Algebraic properties and the finite rank problem for Toeplitz operators on the Segal-Bargmann space |
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Authors: | Wolfram Bauer Trieu Le |
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Affiliation: | a Mathematisches Institut, Georg-August-Universität, Bunsen-str. 3-5, 37073 Göttingen, Germany b Department of Mathematics and Statistics, University of Toledo, Toledo, OH 43606, USA |
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Abstract: | We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the “finite rank problem”. We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role. |
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Keywords: | Commuting operators Finite rank problem Zero-products of Toeplitz operators |
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