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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a proposed characterization of Schatten-class composition operators


Author: Jingbo Xia
Journal: Proc. Amer. Math. Soc. 131 (2003), 2505-2514
MSC (2000): Primary 47B10, 47B33, 47B38
Published electronically: November 27, 2002
MathSciNet review: 1974649
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Abstract: For an analytic function $\varphi $ which maps the open unit disc $D$ to itself, let $C_{\varphi }$ be the operator of composition with $\varphi $ on the Bergman space $L^{2}_{a}(D,dA)$. It has been a longstanding problem to determine whether or not the membership of $C_{\varphi }$ in the Schatten class ${\mathcal{C}}_{p}$, $1 < p < \infty $, is equivalent to the condition that the function $z \mapsto \{(1-\vert z\vert^{2})/(1-\vert\varphi (z)\vert^{2})\}^{p}$ has a finite integral with respect to the Möbius-invariant measure $d\lambda (z) = (1-\vert z\vert^{2})^{-2}dA(z)$ on $D$. We show that the answer is negative when $2 < p < \infty $.


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Additional Information

Jingbo Xia
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: jxia@acsu.buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06891-0
PII: S 0002-9939(02)06891-0
Received by editor(s): March 22, 2002
Published electronically: November 27, 2002
Additional Notes: This work was supported in part by National Science Foundation grant DMS-0100249
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society