Composition and Toeplitz operators on general domains
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.
scientific article; zbMATH DE number 2185488