhood Ω1 of the point λ0 such that for each λ ∈ Ω1 , there is a fixed point x = xλ ∈ D of the mapping Fλ . Moreover, there is a unique vector function xλ , depending holomorphically on λ ∈ Ω1 , which satisfies the initial condition xλ0 = x0 and the equation Fλ (xλ ) = xλ .

5.3. Ergodicity and fixed points

141

Proof. Without loss of generality we may assume that x0 = 0 ∈ B ⊂ X and λ0 = 0 ∈ Ω ⊂ C. Consider the mapping T : D × Ω → D × Ω defined by T (x, λ) := (Fλ (x), λ). It is clear that T is holomorphic on D × Ω and that (0, 0) is a fixed point of T . In addition, it follows from (5.3.8) that 1 sup T n (x, λ)Z → 0 n (x,λ)∈B×Ω

(5.3.9)

as n → ∞, where Z := X × C and ·Z := max{ · X , | · |C }. Let S = T  (0, 0), that is,   A K S := 0 1  λ where K = ∂F ∂λ x=0,λ=0 ∈ X. Note that the Cauchy inequalities for Fr´echet derivatives and our conditions imply that, actually, the linear operator A = (Fλ0 ) (x0 ) is (UPE), and since Im(I − A) is closed, it follows that A is, in fact, uniformly ergodic (see Section 1.5). We will now show that S is also a uniformly ergodic operator on the Banach space Z. First, (5.3.9) and the Cauchy integral formula imply that lim

n→∞

1 n S → 0, uniformly as n → ∞. n

(5.3.10)

Since Im(I − A) is closed by assumption, it follows from the matrix representation of S that Im(I − S) is also closed in Z. Hence, the operator S is uniformly ergodic on Z and, in particular, the point (x0 , λ0 ) = (0, 0) is a quasi-regular fixed point of T. So, Theorem 5.3.1 can be applied. To show the uniform ergodicity of S explicitly and to find the projection P1 onto Ker(I − S) along Im(I − S), we first observe that by direct calculations we have  n  A (I + A + · · · + An−1 )K Sn = . 0 1 Hence 1 n S = n

1

n nA

0

 Mn (A)K . 1/n

Since Mn (A) → P as n → ∞, uniformly on X, where P is the projection of X onto Ker(I − A) along Im(I − A), we get by (5.3.10) that P K = 0, i.e., K ∈ Im(I − A). Therefore if we denote −1  Γ := (I − A)|Im(I−A) ,

142

Chapter 5. Ergodic Theory of Holomorphic Mappings

the element ΓK belongs to Im Γ ⊂ Im(I − A). We claim that the operator defined by   P ΓK P1 := 0 1 is a projection of Z onto Ker(I − S). First, since P ΓK = 0, it follows that P12 = P1 and SP1 = P1 . Similarly, P1 S = P1 . This implies that P1 Z ⊂ Ker(I − S). On the other hand, if (x, λ) ∈ Ker(I − S), then x must satisfy the equation x = Ax + λK, the solution of which is x = u+λΓK, where u is an arbitrary element of Ker(I −A) and P u = u. Consequently, for each z ∈ Ker(I − S), we have         P ΓK x P ΓK u + λΓK u + λΓK P1 (z) = = = = z. 0 1 λ 0 1 λ λ Thus P1 is a projection onto Ker(I − S). We will now show that actually P1 = lim MN (S) in the operator-norm N →∞

topology of Z. In other words, we will show, directly, that S is uniformly ergodic on Z. To this end, we write the element K ∈ Im(I −A) in the form K = (I −A)ΓK. Then n−1 n−1 Aj K = Aj (I − A)ΓK = (I − An ) ΓK. j=0

j=0

Since ΓK ∈ Im(I − A), the Ces` aro averages MN (S) =

 N −1 1 n Mn (A) S = 0 N n=0

converge to P1 =

 P 0

(I − Mn (A))ΓK 1



 ΓK , 1 

as claimed.

5.4 Numerical range and power boundedness Let B be the open unit ball in a complex Banach space X and let h ∈ Hol(B, X) ∩ C(B). By V (h) we denote the numerical range of h, that is, V (h) = { (h(x)) : x ∈ ∂B, ∈ J(x)}. Definition 5.4.1. Let Ω be a convex subset of C, Ω = C. We say that a linear operator A : X → X is Ω-bounded on X if the condition V (An ) ⊂ Ω,

n = 1, 2, . . . ,

5.4. Numerical range and power boundedness

143

implies that A is power bounded on X, that is, An  ≤ M < ∞,

n = 1, 2, . . . ,

for some M ≥ 0. The following theorem shows that the power boundedness of a holomorphic mapping can be derived from the above property of its Fr´echet derivative at a fixed point. Theorem 5.4.1. Let D be a domain in X containing the closure of the open unit ball B of X. Let F be a holomorphic self-mapping of D with F (0) = 0 and F  (0) = A, such that the closed convex hull Ω of the set ∪n≥1 V (F n ) is not all of C. Then the following assertions hold: (i) V (An ) ⊂ Ω; (ii) if A is Ω-bounded, then F is power bounded on a neighborhood of 0. This theorem is a consequence of the following assertion, which is of intrinsic interest. It is an improvement of Theorem 1 in [98]. Theorem 5.4.2 (Distortion form of the Schwarz Lemma). Let h : B → X be a holomorphic mapping on the open unit ball B ⊂ X. Fix r ∈ (0, 1) and let Ωr be the closed convex hull of the set ∪s∈(r,1) V (hs ), where hs (x) := h(sx). The following assertions hold: (i) V (h (0)) ⊂ Ωr ; (ii) if h(0) = 0 and δ := sup{dist( (h (0)x), ∂Ωr ) : x ∈ ∂B, ∈ J(x)}, then h(x) − h (0)x ≤

8x2 δ, (1 − x)2

x ∈ B.

Moreover, if h has a continuous extension to the closure of B, then the set Ωr above can be replaced by Ω = co V (h). Proof. If Ωr = C, then δ = ∞ and both assertions are trivial. Hence, let us assume that Ωr is not the whole complex plane C. Let h ∈ Hol(B, X) and let hs (x) = h(sx). Fix any x ∈ ∂B, ∈ J(x) and s ∈ (r, 1), and consider the function g : D → C on the closed unit disk D ⊂ C defined as follows: g(λ) := s−1 (h(λsx)), λ ∈ D. It is clear that g is holomorphic in a neighborhood of D and g  (0) = (h (0)x). In particular, g is continuous on D and g  (0) is independent of s. We have to show that the point w0 = g  (0) belongs to every closed half-plane that contains the set ∪t∈(r,1) V (ht ). Suppose, to the contrary, that Π is a closed half-plane such that its boundary line separates the point w0 = (h (0)x) from the set ∪t∈(r,1) V (ht ), with w0 ∈ Π.

144

Chapter 5. Ergodic Theory of Holomorphic Mappings

Let Πε = Π + εD be a half-plane in C strictly containing Π, and let T (w) = a + eiα w be a one-to-one transformation of Πε onto the right half-plane Π+ . Define a holomorphic function ϕ on D by ϕ(λ) := λa + eiα g(λ) and note that ϕ is continuous on D since g is. Since λJ(λx) ⊂ J(x), we have λg(λ) ∈ s−1 Π ⊂ Πε ,

|λ| = 1,

where the last inclusion holds for s close to 1, and so Re λϕ(λ) ≥ 0 for all λ : |λ| = 1. Hence ϕ is accretive on D, and by Remark 2.5.3, it can be represented in the form ϕ(λ) = ϕ(0) − ϕ(0)λ2 + λp(λ), λ ∈ D, where p is a holomorphic function on D with Re p ≥ 0 everywhere on D. Then we have ϕ (0) = p(0) = a + eiα g  (0) ∈ Π+ . Hence g  (0) must lie in Πε . Since ε > 0 was arbitrary and the corresponding s close to 1 can always be chosen in the above reasoning (namely, to satisfy the inclusion s−1 Π ⊂ Πε ), we conclude that g  (0) = (h (0)x) ∈ Π, a contradiction. Hence V (h (0)) ⊂ Ωr , as claimed in (i). To prove (ii), consider again the point (h (0)x) of V (h (0)), as above, and assume that h(0) = 0. We already know that this point belongs to the closed convex set Ωr . Let z0 ∈ ∂Ωr be a nearest point to l(h (0)x) in ∂Ωr , i.e., |z0 − (h (0)x)| := inf |z − (h (0)x)| . z∈∂Ωr

Let L be a supporting line to the set Ωr at the point z0 . Let now α be the angle between the line L and the imaginary axis. Then again there is a number a ∈ C such that the function ψ(λ) := a + eiα f (λ), where f (λ) := λ1 g(λ), f (0) = g  (0), maps D into the right half-plane Π+ = {z ∈: Re z > 0}. (Note that ψ(λ) = λ1 ϕ(λ) and Re ψ(λ) ≥ 0 for λ : |λ| = 1, hence for all λ ∈ D.) Moreover, Re ψ(0) = dist(f (0), z0 ) = dist( (h (0))x, ∂Ωr ). It follows from Carath´eodory’s inequality [142, p. 69] that    1 (m)   ψ (0) ≤ 2 Re ψ(0)  m!  for m ≥ 1. Hence

If now

   1 (m)   f (0) ≤ 2 dist(f (0), z0 ).   m! ∞

Pm (u)

m=1

is the power series expansion of h(u), then       1 (m)   1  f (0) =  (Pm+1 (sx)) ≤ 2 dist( (h (0)x), ∂Ωr ).  m!  s 

5.4. Numerical range and power boundedness

145

Thus |V (Pm+1 )| ≤ 2δ,

m = 0, 1, 2, . . . .

Now we have by Theorem 2.3.1 that Pm  ≤ mm/m−1 · 2δ ≤ 4mδ. Therefore h(u) − h (0)u ≤ 4δ

∞

mum ≤

m=2

8δu2 , (1 − u)2

u ∈ B.

In order to obtain the last assertion one can apply the preceding proof with s = 1.  Let now D be a domain in X (not necessarily bounded) such that D contains the closure of the open unit ball B of X. The set n V := ∪∞ n=0 V (F ), is called the numerical orbit of a holomorphic mapping F : D → D. Theorem 5.4.3. Let X = H be a complex Hilbert space and let D be a domain in H that contains the closure of the open unit ball B of H. Suppose that F : D → D with F (0) = 0 has its numerical orbit in a vertical strip: m ≤ Re V ≤ M. Then there is a bounded open neighborhood U of the origin such that F (U ) ⊂ U. If, in particular, H is finite-dimensional, then U can be chosen to be convex. Proof. Since the numerical orbit of A := F  (0) lies in the same strip by Theorem 5.4.1 (i), it follows from Theorem 1.3.3 that A is power bounded and so our theorem is a consequence of Theorem 5.4.1 (ii) and Proposition 5.2.1.  Remark 5.4.1. It was shown in [85] that in the finite-dimensional case the vertical strip can be replaced by any general strip, while in the infinite-dimensional Hilbert space case the still unclear instance is perhaps a horizontal strip. The problem is still open for general Banach spaces and arbitrary strips. However, applying again Theorem 5.4.2 and Proposition 5.2.2 to the mapping Φ = 12 (I + F ), or Theorem 5.3.1 to the mapping F , one can formulate the following assertions. Corollary 5.4.1. Let D be a domain in a complex Banach space X which contains the closure of the open unit ball B of X. Let F be a holomorphic self-mapping

146

Chapter 5. Ergodic Theory of Holomorphic Mappings

of D such that F (0) = 0 and the closed convex hull of the numerical orbit V n (= ∪∞ n=0 V (F )) is not the whole complex plane C. Assume that either (i) the operator S = 12 (I + F  (0)) is power bounded and the fixed point 0 is quasi-regular, or (ii) the operator A = F  (0) is uniformly ergodic. Then there is a convex neighborhood U ⊂ B of the origin such that F = FixU F is a holomorphic retract of U . Remark 5.4.2. The operator T constructed by McCarthy [166] is not power bounded, but the midpoint operator 12 (I + T ) is power bounded by [172, Corollary 1]. The conclusion of Corollary 5.4.1 can also be achieved if we assume the uniform ergodicity of the mapping F . Corollary 5.4.2. Let F : B → X be uniformly ergodic on B, that is, the Ces` aro averages 1 Mn (F ) = (I + F + · · · + F n−1 ) n converge locally uniformly on B. Let F (0) = 0. Then there is a convex neighborhood U ⊂ B of the origin such that F ∩ U is a holomorphic retract of U . Proof. Since F is uniformly ergodic, it follows again from the Cauchy inequalities that the operator A = F  (0) is also uniformly ergodic; hence the fixed point F (0) = 0 is quasi-regular. Moreover, > n sup F (x) n → 0 x∈B

as n → ∞. Now Theorem 5.3.1 applies.



