Abstract
The theory of Toeplitz operators on the Hardy or Bergman space over the unit disk is one of the central topics in operator theory. In this survey article we describe some of the main features concerning the multi-variable case. The main topics are the analysis of Hilbert spaces of holomorphic functions in several complex variables, the C ∗-algebraic structure of Toeplitz operators and aspects of geometric quantization such as the Berezin transform. These topics have been studied for three main classes of domains, namely strongly pseudoconvex domains, bounded symmetric domains and Reinhardt domains. The common intersection of these classes is the unit ball. We also discuss recent developments concerning Hilbert quotient modules, 140related to holomorphic functions vanishing on analytic subvarieties of the underlying domain.
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