Composition and Toeplitz operators on general domains
type: Item
instance of: scholarly article
title: Composition and Toeplitz operators on general domains
zbMATH Open document ID: 1087.47057
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-07-01T00:00:00Z
review text: Let \(M\) be a bounded, open, connected, non-empty subset of \(\mathbb C\), and \({\mathcal T}(M)\) be the algebra generated by the set of Toeplitz operators \(\{T_{\phi}\mid \phi\in C(\overline M)\}\). The aim of the present paper is to examine the problems of invertibility and Fredholm properties of composition operators and Toeplitz operators on general domains of \(\mathbb C\). The characterizations of invertible and Fredholm composition operators are obtained in terms of the topological boundary \(\partial (M)\) and the Bergman essential boundary \(\partial _r(M)\) of the domain \(M\). The isomorphism between Toeplitz algebras and their \(K\)-theories is discussed. One of the main results of the paper is the following. Theorem 9. Suppose that \(M\) and \(N\) are finitely connected domains. If \(K_0({\mathcal T}(M)) \cong K_0({\mathcal T}(N))\), then (1) \(\pi ^1(\partial (M)) \cong \pi ^1(\partial (N))\), (2) \(C(\partial (M)) \cong C(\partial (N))\), (3) \(\partial (M)\) and \(\partial (N)\) are homeomorphic.
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 19K99
zbMATH DE Number: 2185488
zbMATH Keywords: composition operators
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: Bergman essential boundary
author: Guangfu Cao
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113234831
OpenAlex ID: W2034152179
cites work: Toeplitz Operators on Bergman Spaces
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DOI: 10.2748/TMJ/1113234831
description: scientific article; zbMATH DE number 2185488