Generalized Hilbert matrix operators acting on Bergman spaces

TL;DR

This paper characterizes when the generalized Hilbert matrix operator acts boundedly and compactly on Bergman spaces, providing measure conditions, norm estimates, and essential norm calculations.

Contribution

It offers new characterizations of measures for boundedness and compactness of the operator on Bergman spaces, including norm estimates.

Findings

Characterization of measures for boundedness of $\Gamma_\mu$

Norm estimates of the operator

Criteria for compactness via essential norm

Abstract

In this article we study the generalized Hilbert matrix operator $\Gamma_\mu$ acting on the Bergman spaces $A^p$ of the unit disc for $1\leq p<\infty$. In particular, we characterize the measures $\mu$ for which the operator $\Gamma_\mu$ is bounded and we provide estimates of its operator norm. Finally, we also describe when $\Gamma_\mu$ is compact by computing its essential norm.