We proceed with a variant of Theorem 5.1.1 for holomorphic mappings not necessarily preserving the given bounded convex domain. Control of the expansion is expressed in terms of the numerical range. Theorem 5.4.4. Let B be the open unit ball in X and let F ∈ Hol(B, X) have a continuous extension to the closure of B such that Re V (F ) ≤ 1. Then the following assertions hold: (a) For each x ∈ B and each 0 < t < 1, the equation y = tF (y) + (1 − t)x

(5.4.1)

has a unique solution y (= y(x, t)) ∈ B; the mapping G (= Gt ) : B → B defined by Gt (x) := y(x, t) (5.4.2) is holomorphic on B, for each fixed t ∈ (0, 1), and FixB G = FixB F =: F .

5.4. Numerical range and power boundedness

147

(b) If 0 ∈ F is quasi-regular for F , then the iterates {Gn }∞ n=1 converge locally uniformly on B to a retraction ϕ of B onto F , that is, ϕ2 = ϕ and FixB ϕ = F . (c) If F = ∅ and X is either finite-dimensional or a Hilbert space, then the net {Gt }t∈(0,1) defined by (5.4.1)–(5.4.2) strongly converges (with respect to x ∈ B), when t tends to 1− , to a mapping ψ ∈ Hol(B), which is also a holomorphic retraction onto F . Proof. (a) Consider the mapping h := F − I. By assumption, Re V (h) = Re V (F ) − 1 ≤ 0. It then follows from a nonlinear analog of the Lumer–Phillips theorem given in Theorem 4.1.1 that for each r ≥ 0, (I − r(F − I))B ⊃ B and the inverse [I − r(F − I)]−1 is a well-defined holomorphic self-mapping of B. In other words, for each x ∈ B and each r ≥ 0, the equation z − r(F − I)(z) = x

(5.4.3)

has a unique solution z (= z(x, r)) ∈ B, depending holomorphically on x ∈ B. t Setting r = 1−t , 0 < t < 1, in (5.4.3) and  y(x, t) := z x,

t 1−t

 ,

we get equation (5.4.1). Now we just set G(x) = Gt (x) := y(x, t). Furthermore, equation (5.4.1) shows that FixB G = FixB F = F . Indeed, the inclusion FixB G ⊂ FixB F is obvious. Conversely, if x ∈ FixB F , then setting y = x we have equality (5.4.1). On the other hand, (5.4.1) has a unique solution in B. Hence, for such x ∈ F, the point y = G(x) must be x. So, FixB F ⊂ FixB G. To prove (b), suppose that 0 ∈ F is quasi-regular for F . We show that 0 is also quasi-regular for G and that the spectrum of G (0) is contained in D ∪ {1}. Indeed, it follows from the chain rule that T := G (0) = [I − r(A − I)]−1 , t where A := F  (0) and r = 1−t , satisfies the equation r (I − A) T = rT (I − A) = I − T, which shows that Ker(I − T ) = Ker(I − A)

and

Im(I − T ) = Im(I − A).

So, 0 ∈ F is a quasi-regular fixed point of G.

(5.4.4)

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Chapter 5. Ergodic Theory of Holomorphic Mappings

The only inclusion which perhaps requires explanation here is that Im(I − A) ⊂ Im(I − T ). To see this, take any x ∈ Im(I − A), x = (I − A) y, y ∈ X, and set z = 1r y + x. Then it follows from (5.4.4) that (I − T ) z = r (I − A) T z = (I − A) T (y + rx) = (I − A) T (y − r (A − I) y) = (I − A) T [I − r (A − I)] y = (I − A) y = x and we are done. In addition, it follows from Theorem 5.4.2 that the numerical range, hence the spectrum, of the operator A lies in the half-plane Π := {λ ∈ C : Re λ ≤ 1}. Therefore, the analytic function f (λ) =

1 , 1 − r(λ − 1)

λ ∈ Π,

t where r = 1−t > 0, is defined on an open neighborhood of the spectrum σ(A), and we obtain by the Spectral Mapping Theorem the inclusion

σ(T ) = σ(f (A)) ⊂ D ∪ {1}, which implies, by the already mentioned result from [230], that G is power convergent, so the limit lim Gn =: ϕ n→∞

exists in the topology of locally uniform convergence on B and is a holomorphic retraction of B onto FixB G = FixB F = F . This proves (b). Finally, we prove (c). Fix any r ∈ (0, 1) and consider the mapping Gr ∈ Hol(B) defined by (5.4.2). Since Re V (Gr ) ≤ sup Gr (x) ≤ 1, x∈B

it again follows from Theorem 4.1.1 that for each s ∈ (0, 1) and x ∈ B, the equation u = sGr (u) + (1 − s)x

(5.4.5)

has a unique solution u =: Js (x) ∈ B, which is holomorphic in x ∈ B. Moreover, it follows from results in [81, 146] and [203] that if X is either finite-dimensional or a complex Hilbert space, then the net {Js }s∈(0,1) converges to a mapping ψ ∈ Hol(B), when s → 1− , which is a retraction onto FixB Gr (= FixB F ). We show now that the net {Gt }t∈(0,1) converges to the same ψ as t → 1− . Indeed, for a fixed r ∈ (0, 1) and an arbitrary t ∈ (r, 1), we set s=

1 − rt . 1−r

(5.4.6)

It is clear that s ∈ (0, 1) and s → 1− when t → 1− . We claim that for such r, t and s the following relation holds: Gt (x) = Gr (Js (x)).

(5.4.7)

5.4. Numerical range and power boundedness

149

To prove this, we substitute y = Gr (Js (x)) in (5.4.1). We obtain y = rF (y) + (1 − r)Js (x). Then, setting u = Js (x) in (5.4.5), we get the equation y = rF (y) + (1 − r) [sGr (Js (x)) + (1 − s)x] = rF (y) + (1 − r) [sy + (1 − s)x] , which can be rewritten as y=

r (1 − r)(1 − s) F (y) + x. 1 − (1 − r)s 1 − (1 − r)s

Since, by (5.4.6), t=

r ∈ (r, 1), 1 − (1 − r)s

we see that 1−t=

(1 − r)(1 − s) . 1 − (1 − r)s

Hence the latter equation is exactly y = tF (y) + (1 − t)x, which has the unique solution y = Gt (x) by assertion (a). This proves equality (5.4.7), as claimed. Finally, by (5.4.7) and (5.4.5), we obtain Gt (x) − Js (x) = Gr (Js (x)) − Js (x) = Gr (Js (x)) − sGr (Js (x)) − (1 − s)x = (1 − s)Gr (Js (x)) − x ≤ 2(1 − s) → 0 as s → 1− (respectively, t → 1− ). So, we have lim− Gt (x) = lim− Js (x) = ψ(x). t→1

This completes our proof.

s→1



For a linear F , Part (b) of Theorem 5.4.4 follows from [158], pp. 89–90, or [191], p. 43. Indeed, Re V (F ) ≤ 1 implies that the linear operator Gt , 0 < t < 1, calculated directly from (5.4.1), has its spectrum contained in D∪{1} and the point 1 is at most a simple pole of Gt , as shown in our proof of Part (b). Moreover, this shows that the assumption “F (0) = 0 is quasi-regular for F ” is actually necessary for the power convergence of each Gt , 0 < t < 1, already in the linear case.

150

Chapter 5. Ergodic Theory of Holomorphic Mappings

On the other hand, the same conclusion as in Part (c), for a linear F = A, was obtained by Lin in [153], under the assumption that An /n → 0 and Im(I − A) is closed, in place of our Re V (F ) ≤ 1. The two examples below show that there is no general relation (implication) between these two assumptions, even in a finitedimensional Hilbert space. Actually, the common conclusion of both (linear) results (that is, the uniform Abel convergence of the resolvent at 1) is equivalent, on Banach spaces, precisely to the fact that the point 1 is (at most) a simple pole of the resolvent (see [111, Theorem 18.8.1] and Section 1.5). Thus, for a linear F , Parts (b) and (c) in Theorem 5.4.4 are equivalent, even on a Banach space. It may be of interest to note, in the present context, that the latter property (that is, our quasi-regularity condition (1) above) is also a consequence of the Ces` aro boundedness of A (in place of An /n → 0), together with the closedness of Im(I − A). It would be of interest to find analogous results for holomorphic mappings on Banach spaces. Example 5.4.1. Let N be the n × n nilpotent Jordan block, ⎛

0 ⎜0 ⎜ ⎜0 N := ⎜ ⎜. . . ⎜ ⎝0 0

1 0 0 1 0 0 0 0 0 0

⎞ 0 ... 0 0 . . . 0⎟ ⎟ 1 . . . 0⎟ ⎟. ⎟ ⎟ 0 . . . 1⎠ 0 ... 0

It is well known that the numerical range of this operator is the disk centered at the origin of radius cos (π/(n + 1)) < 1; see, e.g., [95, 96, 239]. So the operator A = −I + N has its numerical range in the half-plane Re z < 0 < 1. Since its eigenvalue −1 is a pole of degree n (of the resolvent  k of A), it A  = O(k), follows from [241, Theorem 8] that A is neither Ces`aro bounded, nor   k → ∞, if n > 2. Thus, even Re V (A) < 0 does not imply Ak /k  → 0, in general. For aro boundedness of A does not  n= 2, this example shows that the Ces` imply Ak  /k → 0, either. Example 5.4.2. In the two-dimensional Hilbert space, choose two different unit vectors that are not orthogonal. Define a linear operator A having these two eigenvectors, with eigenvalues 1 and α = 1, |α| ≤ 1, respectively. Then A is power bounded (from the Jordan form, or by a well-known criterion (see, e.g., [241, Theorem 7]): the spectrum of A is contained in the closed unit disk and the unimodular eigenvalues are simple poles of the resolvent of A). If Re V (A) ≤ 1, then the eigenvalue 1 must belong to the boundary of V (A) and, hence, its eigenvector must be orthogonal to the other eigenvector (see [77]

5.4. Numerical range and power boundedness

151

or [114]), a contradiction. Thus, even the power boundedness of A does not imply Re V (A) ≤ 1, in general. Remark 5.4.3. In fact, in the above theorem, the open unit ball B can be replaced by any bounded convex domain D, by using the definition and properties of the numerical range of a holomorphic mapping F on D. Since in general, Re V (F ) ≤ sup p(F (x)), where p is the Minkowski functional for D, we see that Re V (F ) ≤ 1 x∈D

for each holomorphic self-mapping on D. Therefore the above theorem formulated for a bounded convex domain is a generalization of Theorem 5.1.1. Also, as in Theorem 5.1.1, the condition of quasi-regularity (5.1.1) is not necessary if X is reflexive. In particular, this condition can be removed if X = H is a complex Hilbert space. Using Rudin’s theorem ([199, 200], see also [81], pp. 120–121, and [191], p. 145) and the above method, we now prove the following assertion. Theorem 5.4.5. Let B be the open unit ball in a complex Hilbert space H and let a holomorphic F : B → H have a continuous extension to the closure of B, such that Re V (F ) ≤ 1. If F = FixB F = ∅, then it is an affine submanifold of B. In particular, if F  0, then F = Ker(I − F  (0)) ∩ B. Moreover, the retraction ψ defined by ψ = lim− Gt , t→1

is a linear projection of B onto F . Proof. As we have already mentioned, the first part of our theorem is a consequence of Rudin’s theorem and of the fact that, for t ∈ (0, 1), we have FixB F = FixB Gt , where Gt is a holomorphic self-mapping of B. To prove the second part, we use the uniqueness result as well as some auxiliary assertions presented in [203]. Namely, we want to show that for all t ∈ (0, 1), the mappings Gt ∈ Hol(B), hence the retraction ψ (= lim Gt ) are firmly holot→1−

morphic [80] (firmly nonexpansive of the second kind with respect to the Poincar´e hyperbolic metric ρ : B2 → R+ ), that is, the function α(s) := ρ((1 − s)x + sGt (x), (1 − s)y + sGt (y)) is decreasing on [0, 1] (see [81], p. 129).

