Numerical Range of Holomorphic Mappings and Applications

238 Pages·2019·2.527 MB·English

by  Mark Elin, Simeon Reich, David Shoikhet

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 Simeon Reich
Department of Mathematics Department of Mathematics
ORT Braude College The Technion - Israel Institute of Technology
Karmiel, Israel 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

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(1946–2017)
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Semigroups of Linear Operators
1.1 Linear operators. Spectrum and resolvent . . . . . . . . . . . . . . 1
1.2 Continuous semigroups and their generators. . . . . . . . . . . . . 3
1.3 Numerical range of linear operators . . . . . . . . . . . . . . . . . 6
1.4 Analytic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Cesa`ro and Abel averages of linear operators . . . . . . . . . . . . 11
1.6 Abel averages:recent results . . . . . . . . . . . . . . . . . . . . . 17
2 Numerical Range
2.1 Holomorphic mappings in Banach spaces . . . . . . . . . . . . . . 21
2.2 Spectrum and resolvent of holomorphic mappings . . . . . . . . . 25
2.3 Numerical range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Real part estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Holomorphically dissipative and accretive mappings . . . . . . . . 41
2.6 Growth estimates for the numerical range . . . . . . . . . . . . . . 49
2.7 Filtration of dissipative mappings . . . . . . . . . . . . . . . . . . 54
3 Fixed Points of Holomorphic Mappings
3.1 Fixed points in the unit disk . . . . . . . . . . . . . . . . . . . . . 64
3.2 Fixed points in the Hilbert ball . . . . . . . . . . . . . . . . . . . . 69
3.3 Boundary fixed points and the horosphere function . . . . . . . . . 71
3.4 Canonical representation of the fixed point set . . . . . . . . . . . 76
3.5 Around the Earle–Hamilton fixed point theorem . . . . . . . . . . 78
3.6 Inexact orbits of holomorphic mappings . . . . . . . . . . . . . . . 83
3.7 The Bohl–Poincar´e–Krasnoselskiitheorem. . . . . . . . . . . . . . 85
3.8 Fixed points of pseudo-contractive holomorphic mappings . . . . . 90
vii
viii Contents
4 Semigroups of Holomorphic Mappings
4.1 Generated semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Stationary points of semigroups . . . . . . . . . . . . . . . . . . . 100
4.3 Flow invariance conditions . . . . . . . . . . . . . . . . . . . . . . 102
4.4 Semi-complete vector fields on bounded symmetric domains . . . . 107
4.5 Rates of convergence of semigroups . . . . . . . . . . . . . . . . . 110
4.6 Semigroups and pseudo-contractive holomorphic mappings . . . . 115
4.7 Semigroups on the Hilbert ball . . . . . . . . . . . . . . . . . . . . 121
5 Ergodic Theory of Holomorphic Mappings
5.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Power bounded holomorphic mappings. . . . . . . . . . . . . . . . 131
5.3 Ergodicity and fixed points . . . . . . . . . . . . . . . . . . . . . . 133
5.4 Numerical range and power boundedness . . . . . . . . . . . . . . 142
5.5 Dissipative and pseudo-contractive mappings . . . . . . . . . . . . 153
5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6 Some Applications
6.1 Bloch radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2 Radii of starlikeness and spirallikeness . . . . . . . . . . . . . . . . 175
6.3 Analytic extension of one-parameter semigroups . . . . . . . . . . 179
6.4 Analytic extension of semigroups without stationary points . . . . 187
6.5 Composition operators and semigroups . . . . . . . . . . . . . . . 192
6.6 Analytic semigroups of composition operators. . . . . . . . . . . . 197
6.7 Semigroups of composition operators on Hp(Π) . . . . . . . . . . . 200
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 er-
godic theory. In particular, this notion plays a crucial role in establishing expo-
nential and product formulas for semigroups of holomorphic mappings, the study
of flow invariance and range conditions, geometric function theory in finite- and
infinite-dimensionalBanachspaces,the study ofcompleteandsemi-completevec-
torfields andtheirapplicationsto severalclassesofbiholomorphicmappings,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 λ > 0 (hence, for all λ > 0), satisfies the range
0
condition
(λ I −A)D =X. (0.0.1)
0
The first condition in this theorem means that the numerical range of the
operatorAliesintheclosedlefthalf-plane.Ifthesecondconditionholdstoo,then
