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

TL;DR

This paper characterizes Schatten class Toeplitz and composition operators on small Bergman spaces with radial weights, providing criteria based on measure and kernel properties, and explores their basic properties.

Contribution

It introduces new criteria for Schatten class membership of Toeplitz and composition operators on weighted Bergman spaces with doubling weights.

Findings

Characterization of Schatten class Toeplitz operators via measure and kernel conditions

Description of Schatten class composition operators on small Bergman spaces

Analysis of properties of composition operators between different weighted 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 $A^2_\omega$, 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 $A^p_\omega$ to $A^q_v$.