(5.4.8)

152

Chapter 5. Ergodic Theory of Holomorphic Mappings

To do this, we first observe that since F is holomorphic, it is locally bounded on B, hence for each pair u, v ∈ B, there is δ > 0 such that for all r ∈ [0, δ) the elements x = (1 + r)u − rF (u) and y = (1 + r)v − rF (v) belong to B and ρ((1 + r)u − rF (u), (1 + r)v − rF (v)) ≥ ρ(u, v).

(5.4.9)

The last inequality follows from the fact that u and v can be represented as r u = Gp (x) and v = Gp (y), respectively, where p = r+1 , because of the uniqueness of the solutions to the equations u=

1 r x+ F (u) r+1 r+1

v=

1 r y+ F (v), r+1 r+1

and

respectively. So, (5.4.9) is equivalent to the inequality ρ(Gp (x), Gp (y)) ≤ ρ(x, y), which is the content of the Schwarz–Pick Lemma. Fix now t ∈ (0, 1) and set r = t(1 − s)(1 − t)−1 , u = Gt (x) and v = Gt (y), where x, y are in B. Then by simple calculations we have (1 + r)u − rF (u) = (1 − s)x + sGt (x) and (1 + r)v − rF (v) = (1 − s)y + sGt (y). Substituting these relations into (5.4.9), we get by (5.4.8) that α(1) = ρ(Gt (x), Gt (y)) ≤ ρ((1 − s)x + sGt (x), (1 − s)y + sGt (y)) = α(s), whenever s is close enough to 1, or α2 (1) ≤ α2 (s).

(5.4.10)

It was shown in [203, Corollary 2.6] that the function α2 is differentiable at the point s = 1− . Therefore, by (5.4.10) we get that (α2 ) (1) ≤ 0, which means by [203, Proposition 4.3] that α2 , hence α, are decreasing functions on [0, 1], as claimed. So, each Gt and consequently, the retraction ψ are firmly holomorphic mappings. Consider now the linear operator T (= Tt ) defined by equation (5.4.4), where A = F  (0). Since Re V (A) ≤ 1, it follows from the same arguments as above that T (= Tt ) is also firmly holomorphic.

5.5. Dissipative and pseudo-contractive mappings

153

Observe that (5.4.4) is an analogue of (5.4.1), with A = F  (0) in place of F . Hence by Theorem 5.4.4 (c) we have the strong (pointwise) convergence of the linear operators lim− Tt =: P, t→1

where P is a linear projection onto Ker(I − Tt ) = Ker(I − A), which maps B exactly onto F = FixB F . But, once again, P is firmly holomorphic; therefore ψ and P must coincide by the uniqueness result in [203].  Remark 5.4.4. Note that if, in particular, in Theorem 5.4.5 we have F (0) = 0 and F  (0) = I, then F = FixB F = FixB I = B. In other words, F = I, the identity mapping on B. This rigidity property is the content of Cartan’s Uniqueness Theorem for a holomorphic self-mapping of B (see, for example, Corollary 2.5.2 and the books [40, 49, 81] and [191]), which actually holds for each bounded domain in a Banach space X.

5.5 Dissipative and pseudo-contractive mappings Recall that for T ∈ L(X), the Abel average is defined by Aα := (1 − α)

∞

αn T n ,

(5.5.1)

n=1

where α ∈ (0, 1) is such that the series in (5.5.1) converges in the operator norm topology. The study of Abel averages of bounded linear operators goes back to, at least, Hille [110] and Eberlein [57]. Two natural extensions of L(X) are the class of unbounded linear operators T : DT ⊂ X → X and the class of nonlinear holomorphic mappings h : V → X with various choices of the domains V . Note that, for infinite-dimensional Banach spaces, these two classes are disjoint. Separately they are studied extensively. However, in many applications one encounters mappings that are neither linear nor continuous. A typical example is the mapping T + g where T is a linear unbounded operator and g is nonlinear and holomorphic. Such mappings appear, in particular, in evolution equations of reaction-diffusion type; see, for example, [18] and Section 5.6 below. The aim of this section is: 1. to introduce a class of mappings – ω-pseudo-contractive mappings – which includes both unbounded linear operators and nonlinear holomorphic mappings, and to study this class by means of properly defined Abel averages; 2. to study, cf. [25, 101, 103], the Abel averages of nonlinear holomorphic mappings in the spirit of [134], where a number of properties of the Abel averages of unbounded linear operators were obtained.

154

Chapter 5. Ergodic Theory of Holomorphic Mappings

For an unbounded operator T : DT ⊂ X → X, the Abel average is defined by Aα := (1 − α)[I − αT ]−1 ,

(5.5.2)

where α ∈ (0, 1) is such that Aα ∈ L(X). A further extension of this kind consists in defining Abel averages for nonlinear holomorphic mappings in X. The Abel averages of these mappings are also nonlinear holomorphic mappings. They provide effective tools for studying nonlinear holomorphic mappings, and in particular, nonlinear holomorphic semigroups and their fixed point sets [191]. Of course, the condition lim n1 T n = 0 in Theorem 1.5.7 and Proposin→∞

tion 1.5.1 (see also Theorem 1.6.1) is not applicable to unbounded operators. Moreover, the condition is evidently far from necessary for the corresponding convergence to hold; it can be replaced by, e.g., the dissipativity condition used in the classical Lumer–Phillips Theorem 1.3.6. It turns out that the only essential property of T required to guarantee the convergence in Proposition 1.5.1 is that the spectrum σ(T ), except possibly for the point ζ = 1, lies in the half-plane Π1 = {ζ ∈ C : Re ζ < 1}. Note that the condition T n /n → 0 and the closedness of the image Im(I − T ) (see Proposition 1.5.1) imply that Ker(I − T ) ⊕ Im(I − T ) = X.

(5.5.3)

It follows from Theorem 1.6.1 that for a densely defined linear operator T such that (1, +∞) ⊂ ρ(T ), the Abel averages (5.5.2) are uniformly power convergent (to a projection P onto Ker(I − T )) if and only if σ(T ) ⊂ Π1 ∪ {1} and (5.5.3) holds. Definition 5.5.1. Given ω, θ ∈ R and a dense subset D ⊂ BR , a mapping h : D → X is called (ω, θ)-dissipative (or just quasi-dissipative) on D ⊂ BR if for some ε > 0 and ς ∈ R, for each x ∈ D such that R − ε < x < R and for all x∗ ∈ J(x), the following inequality holds: Reeiθ h(x), x∗  ≤ ωx2 + ς(1 − x2 ).

(5.5.4)

For a holomorphic mapping h on the unit ball B the above definition means that for each s ∈ (1 − ε, 1), the closed convex hull of the numerical range of each hs is not the whole complex plane, or which is one and the same, h : B → X is quasi-dissipative if there is ε > 0 such that the closed convex hull of the set Ωε (h) := {h(x), x∗  : 1 − ε < x < 1} is not the whole plane C.

5.5. Dissipative and pseudo-contractive mappings

155

Obviously, a holomorphic (0, 0)-dissipative mapping on the unit ball is holomorphically dissipative in the sense of Definition 2.5.2. Also note that holomorphic (0, π)-dissipative mappings are holomorphically accretive. Below we study the connection between the (ω, 0)-dissipativity of mappings and the existence and contractivity of their Abel averages, which we define as follows. For a mapping h : D ⊂ B → X, and for real ω and α = 1/ω, we set, cf. (5.5.2), Φ0 = I. (5.5.5) Φα := (I − αh)−1 ◦ [(1 − αω)I], Definition 5.5.2. A mapping h : D → X is called ω-pseudo-contractive if there exists δ > 0 such that for each α ∈ (0, δ), the Abel average Φα defined in (5.5.5) is in Hol(B, D). Note that in Definition 5.5.2 we do not require D to be dense in B. Remark 5.5.1. The use of pseudo-contractive nonlinear mappings goes back to papers by Browder [36] and Kirk [128]. In those works, h is said to be pseudocontractive if I − h is accretive and hence h − I is dissipative. As we will see below in Theorem 5.5.2, ω-pseudo-contractivity with ω = 1 is parallel to the pseudocontractivity in the works just mentioned. Definition 5.5.3. A mapping h : D → X is said to be closed in the weak topology if, for each norm-convergent sequence {xn }n∈N ⊂ D such that lim xn =: x ∈ D n→∞

and such that the sequence {h(xn )}n∈N weakly converges to y ∈ X, it follows that h(x) = y. Lemma 5.5.1. Let h : D → X be ω-pseudo-contractive. Then D is dense in B if and only if, for each x ∈ B, lim Φα (x) = Φ0 (x) = x,

α→0+

(5.5.6)

where the convergence is in the norm topology of X. Proof. If (5.5.6) holds for all x ∈ B, then D is dense since Φα (x) ∈ D for all α ∈ (0, δ) and x ∈ B. Let us now prove the converse. First we prove that (5.5.6) holds for all x ∈ D. For α ∈ (0, δ), we define yα =

x − αh(x) , 1 − αω

x ∈ D.

(5.5.7)

Clearly, yα ∈ B for small enough α, and hence we can compute Φα (yα ) and get Φα (yα ) = x. Let ρ be the hyperbolic metric on B. Then ρ(Φα (x), x) = ρ(Φα (x), Φα (yα )) ≤ ρ(x, yα )

(5.5.8)

156

Chapter 5. Ergodic Theory of Holomorphic Mappings

since Φα ∈ Hol(B). As we have already mentioned, ρ is locally equivalent to the norm metric of B. Moreover, for each x ∈ B and r ∈ (0, 1), there exist positive numbers c1 (x, r) and c2 (x, r) such that, for all y ∈ Br , c1 (x, r)x − y ≤ ρ(x, y) ≤ c2 (x, r)x − y.

(5.5.9)

Now we choose some r ∈ (x, 1) and αr > 0 such that yα in (5.5.7) lies in Br for α < αr . Then by (5.5.9) and (5.5.8), ρ(Φα (x), x) ≤

c2 (x, r)α ωx − h(x), 1 − αω

(5.5.10)

which implies (5.5.6) for this x. Since D is dense in B, the general case of x ∈ B can be handled by the triangle inequality and the result just proven.  As a direct consequence of (5.5.6) we have the following result. Corollary 5.5.1. Let D ⊂ B be dense and h : D ⊂ B → X be ω-pseudo-contractive. Then, for each x ∈ B and r ∈ (x, 1), there exists δr < δ such that Φα (x) ∈ Br whenever α ∈ [0, δr ]. Next, we need the following version of [103, Lemma 4]. Proposition 5.5.1. Given ϑ > 0, let f : D × [0, ϑ) → D be holomorphic in the first variable for each fixed t ∈ [0, ϑ), and right-differentiable at 0 in the second variable for each fixed ζ ∈ D. Suppose also that f (ζ, 0) = cζ for all ζ ∈ D and some c ∈ (0, 1]. Then, for each ζ ∈ D, Re ζft (ζ, 0) ≤ (1 − c2 |ζ|2 ) Re ζft (0, 0),

(5.5.11)

where ft (ζ, t) = ∂f (ζ, t)/∂t. By w-lim we denote the limit in the weak topology of X. Recall that a set C ⊂ X is called balanced if ζx ∈ C for each x ∈ C and ζ ∈ D. Lemma 5.5.2. Let D ⊂ B be balanced and dense, and let h : D → X be ω-pseudocontractive for some ω ∈ R. Suppose also that, for each x ∈ D, we have w- lim+ α→0

Φα (x) − x = h(x) − ωx. α

(5.5.12)

Then h is ω-dissipative. Proof. For t > 0, we set αt = t/(1 + ωt) and let ϑδ > 0 be such that αt ∈ [0, δ) for t ∈ [0, ϑδ ). Fix x ∈ D \ {0} and let ζ ∈ D. Then set ut = (1 + ωt)ζx, where t ∈ [0, ϑδ ) is such that ut ∈ B. By (5.5.9) and (5.5.10), and by Corollary 5.5.1, we obtain ρ(Φαt (ut ), ζx) ≤ th(ζx)c(ζx),

5.5. Dissipative and pseudo-contractive mappings

157

which holds for some positive c(ζx). Note that ζx ∈ D since D is balanced. We conclude that, for each r ∈ (x, 1), there exists ϑr < ϑδ such that Φαt (ut ) ≤ r

whenever t ∈ [0, ϑr ].

