Schatten class composition operators

Abstract Let $C_{\varphi }$ be a composition operator on the Bergman space $A^2$ of the unit disc. A well-known problem asks whether the condition $\int _D\big ({1-|z|^2\over 1-|\varphi (z)|^2}\big )^pd\lambda (z) < \infty $ is equivalent to the membership of $C_\varphi $ in the Schatten class ${\mathcal {C}}_p$ , $1 < p < \infty $ . This was settled in the negative for the case $2 < p < \infty $ in [3]. When $2 < p < \infty $ , this condition is not sufficient for $C_\varphi \in {\mathcal {C}}_p$ . In this article, we take up the case $1 < p < 2$ . We show that when $1 < p < 2$ , this condition is not necessary for $C_\varphi \in {\mathcal {C}}_p$ .