On singular values of Hankel operators on Bergman spaces

Abstract

In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{\omega _\varphi}$, where $\omega _\varphi= e^{-\varphi}$ and $\varphi$ is a subharmonic function. We consider compact Hankel operators $H_{\overline {\phi}}$, with anti-analytic symbols ${\overline {\phi}}$, and give estimates of the trace of $h(|H_{\overline \phi}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{\overline {\phi}}))_n$ in terms of decreasing rearrangement of $|\phi '|/\sqrt{\Delta \varphi}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{\overline {\phi}}))_n$ is achieved by $(s_n (H_{\overline{z}}))_n$. Namely, we establish that if $s_n(H_{\overline {\phi}})= o (s_n(H_{\overline {z}}))$, then $H_{\overline {\phi}} = 0$. Then, we show that if $\Delta \varphi (z) \asymp \frac{1}{(1-|z|^2)^{2+\beta}}$ with $\beta \geq 0$, then $s_n(H_{\overline {\phi}}) = O(s_n(H_{\overline {z}}))$ if and only if $\phi '$ belongs to the Hardy space $H^p$, where $p= \frac{2(1+\beta)}{2+\beta}$. Finally, we compute the asymptotics of $s_n(H_{\overline {\phi}})$ whenever $ \phi ' \in H^{p }$.