Since Φαt ∈ Hol(B), yt (ζ) := Φαt (ut ) = ζx +

t h(yt (ζ)), 1 + ωt

y0 (ζ) = ζx,

(5.5.13)

defines a holomorphic mapping D  ζ → yt (ζ) ∈ B ⊂ X for t ∈ [0, ϑr ]. At the same time, by (5.5.12) the mapping t → yt (ζ) ∈ B ⊂ X has a one-sided weak derivative at t = 0+ . For x∗ ∈ J(x), consider f (ζ, t) :=

1 yt (ζ), x∗ . x(1 + ωt)

(5.5.14)

For each t ∈ [0, ϑr ], it is a holomorphic function on D. By (5.5.13), f (ζ, 0) = ζx and yt (ζ) r |f (ζ, t)| < ≤ . |1 + ωt| |1 + ωt| Thus, if ω ≥ 0, |f (ζ, t)| ≤ r < 1 for all t ∈ [0, ϑr ], whereas if ω < 0, |f (ζ, t)| < 1 for sufficiently small t. Hence, for such t, f (·, t) maps D into itself. For each fixed ζ ∈ D, f (ζ, t) has a one-sided derivative at t = 0+ . A direct computation using (5.5.13) and (5.5.14) yields ft (ζ, 0) = [−ωζx2 + h(ζx), x∗ ]/x. Applying this in (5.5.11) with c = x and ζ = ζ = s ∈ (0, 1), we then obtain Reh(sx), x∗  ≤ ωsx2 + (1 − s2 x2 ) Reh(0), x∗ , which holds for all s < 1. Thus, in the limit as s → 1− we get (5.5.4).



Theorem 5.5.1. Let D ⊂ B be dense in B and balanced. Let also h : D → X be an ω-pseudo-contractive mapping, closed in the weak topology. Assume that either X is reflexive, or X is weakly sequentially complete and h is bounded on every K(x) := {ζx : ζ ∈ D}, x ∈ D. Then h is ω-dissipative. Proof. By Lemma 5.5.2, it is enough to show that (5.5.12) holds for every x ∈ D. By (5.5.9)–(5.5.10) and Corollary 5.5.1, for sufficiently small positive α, (Φα (x) − x)/α ≤

c2 (x, r) ωx − h(x). c1 (x, r)(1 − αω)

If X is reflexive, this estimate implies that the set {(Φα (x) − x)/α ⊂ X : α ∈ (0, δ)}

(5.5.15)

158

Chapter 5. Ergodic Theory of Holomorphic Mappings

is relatively weakly compact. For every sequence {αn }n∈N , converging to 0 as n → ∞, Φαn (x) → x by Lemma 5.5.1. By (5.5.5), Φαn (x) − x = h(Φαn (x)) − ωx. αn The assumed closedness (in the weak topology) of h, the weak compactness just mentioned, and the fact that reflexive Banach spaces are weakly sequentially complete then yield: for each αn → 0+ , the sequence {(Φαn (x) − x)/αn }n∈N converges weakly to the right-hand side of (5.5.12). If X is not reflexive but is sequentially complete, D is balanced and h is bounded on each K(x), x ∈ D, we use the following arguments. Take an arbitrary y ∗ ∈ X ∗ and consider 8 9 1 ∗ fα (ζ) := [Φα (ζx) − ζx] , y α (5.5.16) = − ζωx, y ∗  + h(Φα (ζx)), y ∗ , ζ ∈ D, α ∈ (0, δ]. For each α ∈ (0, δ), fα ∈ Hol(D, C). For a fixed ζ ∈ D, the first line in (5.5.16) can be estimated by means of (5.5.15), from which it follows that |fα (ζ)| ≤

y ∗ c2 (ζx, r) ζωx − h(ζx), c1 (ζx, r)c3 (δω)

where c3 (δω) = min{1, (1−δω)}. Note that c2 (ζx, r)/c1 (ζx, r) can be estimated on each K(x). Thus, in view of the assumed boundedness of h on the sets K(x), the family {fα : α ∈ (0, δ)} is uniformly bounded on compact subsets of D. By Montel’s theorem this family contains a sequence {fαn }n∈N , which converges as n → ∞, uniformly on compact subsets of D, to some f ∈ Hol(D, C). Then by (5.5.16) we get that the sequence {h(Φαn (ζx)), y ∗ }n∈N converges to f (ζ) + ζωx, y ∗ . Now we use the convergence in (5.5.6), the weak sequential completeness of X, and the closedness of h to obtain that h(Φαn (ζx)), y ∗  → h(ζx), y ∗ ,

n → ∞,

for all y ∗ ∈ X ∗ .



Remark 5.5.2. From the previous proof it follows that, for weakly sequentially complete spaces X and mappings h as in Theorem 5.5.1, bounded on each K(x), the mapping ζ → h(ζx) is in Hol(D, X) for each x ∈ D. The following statement is an immediate corollary of Theorem 2.5.7. Proposition 5.5.2. If N (h) < 0, then for each y ∈ Br , r = −N (h), the equation y = h(x) has a unique solution x ∈ B. In particular, h has a unique null point in B. Now we prove a more general result.

5.5. Dissipative and pseudo-contractive mappings

159

Lemma 5.5.3. Given ω ∈ R and h ∈ Hol(B, X), assume that N (h) ≤ ω. Let also r > 0 and λ ∈ C be such that Re λ > ω + r. Then for each y ∈ Br , the equation λx − h(x) = y has a unique solution in B. Proof. For a given y ∈ Br , the null points of g(x) := y − λx + h(x) solve the equation in question. For x ∈ ∂B, x∗ ∈ J(x), and s ∈ (0, 1), we obtain Reg(sx), x∗  = Rey, x∗  − s Re λ + Reh(sx), x∗ . Thus, lim sup Reg(sx), x∗  ≤ y − (Re λ − ω) ≤ r − (Re λ − ω) < 0. s→1−

Hence the result follows from Proposition 5.5.2.



The next statement was proven in [25, Theorem 1.5]. Proposition 5.5.3. Let h ∈ Hol(B, X) be ω-dissipative. Then, for each x ∈ B, it follows that κh x h(x) − h(0) ≤ + 4h(0)x2, 1 − x2 where the constant κh > 0 can be calculated explicitly. In particular, h has unit radius of boundedness. Note that densely defined closed linear operators are closed in the weak topology and bounded on the sets K(x). Indeed, for a linear operator T , the graph Γ(T ) ⊂ X × X is a convex set, which is weakly closed whenever T is closed in the usual sense. Moreover, T K(x) = K(T x), which yields that T is bounded on compact subsets of K(x). Set Qω = [0, 1/ω), ω > 0; Q0 = [0, +∞); Qω = (−∞, 1/ω) ∪ [0, +∞), ω < 0.

(5.5.17)

Theorem 5.5.2. Given h ∈ Hol(B, X) and ω ∈ R, the following statements are equivalent: (i) h is ω-dissipative; (ii) N (h) ≤ ω; (iii) h has unit radius of boundedness and is ω-pseudo-contractive. Moreover, for all x ∈ B, lim+ Φα (x) = x,

α→0

lim+

α→0

Φα (x) − x = h(x) − ωx, α

where the convergence is pointwise in the norm topology of X; (iv) h has unit radius of boundedness and Φα ∈ Hol(B) for each α ∈ Qω .

(5.5.18)

160

Chapter 5. Ergodic Theory of Holomorphic Mappings

Proof. (i) =⇒ (ii) is immediate. (i) =⇒ (iv) =⇒ (iii): The part related to the boundedness of h follows from Proposition 5.5.3. For α = 0, 1/ω and z ∈ B, consider (I − αh)(x) = (1 − αω)z. For λ = 1/α, this equation becomes λx − h(x) = y with y = (λ − ω)z. By Lemma 5.5.3, whenever 1/α > ω, the latter equation has a unique solution x ∈ B and this holds for each α ∈ Qω ; see (5.5.17). Thus Φα is defined as a self-mapping of B. If ω = 0, it can be shown by using the Implicit Function Theorem that Φα ∈ Hol(B) whenever α ≥ 0. Similarly, Φα ∈ Hol(B) for all ω = 0 and each α ∈ Qω . The existence of the first limit in (5.5.18) follows from Lemma 5.5.1 as h is clearly ωpseudo-contractive; cf. Definition 5.5.2. The second limit in (5.5.18) follows from the continuity of h. (iii) =⇒ (i) follows directly from Lemma 5.5.2. (ii) =⇒ (iii): The boundedness follows from [103, Corollary 9]. To prove that, for some δ > 0, Φα ∈ Hol(B) for all α ∈ [0, δ) we let g := h− ωI, so that N (g) ≤ 0. Hence, by [103, Theorem 1], (I − tg)−1 ∈ Hol(B) for all t > 0, that is, the mapping B  x → y ∈ B is holomorphic, where y is defined by y − tg(y) = (1 + ωt)y − th(y) = x, which, for t ∈ [0, 1/|ω|), can be rewritten as yαt − αt h(yαt ) = (1 − αt ω)x. Thus, for 0 < δ < 1/|ω|, it follows that Φα ∈ Hol(B) for all α ∈ [0, δ].



As an immediate consequence of Theorem 5.5.1 and Theorem 5.5.2 we obtain the following result: Corollary 5.5.2. Let X be a weakly sequentially complete Banach spaces, ω ∈ R and h ∈ Hol(B, X). Then h is ω-dissipative if and only if it is ω-pseudo-contractive. Let h ∈ Hol(B, X) be such that h(0) = 0. For every nonzero λ ∈ / σ(h) = σ(h (0)), the set U in the definition of σ(h) also contains 0, and hence one can choose r > 0 such that (λI − h)−1 (Br ) ⊂ B. Fix these λ and r. Then, for α = 1/λ and all ω ∈ R such that |1/α − ω| ≤ r, the mapping  Φα =

1 I −h α

−1   1 ◦ −ω I α

is in Hol(B). Note that here we do not assume that h is ω-dissipative. Let Fix Φα := {x ∈ B : Φα (x) = x} be the set of fixed points of Φα . Since Φα (x) = x is equivalent to (I − αh)(x) = (1 − αω)x, we then get Fix Φα = Null(ωI − h) = {x ∈ B : ωx − h(x) = 0}.

5.5. Dissipative and pseudo-contractive mappings

161

Recall that a subset R ⊂ B is called a holomorphic retract if there exists a holomorphic retraction from B onto R, that is, a holomorphic self-mapping φ of B such that φ(B) = R and φ(z) = z for all z ∈ R. If R is a holomorphic retract of B then, in particular, it is a non-singular closed submanifold of B and it is also totally geodesic with respect to the hyperbolic metric of B. Combining classical results of Koliha (see Theorem 1.5.5) and Vesentini [230, Theorem 1], one can obtain the following characterization of the power convergence of holomorphic mappings. Proposition 5.5.4. Let Ψ ∈ Hol(B) be such that 0 ∈ Fix Ψ. Then the following statements are equivalent. (a) The sequence of iterates {Ψn }n∈N converges, in the operator norm topology, uniformly on closed subsets of B, to a holomorphic retraction of B onto Fix Ψ. (b) The sequence {(Ψ (0))n }n∈N is convergent in the operator norm topology. (c) σ(Ψ (0)) ⊂ D ∪ {1} and λ = 1 is at most a simple pole of the resolvent of Ψ (0). Theorem 5.5.3. Let h ∈ Hol(B, X) be such that h(0) = 0. Suppose that for some ω ∈ R, (5.5.19) Ker(ωI − h (0)) ⊕ Im(ωI − h (0)) = X. Assume also that Φα ∈ Hol(B) for a certain α ∈ R \ {0} such that αω = 1. Then the sequence {Φnα }n∈N converges in the norm topology of X, uniformly on closed subsets of B, if and only if the following containment holds:    ( '  1  1     (5.5.20) σ(h (0)) ⊂ Ω(α, ω) := {ω} ∪ ζ ∈ C : ζ −  > ω −  . α α Moreover, the limit of {Φnα }n∈N is a holomorphic retraction φα : B → Null(ωI − h) := {x ∈ B : h(x) = ωx}. Contrary to the case of linear operators, the retractions φα may depend on α; see Section 5.6 below. Proof. Note that (αI − h (0)) is invertible as 1/α ∈ / σ(h) = σ(h (0)). Since Φα ∈  Hol(B), we can compute its Fr´echet derivative Φα (x), x ∈ B. By the chain rule we get from (5.5.5) Φα (x) − αh (Φα (x))Φα (x) = (1 − αω)I. Next, we have Φα (0) = 0 as h(0) = 0, and hence Φα (0) = (1 − αω)(I − αh (0))−1 . Set ψ(ζ) =

1 − αω , 1 − αζ

ζ ∈ C.