the operator A is often referred to as an m-dissipative operator, and condition
(0.0.1)justmeansthattheresolvent(λ I−A)−1isaboundedlinearoperatoronX.
0
AnonlinearanalogoftheLumer–PhillipsTheoremforholomorphicmappings
andits applications was studied in [103]. It shouldbe mentioned that a linear op-
eratoris holomorphic if and only if it is bounded. Therefore the above-mentioned
nonlinearanalogextendstheLumer–PhillipsaswellastheHille–YosidaTheorems
(see,forexample,[240]). Moreover,itcanbe appliedtolocallyuniformly continu-
oussemigroups.Thereforeanaturalideaistoconsidernonlinearmappingsf (not
necessarily holomorphic) the(nonlinear)resolventofwhichexistsandisnonexpan-
sive 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
manydevelopmentshaveoccurred,inparticular,intheareaofnonexpansivesemi-
groups in Banach spaces. As a rule, such semigroups are generated by dissipative
operatorsandcanbeviewedasnonlinearanalogsoftheclassicallinearcontraction
semigroups. See, for example, [32], [17] and [174].
Anotherclassofnonlinearsemigroupsconsistsofthosesemigroupsgenerated
byholomorphicmappingsincomplexmanifoldsandcomplexBanachspaces.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 provedin[1] that eachcon-
tinuous semigroup of holomorphic mappings is everywhere differentiable with re-
spect to its parameter, that is, it is generated by a holomorphic mapping. In
addition,heestablishedacriterionforaholomorphicmappingtobethegenerator
ofaone-parametersemigroup.Earlier,intheone-dimensionalcase,similarresults
were presented by Berkson and Porta in their study [19] of linear C -semigroups
0
of composition operators on Hardy spaces. Vesentini investigated semigroups of
those fractional-linear transformations which are isometries with respect to the
infinitesimal hyperbolicmetric onthe unitball ofa Banachspace(see [231, 232]).
Heusedthis approachtostudy severalimportantproblemsinthe theoryoflinear
operators on indefinite metric spaces. Note that, generally speaking, such semi-
groups are not everywhere differentiable in the infinite-dimensional case. Since
holomorphic self-mappings of a domain D in a complex Banach space are non-
expansive 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
usingtheEarle–HamiltonFixedPointTheoremandthegeneralizedSchwarz–Pick
Lemma in Banach and Hilbert spaces.
Over the last thirty years these results have been developed in many direc-
tions. One of them concerns increasing the dimension of the underlying space.
Finite-dimensionalextensionsaretobe found,forinstance,inthe papersbyKub-
ota [143], MacCluer [161], Chen [42], Abate [1, 2] and Mercer [168]. In this con-
nection see also [6, 4, 159, 106] and [218]. Infinite-dimensional generalizations
are due, for example, to Fan [75, 76], Wl(cid:4)odarczyk [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.
AnotherdirectionisconcernedwithanaloguesoftheclassicalDenjoy–WolffTheo-
Preface xi
remforcontinuoussemigroups.Thisapproachhasbeenusedbyseveralauthorsto
study the asymptotic behaviorofsolutions to Cauchyproblems (see, for example,
[19, 3, 46] and [186]).
Italsoturnsoutthattheasymptoticbehaviorofsolutionstoevolutionequa-
tions 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(cid:2)(0) (cid:4)= 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(cid:2)(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]anddevelopedbyBrickman[33],whointroducedtheconcept
of Φ-like functions as a generalization of starlike and spirallike functions (with
respectto the origin)ofasingle 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 viewofall ofthis, inthis book we describerecentdevelopmentsaswell 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 inmore detail the numerical rangeandthe spectrumof closeddensely
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 an-
alytic 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 non-
linear semigroups of holomorphic self-mappings of the unit disk.
In the second chapter we first discuss the basic notions and facts in infinite-
dimensional holomorphy and hyperbolic geometry in Banach and Hilbert spaces.
We then presentand generalize Harris’ theory of the numerical range of holomor-
phic 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–
Hamiltonand Bohl–Poincar´e–KrasnoselskiiTheorems including their connections
with Schwarz–Pick systems of pseudometrics and pseudo-contractive mappings.
Wealsopresentasolutiontotheso-calledcoefficientprobleminbranchingstochas-