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

34 pages