(5.5.21)

162

Chapter 5. Ergodic Theory of Holomorphic Mappings

This function is holomorphic on Ω(α, ω) and ψ(Ω(α, ω)) = D ∪ {1}. On the other hand, (5.5.21) and the Spectral Mapping Theorem (Theorem 1.1.2) imply that σ(Φα (0)) = σ (ψ(h (0))) = ψ (σ(h (0))) .

(5.5.22)

Suppose now that σ(h) ⊂ Ω(α, ω). Then it follows from (5.5.22) that σ (Φα (0)) ⊂ D ∪ {1}. Taking into account that 1/α ∈ / σ(h (0)), and hence Im(I − αh (0)) = X, direct computations yield Ker(I − Φα (0)) = Ker(ωI − h (0)), Im(I − Φα (0)) = Im(ωI − h (0)).

(5.5.23)

By (5.5.23) and (5.5.19), we then get Ker(I − Φα (0)) ⊕ Im(I − Φα (0)) = X, which means that 1 is at most a simple pole of the resolvent of Φα (0). Then by Proposition 5.5.4, the sequence {Φnα }n∈N converges, uniformly on closed subsets of B, to a holomorphic retraction φα : B → Fix Φα = Null(ωI − h). The converse statement follows directly from Proposition 5.5.4.  Remark 5.5.3. It follows from Theorem 5.5.3 that a necessary condition for the sequence {Φnα }n∈N to converge is that Null(ωI − h) be a holomorphic retract of B.

5.6 Examples Dissipative mappings. As a typical example of a mapping h : D ⊂ B → X described by Theorem 5.5.1 we consider h = T + g,

(5.6.1)

where T : D(T ) ⊂ X → X is a closed densely defined linear operator with a nonempty resolvent set and g ∈ Hol(B, X). In this case, D = D(T ) ∩ B. It is known that T is also closed in the weak topology. Hence so is h. For each x ∈ B, h is bounded on compact subsets of K(x). Moreover, by Remark 5.5.2, the mapping ζ → h(ζx) is in Hol(D, X). Proposition 5.6.1. Let T : D(T ) ⊂ X → X be closed and densely defined, and such that ReT x, x∗  ≤ 0 for each x ∈ D(T ) and x∗ ∈ J(x). Assume also that g ∈ Hol(B, Bω ) for some ω > 0. Then h defined in (5.6.1) is ω-dissipative.

5.6. Examples

163

Proof. The assumed properties of T imply that its Abel average Aα , α ≥ 0, defined in (5.5.2) exists and satisfies (1.5.7). For α > 0, the Abel average of h (5.5.5) maps x ∈ B to y ∈ X given by the unique solution of the equation y − αT y − αg(y) = (1 − αω)x. This can be rewritten as y = α(Aα ◦ g)(y) + (1 − αω)Aα x. By Proposition 5.5.2, the mapping y → α(Aα ◦ g)(y) + (1 − αω)Aα x − y has a unique null point in B. Thus, the mapping x → y is in Hol(B). The proof now follows from Theorem 5.5.1.  A concrete example of h as in Proposition 5.6.1 is given by the following integro-differential mapping, which appears in nonlinear and nonlocal evolution equations of the Fisher-KPP type [18]. Here X is the complex Hilbert space L2 (R). Let a : R × R → (0, +∞) be symmetric and such that the operator

a(·, s)x(s)ds L2 (R)  x → R

maps L2 (R) into L∞ (R) and      a(·, s)x(s)ds   R

L∞ (R)

≤ axL2 (R)

for some a > 0. The integro-differential mapping h = T + g, where !

d2 T = 2, g(x) = bx(·) 1 − a(·, s)x(s)ds , b > 0, dt R is 1-dissipative for b < 1/(1 + a). For D(T ), one can take the Sobolev space W 2,2 (R); see, for example, [151, Chapters 6 and 7]. Nonlinear Abel averages. For a linear operator with Abel average Aα , α ∈ (0, 1), the limit of the sequence {Anα }n∈N , if it exists, is one and the same for all α; see Theorem 1.6.1. In the nonlinear case, this is no longer true. This is related to the non-uniqueness of holomorphic retractions for holomorphic retracts of B. In [29, Section 3] it was proved that any one-dimensional retract of a bounded convex domain in Cn with smooth boundary admits a unique holomorphic retraction whose fibers are affine. However, in general, there can also be non-affine retractions. Using Abel averages for nonlinear holomorphic mappings, one can construct such non-affine holomorphic retractions, as the following example shows.

164

Chapter 5. Ergodic Theory of Holomorphic Mappings

Let B be the open unit ball in C2 with respect to the standard Euclidean norm  ·  and set h(z) = h(ξ, η) = (λξ + η 2 , 0), 0 <  < 1. Note that σ(h) = σ(h (0)) = {0, λ}. Put ω = λ. Then σ(h) ⊂ Ω(α, λ) for |1 − αλ| < 1. A direct computation shows that Φα (ξ, η) = (ξ + α(1 − αλ)η 2 , (1 − αλ)η).

(5.6.2)

It is easy to check that Φα is a holomorphic self-mapping of B2 for small enough α. Since (5.3.10) is always satisfied in finite-dimensional Banach spaces, Theorem 5.5.3 applies and {Φnα } converges to a holomorphic retraction φα of B onto Null(λI − h) = {(ξ, 0) : ξ ∈ D}. By (5.6.2), for n ∈ N, we then get Φnα (ξ, η) = (ξ + α(1 − αλ)η 2

n−1

(1 − αλ)2j , (1 − αλ)n η),

j=0

which yields φα (ξ, η) = lim

n→∞

Φnα (ξ, η)

  1 − αλ 2 = ξ+ η ,0 . 2 − αλ

In particular, φα depends on α. It is also of interest to note that φα can be extended to all α ∈ R such that |1 − αλ| < 1. However, the mapping Φα may no longer be a self-mapping of B for some α. For instance, take λ = 1,  = 1/2 and α ∈ (0, 2). Then lim Φα (0, η) = 2

|η|→1

(1 + α4 )(1 − α)2 and, for α close to 2, this number is bigger than 1. Thus in this case φα is not a self-mapping of B, so that it is not a holomorphic retraction of B onto Null(λI − h). For ω = 0, it follows that σ(h) ⊂ Ω(α, 0) for all α ∈ R such that |1 − λα| > 1. In this case,   ξ + αη 2 ,η . Φα (ξ, η) = 1 − αλ It is easy to check that, for any α such that |1−λα| > 1, the points Φα (− λ η 2 , η) are not in B for |η| → 1. This is not surprising, because otherwise {Φnα } would converge to a holomorphic retraction of B2 onto Null(h) = {(ξ, η) ∈ B : λξ = −η 2 }, which would imply that Null(h) is a one-dimensional holomorphic retract of B (a socalled complex geodesic of B), while one-dimensional holomorphic retracts of B are known to be just the intersection of affine complex lines with B. Open question. Let D be a balanced dense subset of B and let h be a closed (weakly closed) mapping on D with values in X. Assume that h is 0-dissipative and for some α0 > 0 its Abel average exists, is holomorphic and maps B into itself. Do the Abel averages exist for all positive α?

Chapter 6

Some Applications The crucial point in our subsequent considerations is the fact that for a holomorphic mapping, even in an infinite-dimensional space, one-sided boundedness of the numerical range already implies that the mapping has unit radius of boundedness. This allows us to study diverse local and global geometric properties and characteristics of holomorphic mappings like Bloch radii, radii of starlikeness and spirallikeness, as well as the problem of analytic extension of semigroups of holomorphic mappings and composition operators discussed below. For a continuous semigroup of bounded linear operators on a complex Banach space the problem of analytic extension with respect to the parameter goes back to the pioneer works [111] and [240]. In these works some criteria of analytic continuation were established along with estimates of those sectors in the complex plane to which analytic continuation is possible. This study was developed by many mathematicians (see, for instance, [38, 174, 117]). The recent paper [16] presents also specific criteria for analytic extension of a continuous semigroup of composition operators. On the other hand, a linear semigroup of composition operators is induced by a nonlinear semigroup of holomorphic self-mappings of a domain in an underlying complex space. The relations between those semigroups and the topological structure of their domains play a crucial role in the study of analytic extension. Although nonlinear semigroup theory for holomorphic self-mappings of a bounded convex domain in a complex Banach space was developed very intensively in the last thirty years, little has been known regarding the analytic extension of such semigroups with respect to a complex parameter. This problem is very well motivated by diverse applications in geometric function theory, such as the radii problem for starlike and spirallike mappings, Bloch constants for locally biholomorphic mappings and the continuous Newton method for solving algebraic or more general nonlinear equations; see [28]. Specifically, for a generated nonlinear semigroup of holomorphic mappings, there arises the additional problem of studying local analytic extension with respect to the complex parameter. More precisely, if the semigroup generator has a singular point inside the domain, then there ex© Springer Nature Switzerland AG 2019 M. Elin et al., Numerical Range of Holomorphic Mappings and Applications, https://doi.org/10.1007/978-3-030-05020-7_6

165

166

Chapter 6. Some Applications

ists a family of subdomains which are invariant under the semigroup action, e.g., hyperbolic balls around the critical point. It turns out that although there are semigroups which have no analytical extension with respect to the parameter, for an invariant subdomain, all semigroups can be extended analytically into a sector in the complex plane with vertex at the origin. Section 6.3 is devoted to the study of this phenomenon. It should also be mentioned that our approach is mostly based on analytic extension with respect to the parameter of the so-called nonlinear resolvent which is of independent interest by itself. We show that the existence domain of the resolvent which preserves some subdomain is much wider than the domain of analyticity of the semigroup generated via the nonlinear exponential formula. Returning to linear continuous semigroups of composition operators, we apply these results to study the problem of analytic extension for suitable Banach spaces of holomorphic self-mappings on shrinking invariant subdomains.

6.1 Bloch radii Let F ∈ Hol(B, X) be such that F (0) = 0 and F  (0) is an invertible operator on X. In other words, F is locally biholomorphic around the origin. One says that the positive numbers r and ρ are Bloch radii for F if F (Br ) ⊇ Bρ and F −1 : Bρ → Br is a well-defined holomorphic mapping on Bρ . A deficiency of this definition is that the pair (r, ρ) is not uniquely defined. If, for example, we find the maximal ρ for which F −1 is holomorphic on Bρ , then for each r ∈ [r, 1], the pair ( r , ρ) constitutes Bloch radii. However, in this case it is often desirable to find the minimal r for which F (Br ) ⊇ Bρ . Sometimes it is preferable to find a number 0 < r∗ ≤ 1 and a continuous function ρ(r) on [0, r∗ ] (if it exists) such that all the pairs (r, ρ(r)) are Bloch radii. In this case, one can investigate the distortion (dilation) coefficient ε(r) =

r >0 ρ(r)

on the interval [0, r∗ ] and look for its bounds. For example, if F (0) = 0 and F  (0) = I, then in the one-dimensional case it follows from Koebe’s 1/4-theorem that if r∗ is the radius of univalence of F in B, then ε(r) = 4 for each r ∈ (0, r∗ ]. In general, under the above normalization, the Inverse Function Theorem shows that Bloch radii exist for F . In this case, one can write F (x) = x − h(x), where h (0) = 0. However, this condition is not necessary: one can just require that I − h (0) be an invertible linear operator. In particular, in order to get estimates in terms of the numerical range we can assume that K = sup Reh (0)x, x∗  < 1. x=1

6.1. Bloch radii

167

Consider the equation x − h(x) = z,

z ∈ X,

x < 1.

