Schatten classes of generalized Hilbert operators

TL;DR

This paper characterizes the Schatten class membership of generalized Hilbert operators acting on Dirichlet-type spaces induced by radial weights with doubling and Muckenhoupt-type conditions, providing integral criteria for $p ext{-}Schatten$ classes.

Contribution

It provides a complete characterization of Schatten class membership for generalized Hilbert operators on weighted Dirichlet spaces under specific weight conditions.

Findings

Schatten class membership characterized by integral conditions on the symbol's derivative.

Operators belong to $S_p$ if and only if a specific integral involving $g'$ converges.

Results extend understanding of operator theory on weighted analytic function spaces.

Abstract

Let $\mathcal{D}_v$ denote the Dirichlet type space in the unit disc induced by a radial weight $v$ for which $\widehat{v}(r)=\int_r^1 v(s)\,ds$ satisfies the doubling property $\int_r^1 v(s)\,ds\le C \int_{\frac{1+r}{2}}^1 v(s)\,ds.$ In this paper, we characterize the Schatten classes $S_p(\mathcal{D}_v)$ of the generalized Hilbert operators \begin{equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt \end{equation*} acting on $\mathcal{D}_v$, where $v$ satisfies the Muckenhoupt-type conditions $$ \sup_{0<r<1}\left(\int_r^1 \frac{\widehat{v}(s)}{(1-s)^2} \,ds\right)^{1/2} \left(\int_0^r \frac{1}{\widehat{v}(s)} \,ds\right)^{1/2}<\infty $$ and $$\sup_{0< r<1}\left(\int_{0}^r \frac{\widehat{v}(s)}{(1-s)^4}\,ds\right)^{\frac{1}{2}} \left(\int_{r}^1\frac{(1-s)^2}{\widehat{v}(s)}\,ds\right)^\frac{1}{2}<\infty. $$ For $p\ge 1$, it is proved that $\mathcal{H}_{g}\in S_p(\mathcal{D}_v)$ if and only if \begin{equation*} \int_0^1 \left((1-r)\int_{-\pi}^\pi |g'(re^{i\theta})|^2\,d\theta\right)^{\frac{p}{2}}\frac{dr}{1-r} <\infty. \end{equation*}