Toeplitz operators in several complex variables |
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Authors: | AM Davie NP Jewell |
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Institution: | Department of Mathematics, University of Edinburgh, 20, Chambers Street, Edinburgh UK |
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Abstract: | Let S be the unit sphere in Cn. We investigate the properties of Toeplitz operators on S, i.e., operators of the form Tφf = P(φf) where φ?L∞(S) and P denotes the projection of L2(S) onto H2(S). The aim of this paper is to determine how far the extensive one-variable theory remains valid in higher dimensions. We establish the spectral inclusion theorem, that the spectrum of Tφ contains the essential range of φ, and obtain a characterization of the Toeplitz operators among operators on H2(S) by an operator equation. Particular attention is paid to the case where φ ? H∞(S) + C(S) where C(S) denotes the algebra of continuous functions on S. Finally we describe a class of Toeplitz operators useful for providing counterexamples—in particular, Widom's theorem on the connectedness of the spectrum fails when n > 1. |
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