(6.1.1)

Our goal is to find numbers 0 < r < 1 and ρ (= ρ(r)) such that for all z ∈ Bρ , equation (6.1.1) has a unique solution x = x(z) ∈ Br , which is holomorphic in z ∈ Bρ . Let us assume that h(0) = 0 and that for some θ ∈ R, the mapping h satisfies the condition " # sup Re eiθ h(x), x∗ = N (θ) < ∞. (6.1.2) x∈B

Suppose that the following numbers are given: " # " # K(θ) := sup Re eiθ h (0)x, x∗ , k(θ) := inf Re eiθ h (0)x, x∗ . x=1

x=1

(6.1.3)

We let K := K(0). Since the Fr´echet derivative of a holomorphic mapping is a bounded linear operator, K(θ) and k(θ) are finite for all θ ∈ R. We use the following version of the Implicit Function Theorem. Lemma 6.1.1 ([186, 122]). Let G (= G(x, z)) be a holomorphic mapping in the domain D = Br × Bρ with values in X and assume that for each z ∈ Bρ and x = su, u = r, 0 < s < 1, lim sup Re G(su, z), u∗  < 0. s→1−

Then (i) for each z ∈ Bρ , there is a unique solution x (= x(z)) ∈ Br of the equation G(x, z) = 0, which holomorphically depends on z ∈ Bρ ; (ii) for each z ∈ Bρ , the linear operator Gx (x(z), z) is invertible in X. We now consider the mapping G : B × X → X defined by G(x, z) := z − x + h(x) and note that equation (6.1.1) is equivalent to G(x, z) = 0.

(6.1.4)

In view of Lemma 6.1.1, our aim becomes to find r ∈ (0, 1) and ρ = ρ(r) > 0 such that the following inequality holds whenever x = r and z < ρ (= ρ(r)): sup Re G(x, z), x∗  < 0.

x=r

(6.1.5)

168

Chapter 6. Some Applications

Equation (6.1.5) is a sufficient condition for (6.1.4) to have a unique solution x = x(z) in the ball Br , r ∈ (0, 1). Since sup Re G(x, z), x∗  ≤ z · r − r2 + sup Re h(x), x∗  ,

x=r

(6.1.6)

x=r

we have, as a matter of fact, to use an appropriate growth estimate for the last term in (6.1.6). Now it follows from Theorem 2.6.2 with R = 1 that sup Re h(x), x∗  ≤ r2 [K + L(θ, r)(Nr (θ) − k(θ))] ,

(6.1.7)

x=r

2r (1 − r cos θ) where L(θ, r) = , 1 − r2 " iθ  # inf Re e h (0)x, x∗ .

" # Nr (θ) := sup Re eiθ h(x), x∗ and k(θ) := x k(θ). In this case sup ReG(x, z), x∗  ≤ 0

x=r

if ρ < ρ(r) := r(1 − K − δ(θ)L(θ, r)),

z < ρ,

x = r.

(6.1.9)

It would, of course, be pertinent to look for conditions which ensure that ρ(r) > 0 for some r ∈ (0, 1] and to find the maximum of this function on this interval.

6.1. Bloch radii

169

Writing down explicitly (6.1.9), we get  r

ρ(r) = (1 − K)(1 − r2 ) − δ(θ)2r(1 − r cos θ) . 1 − r2

(6.1.10)

As above we assume in the sequel that K < 1. Since ρ(0) = 0, we have ρ (0) = ρ(r) lim = 1 − K > 0. In addition, lim− ρ(r) = −∞ whenever θ = 0, and for all r→0+ r r→1 r ∈ (0, 1), ρ (r) =

4δ(r3 cos θ − 3r2 + 3r cos θ − 1) −4δ < < 0. 2 3 (1 − r ) (1 + r)3

Thus, again, the condition K < 1 ensures that ρ has a positive maximum ρ(r0 ) at some point r0 ∈ (0, 1). If θ = 0, it is clear that r0 < r∗ < 1, where r∗ is the minimal (positive) root of the equation ρ(r) = 0 or, which is one and the same, of the equation ϕ(r) := r2 (2δ(θ) cos θ − (1 − K)) − 2rδ(θ) + 1 − K = 0.

(6.1.11)

Since ϕ(0) = 1 − K > 0 and ϕ(1) = 2δ(θ)(cos θ − 1) < 0 whenever θ = 0, we see that if 2δ(θ) cos θ = 1 − K, then the unique root of equation (6.1.11) in the interval (0, 1) is δ(θ) − δ 2 (θ) + [(1 − K) − 2δ(θ) cos θ] (1 − K) r∗ = 2δ(θ) cos θ − 1 + K ; δ(θ) − [δ(θ) − (1 − K)]2 + 2δ(θ)(1 − K)(1 − cos θ) = , 2δ(θ) cos θ − (1 − K)

(6.1.12) θ = 0,

because the numerator and denominator of the last expression have the same sign. Finally, if 2δ(θ) cos θ = 1 − K, (6.1.13) we see that ρ(r) =

r (1 − K − 2rδ(θ)) 1 − r2

and r∗ =

1−K = cos θ < 1 2δ(θ)

(6.1.14)

(6.1.15)

whenever θ = 0. Proposition 6.1.1. Let h : B → X be holomorphic with h(0) = 0, let the functions Nr (θ), k(θ) and K(θ) be defined by (6.1.2) and (6.1.3), and let ρ(r) be defined by (6.1.10). Then for all r ∈ (0, r∗ ), where r∗ is defined by (6.1.12) (or (6.1.15) in the case of (6.1.13)), the numbers r and ρ(r) are Bloch radii for the mapping F = I−h. Moreover, the equation ρ (r) = 0 has a unique solution r0 ∈ (0, r∗ ), and so the function ρ(r) attains its maximum ρ0 at this interior point r0 ∈ (0, r∗ ) ⊂ (0, 1).

170

Chapter 6. Some Applications

To find some explicit estimates for r0 and ρ0 , we again exploit Theorem 2.6.2, but using another approach in order to simplify our calculations. Namely, applying Theorem 2.6.2 with R = 1, we see that for any fixed s ∈ (0, r∗ ), Ns (h) = sup Reh(x), x∗  ≤ s2 [K + L(θ, s)δ(θ)] , x=s

where δ(θ) = N (θ) − k(θ) and N (θ), k(θ) and K are given by (6.1.2) and (6.1.3). On the other hand, if we set R = s in Lemma 2.6.1, we get that for any x such that x = r < s, Re h(x), x∗  ≤ r2 sup Re h (0)u, u∗  · (1 − L(0, r)) + u=1

  ! 2r 1 2r ≤ r2 K 1 − · Ns (h) + 2 s+r s s+r ! 2r N s − r s (h) = r2 K + · s+r s+r s2 ! s−r 2r ≤ r2 K + [K + L(θ, s)δ(θ)] . s+r s+r

r2 L(0, r)Ns (h) s2

(6.1.16)

To simplify further our calculations we denote K1 (= K1 (θ, s)) := K + L(θ, s)δ(θ) = K + Q, Q (= Q(θ, s)) := L(θ, s)δ(θ) > 0, and set R = s < 1. Then (6.1.16) becomes sup Re h(x), x∗  ≤ r2

x=r

s−r 2r K+ K1 s+r s+r

!

r2 = [(s − r)K + 2rK1 ] . s+r

(6.1.17)

Now we again consider inequality (6.1.6) for r ∈ (0, s], taking into account (6.1.17). We then obtain sup Re G(x, z), x∗  ≤ z · r − r2 +

x=r

r2 [(s − r)K + 2rK1 ] < 0 s+r

whenever z < ρs (r) =

r2 (1 + K − 2K1 ) + rs(1 − K) Ar2 + Br = , s+r s+r

(6.1.18)

where A (= A(s)) = 1 + K − 2K1 = 1 + K − 2(K + Q) = 1 − K − 2Q and B (= B(s)) = s(1 − K). It is important to observe that for each fixed s ∈ (0, r∗ ), the following relations hold:

6.1. Bloch radii

171

(i) ρ(s) ≥ ρs (r), while (ii) ρ(s) = ρs (s). However, the investigation of the function ρs (r), r ∈ (0, s], in order to find its maximum value can be done explicitly via quadratures. Consider the function A(s) = 1 − K − 2Q(s) =

(4δ(θ) cos θ − (1 − K))s2 − 4δ(θ)s + (1 − K) . 1 − s2

Note that A(0) = 1 − K > 0, while (1 − K)(1 − r∗2 ) − 4δ(θ)r∗ (1 − r∗ cos θ) 1 − r∗2 2 (1 − K)(1 − r∗ ) − 2δ(θ)r∗ (1 − r∗ cos θ) 2δ(θ)r∗ (1 − r∗ cos θ) = − 1 − r∗2 1 − r∗2 ρ(r∗ ) 2δ(θ)r∗ (1 − r∗ cos θ) 2δ(θ)r∗ (1 − r∗ cos θ) = − =− < 0, 2 r∗ 1 − r∗ 1 − r∗2

A(r∗ ) =

where r∗ is the unique positive root of the equation ρ(r) = 0 in (0, 1) defined by (6.1.12). So the minimal positive root 2δ(θ) − 4δ(θ)2 − (4δ(θ) cos θ − (1 − K))(1 − K) s∗ := 4δ(θ) cos θ − (1 − K) of the equation A(s) = 0 belongs to (0, r∗ ). Moreover, for s ∈ (0, s∗ ), A(s) > 0 or,  1−K which is one and the same, Q < 1−L . 2 , and for s ∈ (s∗ , r∗ ), A(s) < 0 Q > 2 As we have mentioned above, for all s ∈ (0, 1), we have ρ(s) = ρs (s). In s∗ (1 − K) B(s∗ ) particular, ρ(s∗ ) = ρs∗ (s∗ ) = = > 0. 2 2 Since A(s∗ ) = 0, the inequality sup ReG(x, z), x∗  < 0

x=r

holds whenever z < ρs∗ (r) = Note that in this case, ε(r) =

s∗ (1 − L)r , s∗ + r

x = r ∈ (0, s∗ ].

s∗ + r r = is an affine function. ρ(r) B(s∗ )

Since the derivative ρs∗  (r) =

2s2∗ Q >0 (s∗ + r)2

(Q > 0),

(6.1.19)

172

Chapter 6. Some Applications

the increasing function ρs∗ (r), r ∈ [0, s∗ ], attains its maximum on [0, s∗ ] at the point s∗ , that is, max ρs∗ (r) = ρs∗ (s∗ ) = ρ(s∗ ) =

r∈[0,s∗ ]

(1 − K)s∗ , 2

  ∗ and the pair s∗ , (1−K)s constitutes Bloch radii for F with ε(s∗ ) = 2

2 1−K .

Now we fix s ∈ (0, s∗ ). In this case, inequality (6.1.19) holds whenever z < ρs (r) =

A(s)r2 + B(s)r , s+r

r ∈ (0, s).

The derivative dρs (r) A(s)(s + r)2 + 2s2 Q = > 0, dr (s + r)2

r ∈ (0, s),

because A(s) > 0 for s ∈ (0, s∗ ). Hence, ρ is increasing on [0, s], attains its maximum on [0, s] at the point s, that is, 2δ(θ)s2 (1 − s cos θ) , 1 − s2 r∈[0,s]   cos θ) and for each s ∈ (0, s∗ ), the pair s, (1 − K)s − 2δ(θ)(1−s constitutes Bloch 1−s2 radii for F . Next we fix s ∈ (s∗ , r∗ ) and let max ρs (r) = ρs (s) = ρ(s) = (1 − K)s −

z < ρs (r) =

A(s)r2 + B(s)r , s+r

r ∈ (0, s).

