Generalized Hilbert Operators

TL;DR

This paper investigates the properties of generalized Hilbert operators defined via analytic functions, focusing on their boundedness and compactness on various classical spaces of analytic functions in the unit disk.

Contribution

It characterizes the functions g for which the generalized Hilbert operator is bounded or compact on Hardy, weighted Bergman, and Dirichlet-type spaces.

Findings

Characterization of g for boundedness on Hardy spaces

Criteria for compactness on weighted Bergman spaces

Extension of classical Hilbert operator results to generalized operators

Abstract

If $g$ is an analytic function in the unit disc $\D $ we consider the generalized Hilbert operator $\hg$ defined by {equation*}\label{H-g} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt. {equation*} We study these operators acting on classical spaces of analytic functions in $\D $. More precisely, we address the question of characterizing the functions $g$ for which the operator $\hg $ is bounded (compact) on the Hardy spaces $H^p$, on the weighted Bergman spaces $A^p_\alpha $ or on the spaces of Dirichlet type $\mathcal D^p_\alpha $.