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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 $.

References [Enhancements On Off] (What's this?)

  • 1. C. Cowen and B. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, 1995. MR 97i:47056
  • 2. J. Garnett, Bounded analytic functions, Academic Press, New York-London, 1981. MR 83g:30037
  • 3. S. Li, Trace ideal criteria for composition operators on Bergman spaces, Amer. J. Math. 117 (1995), 1299-1323. MR 96g:47023
  • 4. D. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), 345-368. MR 88m:47046
  • 5. D. Luecking and K. Zhu, Composition operators belonging to the Schatten ideals, Amer. J. Math. 114 (1992), 1127-1145. MR 93i:47032
  • 6. B. MacCluer and J. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), 878-906. MR 87h:47048
  • 7. K. Zhu, Operator theory in function spaces, Marcel Dekker, New York, 1990. MR 92c:47031
  • 8. K. Zhu, Schatten class composition operators on weighted Bergman spaces of the disk, J. Operator Theory 46 (2001), 173-181. MR 2002h:47039

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Additional Information

Jingbo Xia
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260

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

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