Trace estimation of commutators of multiplication operators on function spaces

Abstract

Let $A=\sum_{k\geq1}T_{\varphi_{k}}T_{\varphi_{k}}^{*}$ be a bounded linear operator on Bergman space $L_{a}^{2}(B_{d})$ or Hardy space $H^{2}(B_{d})$, where $\varphi_{k}$ is a multiplier for each $k$. We will show by trace estimation that for such an operator, $[A,T_{z_{i}}]$ belongs to Schatten class $\mathcal{L}_{2p}$ for $p>d$, and satisfies $\|[A,T_{z_{i}}]\|_{2p}\leq C\|A\|$ for some constant $C$ depending only on $p$ and $d$.