Trace class criteria for Toeplitz and composition operators on small Bergman spaces

Abstract

We characterize the Schatten class Toeplitz operators induced by a positive Borel measure on the unit disc and the reproducing kernel of the Bergman space $A_{\omega }^2$, where $\omega$ is a radial weight satisfying the doubling property ${\int }_r^1\omega (s)ds\le C{\int }_{\frac{1+r}{2}}^1\omega (s)ds$. By using this, we describe the Schatten class composition operators. We also discuss basic properties of composition operators acting from $A_{\omega }^p$ to $A_v^q$.

arXiv admin note: text overlap with arXiv:1406.2857