Toeplitz Operators and Index Theory in Several Complex Variables

ISBN-13: 9783764352820 / Angielski / Twarda / 1995 / 483 str.

H. Upmeier; Harald Upmeier

Opis

This book gives a comprehensive treatment of Toeplitz operators arising in multivariable complex analysis. The first part describes in detail the underlying geometric structures (strongly pseudoconvex domains, Reinhardt domains, multivariable upper half-planes, symmetric domains and generalizations) as well as the harmonic analysis of the associated Hilbert spaces of holomorphic functions (of Hardy or Bergman type). It is also of interest to mathematicians not primarily interested in operator theory. The second part of the book determines the structure of Toeplitz operators, their C*-algebras and index theory, using modern techniques such as groupoid C*-algebras, co-crossed products and operator K-theory.

Spis treści

1. Multi-variable Complex Analysis and Domains of Holomorphy.- 1.0 Introduction.- 1.1 Holomorphic Functions in Several Complex Variables.- 1.2 Pseudoconvex Domains.- 1.3 Tubular Domains.- 1.4 Polycircular Domains.- 1.5 Symmetric Domains.- 1.6 K-circular Domains.- 1.7 S-bicircular Domains.- 2. Harmonic Analysis on Hilbert Spaces of Holomorphic Functions.- 2.0 Introduction.- 2.1 Bergman Spaces Over Pseudoconvex Domains.- 2.2 Hardy Spaces Over Strictly Pseudoconvex Domains.- 2.3 Hardy Spaces Over Tubular Domains.- 2.4 Bergman Spaces Over Tubular Domains.- 2.5 Hardy Spaces Over Polycircular Domains.- 2.6 Bergman Spaces Over Polycircular Domains.- 2.7 The Segal-Bargmann Space of a Hermitian Vector Space.- 2.8 Hardy Spaces Over Symmetric Domains.- 2.9 Bergman Spaces Over Symmetric Domains.- 2.10 Hardy Spaces Over K-circular Domains.- 2.11 Hardy Spaces Over S-bicircular Domains.- 3. Multiplier C*-Algebras and Their Representations.- 3.0 Introduction.- 3.1 Hardy Multipliers Over Tubular Domains.- 3.2 Bergman Multipliers Over Tubular Domains.- 3.3 Hardy Multipliers Over Polycircular Domains.- 3.4 Bergman Multipliers Over Polycircular Domains.- 3.5 Hardy Multipliers Over K-circular Domains.- 3.6 Hardy Multipliers Over Symmetric Domains.- 3.7 Hardy Multipliers Over S-bicircular Domains.- 4. Toeplitz Operators and Toeplitz C*-Algebras.- 4.0 Introduction.- 4.1 Bergman-Toeplitz Operators Over Bounded Domains.- 4.2 Hardy-Toeplitz Operators Over Strictly Pseudoconvex Domains.- 4.3 Groupoid C*-Algebras.- 4.4 Hardy-Toeplitz Operators Over Tubular Domains.- 4.5 Bergman-Toeplitz Operators Over Tubular Domains.- 4.6 Hardy-Toeplitz Operators Over Polycircular Domains.- 4.7 Bergman-Toeplitz Operators Over Polycircular Domains.- 4.8 Hopf C*-Algebras.- 4.9 Actions and Coactions on C*-Algebras.- 4.10 Hardy-Toeplitz Operators Over K-circular Domains.- 4.11 Hardy-Toeplitz Operators Over Symmetric Domains.- 4.12 Bergman-Toeplitz Operators Over Symmetric Domains.- 5. Index Theory for Multivariable Toeplitz Operators.- 5.0 Introduction.- 5 .1 K-Theory for Topological Spaces.- 5.2 Index Theory for Strictly Pseudoconvex Domains.- 5.3 K-Theory for C*-Algebras.- 5.4 Index Theory for Symmetric Domains.- 5.5 Index Theory for Tubular Domains.- 5.6 Index Theory for Polycircular Domains.- References.- Index of Symbols and Notations.