Numerical Range of Holomorphic Mappings and Applications

Description

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. Front Matter ....Pages i-xiv Semigroups of Linear Operators (Mark Elin, Simeon Reich, David Shoikhet)....Pages 1-20 Numerical Range (Mark Elin, Simeon Reich, David Shoikhet)....Pages 21-62 Fixed Points of Holomorphic Mappings (Mark Elin, Simeon Reich, David Shoikhet)....Pages 63-95 Semigroups of Holomorphic Mappings (Mark Elin, Simeon Reich, David Shoikhet)....Pages 97-128 Ergodic Theory of Holomorphic Mappings (Mark Elin, Simeon Reich, David Shoikhet)....Pages 129-164 Some Applications (Mark Elin, Simeon Reich, David Shoikhet)....Pages 165-205 Back Matter ....Pages 207-229