On a proposed characterization of Schatten-class composition operators

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 $z \mapsto \{(1-|z|^{2})/(1-|\varphi (z)|^{2})\}^{p}$ has a finite integral with respect to the Möbius-invariant measure $d\lambda (z) = (1-|z|^{2})^{-2}dA(z)$ on $D$. We show that the answer is negative when $2 < p < \infty$.