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On a proposed characterization of Schatten-class composition operators

Authors:	Jingbo Xia

Institution:	Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260

Abstract:	For an analytic function $\varphi$ which maps the open unit disc 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 . We show that the answer is negative when $2 < p < \infty$ .

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