In this case, A(s) < 0 and so the equation A(s)(s + r)2 + 2s2 Q dρs (r) = =0 dr (s + r)2 or, which is one and the same, (s + r)2 = −

2s2 Q , A(s)

makes sense. It can be seen that its minimal positive solution 6< 7 2Q r0 = −1 s 2Q − (1 − K) belongs to (0, s) if and only if Q > 1 − K > 0 and

2

d ρs (r) dr 2

2

2 3 (1

− K). Since ρs (0) = 0,



dρs (r)  dr 

r=0+

=

4s Q = − (s+r) 3 < 0 in (0, s), the function ρs (r) attains its

6.1. Bloch radii

173

0 0 maximum

0 on0 [0,  s] at the point r , is positive and increasing on [0, r ], and the pair r , ρs (r ) constitutes Bloch radii for F . If Q ≤ 23 (1 − K), ρs (r) does not vanish in (0, s), and since ρs (0) = 1 − K > 0, ρs is increasing on [0, s] and attains its maximum on [0, s] at the point s, that is,

max ρs (r) = ρs (s) = ρ(s) = s(1 − K − Q) ≥

r∈[0,s]

s (1 − K) > 0, 3

and the pair (s, s(1 − K − Q(s))) constitutes Bloch radii for F . Now we summarize our conclusions in the following assertion. Theorem 6.1.1. Let F = I − h, where h ∈ Hol(B, X) with h(0) = 0, the functions Nr (θ), k(θ) and K(θ) be defined by (6.1.2) and (6.1.3), and let K = K(0) < 1. Then for each s ∈ (0, r∗ ), where r∗ is defined by (6.1.12) (or (6.1.15) in the case of (6.1.13)), the function ρs (r) defined by (6.1.18) is positive on the interval (0, r∗ ) and satisfies the conditions ρ(s) ≥ ρs (r) and ρ(s) = ρs (s). Hence the pair (r, ρs (r)) constitutes Bloch radii for F . Moreover, the following assertions hold. 2δ(θ) − 4δ(θ)2 − (4δ(θ) cos θ − (1 − K)) (1 − K) (a) if s∗ := , then 4δ(θ) cos θ − (1 − K) r s∗ + r ε(r) = is an affine function, namely, ε(r) = ; ρs∗ (r) s∗ (1 − K) (b) if s ∈ (0, s∗ ), then ρs (r), r ∈ (0, s], is strictly increasing and hence, max ρs (r) = ρs (s) = ρ(s) = (1 − K)s −

r∈(0,s]

2δ(θ)s2 (1 − s cos θ) ; 1 − s2

(c) if s ∈ (s∗ , r∗ ), then ⎧ ⎪ ⎨ρs (r0 ), max ρs (r) =

r∈(s∗ ,s]

6< where r0 =

⎪ ⎩ρ(s),

2 (1 − K) 3 , 2 Q ≤ (1 − K) 3 Q>

7 2Q − 1 s. 2Q − (1 − K)

(1 − K)s∗ ≤ ρ0 (where ρ0 is the Bloch radius given 2 by Proposition 6.1.1) is sharp as the following example shows. Note that the estimate

Example 6.1.1. For θ = r∗ =

1 . Then 2

r(1 − 2r) π , K = 0 and δ(θ) = 1, we have ρ(r) = and so 3 1 − r2

ρs (r) =

Ar2 + Br , s+r

174

Chapter 6. Some Applications

s(2 − s) s2 − 4s + 1 where A = 1 − 2Q with Q = . Hence A = and the unique 2 1−s 1 − s2 √ 1 zero of A in (0, 1) is s∗ = 2 − 3 < r∗ = . 2

For 0 < s < s∗ , A(s) > 0 0 < Q < 12 and, in this case, ρs (r) =

A(s + r)2 + 2s2 Q ≥ 0. (s + r)2

Consequently, the function ρs is increasing on (0, s) and max ρs (r) = ρs (s) = ρ(s) =

r∈[0,s]

Furthermore, ρ (r) =

s(1 − 2s) . 1 − s2

(r − 2)2 − 3 (1 − r2 )2

√ and so the unique maximum of ρ in (0, 1) is achieved at r0 = 2 − 3 and equals √ √ 2− 3 ρ(2 − 3) = . 2 √ √ On the other hand, it is easy to calculate that s∗ = 2 − 3 and ρs∗ (s∗ ) = 2−2 3 = √ ρ(2 − 3). 2r Remark 6.1.1. If θ = 0, the factor eiθ − L(θ, r) = 1 − 1+r = 1−r 1+r in (6.1.7) is real and this estimate can, in fact, be replaced with a sharper one than (6.1.7), namely, ! 2r ∗ 2 1−r sup Reh(x), x  ≤ r K +N , 1+r 1+r x=r

so that we do not need in this case the value k(0) = inf Reh (0)x, x∗  ≤ K. x=1

In its turn formula (6.1.10) becomes  r

(1 − K)(1 − r2 ) − δ(θ)2r(1 − r) 2 1−r r = ((1 − K)(1 + r) − 2δ(θ)r) . 1+r

ρ(r) =

In this situation ρ(0) = 0, ρ(1) = 1 − K − δ(θ) = 1 − N and ρ (0) = 1 − K > 0. So, ρ(r) vanishes in (0, 1) if and only if N > 1. Otherwise, if N ≤ 1, then ρ(r) ≥ 0 for all r ∈ (0, 1).

6.2. Radii of starlikeness and spirallikeness

In particular, if N ≤

175

1 (1 + K) < 1, then also ρ (r) > 0 for all r ∈ (0, 1) and 2

max ρ(r) = ρ(1) = 1 − N > 0.

r∈[0,1]

Finally, if N = 1, then ρ(r) = and

r r(1 − r) ((1 − K)(1 + r) − 2(1 − K)r) = (1 − K) 1+r 1+r √ √ max ρ(r) = ρ( 2 − 1) = ( 2 − 1)2 (1 − K).

r∈[0,1]

In general, setting in this case s = 1, we arrive at the following assertion. Theorem 6.1.2. Let h be a holomorphic mapping on the open unit ball B of X with h(0) = 0 and K = sup Re h (0)x, x∗  < 1. x=1

If N = sup Re h(x), x∗  , x 0, then h is said to be planispirallike, or μ-spirallike. In particular, if A = −I, then h is said to be starlike. Observe that a locally biholomorphic mapping h : B → X with h(0) = 0 is A-spirallike if and only if it satisfies the differential equation Ah(x) = h (x)f (x)

(6.2.1)

for some holomorphically dissipative mapping f : B → X (see, for example, [217, 66, 92, 191]). This equation provides the one-to-one correspondence between spirallike mappings h and holomorphically dissipative mappings f . Sometimes h is called the Kœnigs function associated with the semigroup S generated by f . Since h(0) = 0 and h (0) is an invertible linear operator, we get that f (0) = 0

and h (0)f  (0) = Ah (0).

(6.2.2)

Since in this section we essentially deal with planispirallike mappings, we set A = −μI in (6.2.1) and (6.2.2). This implies that f  (0) = −μI. Moreover, we assume without loss of generality that μ = eiθ , where |θ| < π2 . The classical one-dimensional result of Grunsky (see, for example, [84]) asserts that any univalent function on the open unit disc is starlike on the disc centered at the origin with radius r∗ = tanh π4 ≈ 0.65. Appealing to arguments which are similar to those used in the standard proof of Grunsky’s theorem, we can generalize it as follows. Theorem 6.2.1. Let h ∈ Hol(D, C), h(0) = 0, be a univalent function on the open unit disk D, and let θ ∈ − π2 , π2 . Then the function hr defined by hr (z) = 1r h(rz) is μ-spirallike whenever   π |θ| −1 r ≤ tanh − . (6.2.3) 4 2 In multi-dimensional settings radii problems seem to be more complicated. For instance, the radius of starlikeness is studied in [92] for a very special class of mappings obtained as an extension of one-dimensional functions (so-called extension operators). Below we discuss the following problem. Let μ1 , μ2 ∈ C with Re μi > 0 be given. Let h ∈ Hol(B, X) be a μ1 -spirallike mapping on B. Find r ∈ (0, 1) such that h is μ2 -spirallike on the ball Br . In the particular case where μ2 is real we find the radius of starlikeness. From the point of view of equation (6.2.1), we can reformulate our problem as follows.

6.2. Radii of starlikeness and spirallikeness

177

Let θ1 , θ2 satisfy |θi | < π2 . Let h : B → X be a locally biholomorphic mapping on B with h(0) = 0, which satisfies equation (6.2.1) with A = −eiθ1 I and a holomorphically dissipative f , that is, Ref (x), x∗  ≤ 0,

x ∈ B, x∗ ∈ J(x).

(6.2.4)

Find r ∈ (0, 1) (depending on θ1 , θ2 ) and a mapping f1 : Br → X with Ref1 (x), x∗  ≤ 0,

x ∈ Br , x∗ ∈ J(x),

(6.2.5)

such that h also satisfies the equation −eiθ2 h(x) = h (x)f1 (x) whenever x ∈ Br . It is clear that due to the uniqueness property of holomorphic solutions of differential equations, e−iθ2 f1 (x) = e−iθ1 f (x), whence we see that for all x ∈ B and x∗ ∈ J(x), Reei(θ1 −θ2 ) f1 (x), x∗  ≤ 0. Fix u ∈ ∂B and consider the function g ∈ Hol(D, C) defined by g(z) = f (zu), u∗. It follows from (6.2.4) that g is holomorphically dissipative on the open unit disk D, and satisfies g(0) = 0 and g  (0) = f  (0)u, u∗  = −eiθ1 . Hence, z by Remark 2.5.3, it can be represented in the form g(z) = − p(z) , where Re p(z) > −iθ1 0, z ∈ D. In addition, p(0) = e . Thus, by the Riesz–Herglotz formula, there is a probability measure dσ on the unit circle ∂D such that B p(z) = cos θ1 ∂D

1 + zζ dσ(ζ) − i sin θ1 . 1 − zζ

Then for any r ∈ (0, 1), ei(θ1 −θ2 ) p(rz) = ei(θ1 −θ2 )

B ∂D

cos θ1 (1 + rzζ) − i sin θ1 (1 − rzζ) dσ(ζ). 1 − rzζ

This expression has positive real part in D for all measures dσ (that is, for any dissipative f ∈ Hol(B, X) with the same arg(−f  (0)) = θ1 and any u ∈ ∂B) if and only if 

 Re ei(θ1 −θ2 ) (1 − rzζ) cos θ1 (1 + rzζ) − i sin θ1 (1 − rzζ) > 0

178

Chapter 6. Some Applications

for all ζ ∈ ∂D and z ∈ D. Dividing by cos θ2 , we estimate as follows: 

 1 Re ei(θ1 −θ2 ) (1 − rzζ) e−iθ1 + rzζeiθ1 cos θ2 



  1 = Re ei(θ1 −θ2 ) e−iθ1 + 2ir Im zζeiθ1 − r2 |z|2 eiθ1 cos θ2 |sin(θ1 − θ2 )| cos(2θ1 − θ2 ) 2 2 ≥1−2 r|z| − r |z| cos θ2 cos θ2    | sin(θ1 − θ2 )| + cos θ1 | sin(θ1 − θ2 )| − cos θ1 = 1− r|z| 1− r|z| . cos θ2 cos θ2 The latter product is non-negative for all z ∈ D if and only if r≤

cos θ2 . | sin(θ1 − θ2 )| + cos θ1

If this is the case, then Ref1 (rzu), u∗  ≤ 0, so (6.2.5) holds for all x ∈ Br , x∗ ∈ J(x). Thus we have proved the following result. Theorem 6.2.2. Let μ1 , μ2 ∈ C with Re μi > 0 be given. Let h ∈ Hol(B, X) be a μ1 -spirallike mapping on B. Then the mapping hr defined by hr (x) = 1r h(rx) is μ2 -spirallike whenever r ≤ r(θ1 , θ2 ), where θ1 = arg μ1 , θ2 = arg μ2 and r(θ1 , θ2 ) :=

cos θ2 . | sin(θ1 − θ2 )| + cos θ1

The following example shows that the radius of eiθ2 -spirallikeness obtained in this theorem is sharp. Example 6.2.1. Let μi = eiθi , i = 1, 2. Let u ∈ ∂B and u∗ ∈ J(u). Consider 1 the mapping h : B → X, defined by h(x) = x. It can be easily (1 − x, u∗ )1+μ1 2 seen that h is μ1 -spirallike on B. A direct calculation shows that the mapping hr := 1r h(r·) is not μ2 -spirallike whenever r > r(arg μ1 , arg μ2 ). The particular cases where θ1 = 0 or θ2 = 0 are of special interest since they give us the radius of spirallikeness for starlike mappings and, respectively, vice versa. Corollary 6.2.1. Let h ∈ Hol(B,X) be a locally biholomorphic mapping with h(0) = 0. (a) Suppose that h is starlike. Let |θ|
0 be given. Let h ∈ Hol(B, X) be a μ-spirallike mapping on B. Then the mapping hr defined by hr (x) = 1r h(rx) is μ-spirallike whenever 1 r≤ . (6.2.6) 2 sin θ + 1 Consequently, for any μ ∈ C with Re μ > 0, for each μ-spirallike mapping h, the mapping h1/3 is ν-spirallike when | arg ν| ≤ | arg μ|.

