Hilbert-Schmidt Hankel Operators and Berezin Iteration

Abstract

Let $H$ be a reproducing kernel Hilbert space contained in a wider space $L^2(X,\mu)$. We study the Hilbert-Schmidt property of Hankel operators $H_g$ on $H$ with bounded symbol $g$ by analyzing the behavior of the iterated Berezin transform. We determine symbol classes $\mathcal{S}$ such that for $g\in \mathcal{S}$ the Hilbert-Schmidt property of $H_g$ implies that $H_{\bar{g}}$ is a Hilbert-Schmidt operator as well and there is a norm estimate of the form $\|H_{\bar{g}}\|_{\text{HS}}\leq C\cdot \| H_g\|_{\text{HS}}$. Finally, applications to the case of Bergman spaces over strictly pseudo convex domains in $\mathbb{C}^n$, the Fock space, the pluri-harmonic Fock space and spaces of holomorphic functions on a quadric are given.