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Hilbert-Schmidt Hankel operators on the Segal-Bargmann space


Author: Wolfram Bauer
Journal: Proc. Amer. Math. Soc. 132 (2004), 2989-2996
MSC (2000): Primary 47B35
DOI: https://doi.org/10.1090/S0002-9939-04-07264-8
Published electronically: June 2, 2004
MathSciNet review: 2063120
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Abstract: This paper considers Hankel operators on the Segal-Bargmann space of holomorphic functions on $\mathbb{C}^n$ that are square integrable with respect to the Gaussian measure. It is shown that in the case of a bounded symbol $g \in L^{\infty}(\mathbb{C}^n)$ the Hankel operator $H_g$ is of the Hilbert-Schmidt class if and only if $H_{\bar{g}}$ is Hilbert-Schmidt. In the case where the symbol is square integrable with respect to the Lebesgue measure it is known that the Hilbert-Schmidt norms of the Hankel operators $H_g$ and $H_{\bar{g}}$ coincide. But, in general, if we deal with bounded symbols, only the inequality $\Vert H_g\Vert _{HS}\leq2\Vert H_{\bar{g}}\Vert _{HS}$ can be proved. The results have a close connection with the well-known fact that for bounded symbols the compactness of $H_g$ implies the compactness of $H_{\bar{g}}$.


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Additional Information

Wolfram Bauer
Affiliation: Department of Mathematics, State University of New York, Buffalo, New York 14260
Address at time of publication: Johannes Gutenberg Universität Mainz, Fachbereich Mathematik und Informatik, Staudinger Weg 9, 55128 Mainz, Germany
Email: BauerWolfram@web.de

DOI: https://doi.org/10.1090/S0002-9939-04-07264-8
Received by editor(s): July 10, 2002
Received by editor(s) in revised form: February 15, 2003
Published electronically: June 2, 2004
Additional Notes: This work was supported by a fellowship of the \itshape“Deutscher akademischer Austauschdienst” (DAAD)
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society

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