6.3 Analytic extension of one-parameter semigroups In this section we discuss the analytic extension of nonlinear semigroups of holomorphic mappings with respect to their parameter to a domain in the complex plane. If we require the extended family to preserve the semigroup property, this domain should be a sector in the complex plane with vertex at the origin. Of course, not every semigroup admits an analytic extension. So, our first aim is to find conditions providing such an extension. If the semigroup generator has an isolated null point inside the domain, then there exists a family of subdomains which are invariant under the semigroup action, that is, hyperbolic balls around this null point. It turns out that in this case the restriction of any generated semigroup to these subdomains extends analytically to some sector. At the same time, if the semigroup generator has no interior null point, the situation is absolutely different. Namely, there are semigroups such that their restrictions to invariant domains do not possess an analytic extension with respect to the parameter (see the next Section 6.4). It should also be mentioned that our approach is partially based on the analytic extension with respect to the parameter of the so-called nonlinear resolvent which is of independent interest by itself. We show that the existence domain of the resolvent which preserves some subdomain is much wider than the domain of analyticity of the semigroup induced

by  using the nonlinear exponential formula. π Recall that for given θ1 , θ2 ∈ 0, 2 , we have set Λ(θ1 , θ2 ) = {ζ ∈ C : −θ1 < arg ζ < θ2 }

(6.3.1)

and Λ(α) = Λ(α, α). Definition 6.3.1 (cf. Definition 1.4.1). A family S = {Fζ }ζ∈Λ of holomorphic selfmappings of a domain D ⊂ X indexed by a parameter ζ in a sector Λ = Λ(θ1 , θ2 )∪ {0} of the complex plane is said to be a one-parameter analytic semigroup if (i) ζ → Fζ is analytic in Λ; (ii) lim Fζ = F0 = I; Λ ζ→0

(iii) Fζ1 +ζ2 = Fζ1 ◦ Fζ2 whenever ζ1 , ζ2 ∈ Λ.

180

Chapter 6. Some Applications

Multiplying the semigroup parameter by a unimodular complex constant, we do not lose any generality if we consider semigroups which are analytic only in symmetric sectors of the form Λ(θ). Our approach to the analytic extension of semigroups of holomorphic selfmappings is based on the following two results. Theorem 6.3.1. Let S = {Ft }t≥0 be a semigroup of holomorphic self-mappings of B generated by f : B → X with f (0) = 0 and f  (0) = −I. Then S extends holomorphically to a sector Λ = {| arg λ| < α ≤ π/2} in C if and only if, for each ϕ satisfying |ϕ| < α, the mapping eiϕ f is holomorphically dissipative on B. As a matter of fact, it is sufficient to verify that the mappings e±iα f are dissipative on B. Proof. Assume that, for any ϕ satisfying |ϕ| < α, the holomorphic mapping eiϕ f is dissipative on B. By Theorem 4.1.1, for each t ≥ 0 one can define the resolvent J = Jt (eiϕ f ) = (I − teiϕ f )−1 , which is a holomorphic self-mapping of B. In other words, for each λ = teiϕ in Λ, the equation x − λf (x) = y has a unique solution x = x(y, λ) ∈ B for any y ∈ B. One defines J(y) := x(y, λ) for y ∈ B. For each t > 0, the fixed point set of the resolvent J is known to coincide with the null point set of the generator eiϕ f (see Section 4.2). Therefore, in our situation the value x(0, λ) just amounts to zero for all λ ∈ Λ. On the other hand, the mapping g(·, y, λ) : B → X defined by g(x, y, λ) = y − (x − λf (x)) for fixed (y, λ) ∈ B × Λ is dissipative on B, since Reg(x, y, λ), x∗  = Rey, x∗  − x2 + Reλf (x), x∗  < 0 on each sphere x = r with y < r < 1. For each λ ∈ Λ, we get g(0, 0, λ) = 0 and the linear operator gx (0, 0, λ, 0) = −I + λf  (0) = −(1 + λ)I is invertible. Hence it follows from a global version of the Implicit Function Theorem for holomorphic generators (see, for instance, [191] or Lemma 6.1.1) that the (unique) solution x(y, λ) = (I −λf )−1 of the equation g(x, y, λ) = 0 is holomorphic in (y, λ) ∈ B×Λ. On applying Theorem 4.1.1 we deduce that the limit −n  λ lim I − f (y) =: Fλ y n→∞ n exists for all (y, λ) ∈ B × Λ and thus defines an operator family which is analytic in λ ∈ Λ and satisfies lim Fteiϕ (y) = y, t→0+

F(t+s)eiϕ = Fteiϕ ◦ Fseiϕ for all y ∈ B and ϕ with |ϕ| < α. The last equality means that Fλ preserves the semigroup property on any ray λ = teiϕ in Λ. Since a ray is a uniqueness set for

6.3. Analytic extension of one-parameter semigroups

181

analytic functions, we readily conclude that Fλ+κ = Fλ ◦ Fκ is actually valid for all κ and λ in Λ. Moreover, if |ϕ| < α, then F (teiϕ )(x) − x t→0+ t

eiϕ f (x) = lim or, what is one and the same,

f (x) = lim

λ→0

Fλ (x) − x , λ

where λ tends to zero along the ray λ = teiϕ . Converse considerations, where we use the last two formulas complete the proof.  To formulate our next result we need the definition of a starlike mapping. For mappings of the complex plane it was first introduced in [214, 30]. Definition 6.3.2. A biholomorphic mapping h ∈ Hol(B, X) satisfying h(0) = 0 and h (0) = I is said to be strongly starlike of order α, where 0 ≤ α < 1, if it satisfies | arg(h (x))−1 h(x), x∗ | ≤ (1 − α)

π 2

(6.3.2)

for all x ∈ B \ {0}. Theorem 6.3.2. Let S = {Ft }t≥0 be a semigroup of holomorphic self-mappings of B generated by a mapping f : B → X satisfying f (0) = 0 and f  (0) = −I. Then S can be analytically extended to a sector Λ = {| arg λ| < α(π/2)} in the complex plane with 0 < α < 1 if and only if the Kœnigs function h associated with S is strongly starlike of order α. Proof. In our settings, equation (6.2.2) becomes h (x)f (x) = −h(x). So we get by (6.3.2), | arg−f (x), x∗ | ≤ (1 − α) whence

π , 2

| arg−eiϕ f (x), x∗ | = |ϕ + arg−f (x), x∗ | ≤

π . 2

Therefore, Reeiϕ f (x), x∗  ≤ 0, which means that the mapping e∗ϕ f is dissipative. The converse arguments, when combined with Theorem 6.3.1, complete our proof.  Recall that by Proposition 2.5.4, a mapping f ∈ Hol(B, X) is holomorphically dissipative on B if and only if its restriction to every complex line, that is, the function g : D → C, given by g(z) = f (zx), x∗ , is dissipative on the disk D. In

182

Chapter 6. Some Applications

the case of strongly convex domains in Cn , this fact holds for the restriction to any complex geodesic (see Proposition 4.5 in [23]). We now consider the following problem. Let f : B → X be a holomorphically dissipative mapping of the ball B with f (0) = 0 and f  (0) = −I. For a given r ∈ (0, 1), find the set of all complex numbers λ such that the associated resolvent (λI − f )−1 (λI) is a well-defined holomorphic self-mapping of Br . (Here Br stands for the open ball of center 0 and radius r in X.) To this end, fix y ∈ Br and consider the resolvent equation λx − f (x) = λy. This equation has a unique solution in Br if and only if the mapping g defined by g(x) = λy −(λx−f (x)) has a unique null point in Br . (By an abuse of notation, we write g(x) instead of g(x, y, λ).) By Theorem 2.5.7, the latter condition is satisfied if there is ϕ ∈ [0, 2π) depending on λ, such that the inequality Reeiϕ g(x), x∗  < 0 holds whenever x = x∗  = r. We compute Reeiϕ g(x), x∗  = Reeiϕ (λy − λx + f (x)), x∗  = Reeiϕ λy, x∗  − Reeiϕ λx, x∗  + Reeiϕ f (x), x∗  ≤ |λy, x∗ | − Reeiϕ λx, x∗  + Reeiϕ f (x), x∗ . Using Proposition 2.5.1, we get   1 + r2 2r iϕ ∗ 2 iϕ Ree g(x), x  ≤ r |λ| − Re(e λ) − cos ϕ + . 1 − r2 1 − r2 So, the inequality Reeiϕ g(x), x∗  < 0 is fulfilled provided that cos ϕ Re λ − sin ϕ Im λ + cos ϕ Set a := Re λ +

1 + r2 2r > |λ| + . 2 1−r 1 − r2

(6.3.3)

1 + r2 , 1 − r2

b := Im λ. Now, for some fixed λ ∈ C, we wish to check whether there is ϕ such that inequality (6.3.3) holds. Since   a b 2 2 a cos ϕ − b sin ϕ = a + b √ cos ϕ − √ sin ϕ , a2 + b 2 a2 + b 2

6.3. Analytic extension of one-parameter semigroups

183

then, on denoting a cos α := √ 2 a + b2

and

we get a cos ϕ − b sin ϕ =

b sin α := √ , 2 a + b2

a2 + b2 cos(ϕ + α).

Choosing ϕ = −α, we conclude that the inequality Reeiϕ g(x), x∗  < 0 is fulfilled if 2r a2 + b2 > |λ| + . 1 − r2 Substituting the formulas for a and b, we get 2 Re λ or

4r 1 + r2 +1> |λ| 1 − r2 1 − r2

 2 1 + r2 Re λ + (Im λ)2 2(1 − r2 ) − > 1. 2 1 r 4 (1 − r2 )2

(6.3.4)

The corresponding domain Ω in the plane of the complex variable λ is bounded by a hyperbola. Summing up, we arrive at the following theorem. Theorem 6.3.3. Let f : B → X be a dissipative holomorphic mapping on B satisfying f (0) = 0 and f  (0) = −I. Then, for each r ∈ (0, 1) and λ in the domain Ω given by (6.3.4), the nonlinear associated resolvent Jλ (f ) := (λI − f )−1 λI is a well-defined holomorphic self-mapping of the ball Br . It is easy to see that Ω contains a keyhole domain of the form Ω ∪ Ω , where ' ( 1 1−r  Ω = λ ∈ C : |λ| < , 2 1+r ' ( 1 − r2  Ω = λ ∈ C \ {0} : | arg λ| < arcsin 1 + r2 are a disk around the origin and a sector around the non-negative semiaxis in the complex plane, respectively. Theorem 6.3.4. Under the hypotheses of Theorem 6.3.3, the associated resolvent mapping Jλ (f ) is holomorphic in λ ∈ Ω ∪ Ω . Proof. The mapping g defined by g(x, y, λ) = λy − (λx − f (x)) depends holomorphically on (x, y, λ) ∈ Br × Br × Ω. Since x = Jλ (f )y is a solution of g(x, y, λ) = 0, we want to show that the mapping g(·, y, λ) is dissipative on Br , that is, sup Reg(x, y, λ), x∗  < 0

x=r

for each fixed pair (y, λ) ∈ Br × (Ω ∪ Ω ).

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Chapter 6. Some Applications

Indeed, if λ ∈ Ω , then on applying Proposition 2.5.1 with ϕ = 0 we get sup Reg(x, y, λ), x∗  ≤ sup |λ|y − xr − r2

x=r

x=r