On singular values of Hankel operators on Bergman spaces

TL;DR

This paper investigates the asymptotic behavior of singular values of Hankel operators on weighted Bergman spaces, establishing conditions for their decay rates and connections to Hardy space membership.

Contribution

It provides new estimates and asymptotic formulas for singular values of Hankel operators on Bergman spaces, linking decay rates to properties of the symbol and weight functions.

Findings

Critical decay of singular values is achieved by the model operator with symbol z.

Decay rate of singular values relates to Hardy space membership of the symbol derivative.

Asymptotics of singular values are computed for symbols with derivatives in Hardy 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 }$.