Generalized Hilbert matrix operators acting on Bergman spaces

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 $1\le p<\infty$. In particular, we characterize the measures μ for which the operator ${\Gamma }_{\mu }$ is bounded, determine the exact value of the norm for $p\ge 4$, and provide norm estimates for the other values of p. Additionally, we observe an unexpected behavior in the case $p=2$. Finally, we characterize the measures μ for which ${\Gamma }_{\mu }$ is compact by calculating its exact essential norm.