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2019 | Book

Semigroups of Linear Operators Read chapter Read first chapter

Numerical Range of Holomorphic Mappings and Applications

Authors: Prof. Dr. Mark Elin, Prof. Simeon Reich, David Shoikhet

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book describes recent developments as well as some classical results regarding holomorphic mappings. The book starts with a brief survey of the theory of semigroups of linear operators including the Hille-Yosida and the Lumer-Phillips theorems. The numerical range and the spectrum of closed densely defined linear operators are then discussed in more detail and an overview of ergodic theory is presented. The analytic extension of semigroups of linear operators is also discussed. The recent study of the numerical range of composition operators on the unit disk is mentioned. Then, the basic notions and facts in infinite dimensional holomorphy and hyperbolic geometry in Banach and Hilbert spaces are presented, L. A. Harris' theory of the numerical range of holomorphic mappings is generalized, and the main properties of the so-called quasi-dissipative mappings and their growth estimates are studied. In addition, geometric and quantitative analytic aspects of fixed point theory are discussed. A special chapter is devoted to applications of the numerical range to diverse geometric and analytic problems.

Table of Contents

Frontmatter
Chapter 1. Semigroups of Linear Operators
Abstract
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].
Mark Elin, Simeon Reich, David Shoikhet
Chapter 2. Numerical Range
Abstract
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.
Mark Elin, Simeon Reich, David Shoikhet
Chapter 3. Fixed Points of Holomorphic Mappings
Abstract
In this chapter we study conditions on a holomorphic mapping F :D →X which ensure that the set D∩F(D) contains a fixed point of F. A standard situation in this study is when F is a holomorphic self-mapping of D.
Mark Elin, Simeon Reich, David Shoikhet
Chapter 4. Semigroups of Holomorphic Mappings
Abstract
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.
Mark Elin, Simeon Reich, David Shoikhet
Chapter 5. Ergodic Theory of Holomorphic Mappings
Abstract
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.
Mark Elin, Simeon Reich, David Shoikhet
Chapter 6. Some Applications
Abstract
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.
Mark Elin, Simeon Reich, David Shoikhet
Backmatter
Metadata
Title
Numerical Range of Holomorphic Mappings and Applications
Authors
Prof. Dr. Mark Elin
Prof. Simeon Reich
David Shoikhet
Copyright Year
2019
Electronic ISBN
978-3-030-05020-7
Print ISBN
978-3-030-05019-1
DOI
https://doi.org/10.1007/978-3-030-05020-7

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