Operators of Hilbert type acting on some spaces of analytic functions

TL;DR

This paper characterizes when a generalized Hilbert operator acts boundedly or compactly on various spaces of analytic functions, extending classical results and exploring new spaces like logarithmic Bloch and Korenblum spaces.

Contribution

It provides a complete characterization of the boundedness and compactness of the generalized Hilbert operator on multiple analytic function spaces, generalizing known classical results.

Findings

Characterization of boundedness of $\,\mathcal{H}_g$ on Dirichlet spaces

Conditions for compactness of $\,\mathcal{H}_g$

Extension of results to logarithmic Bloch and Korenblum spaces

Abstract

Let $H(\mathbb{D})$ be the space of all analytic functions in the unit disc $\mathbb{D}$. For $g\in H(\mathbb{D})$, the generalized Hilbert operator $\mathcal{H}_{g}$ is defined by $$\mathcal{H}_{g}(f)(z)=\int_{0}^{1}f(t)g'(tz)dt, \ \ z\in \mathbb{D}, f\in H(\mathbb{D}).$$ In this paper, we study the operator $\mathcal{H}_{g}$ acting on some spaces of analytic functions in $\mathbb{D}$. Specifically, we give a complete characterization of those $g\in H(\mathbb{D})$ for which the operator $\mathcal{H}_{g}$ is bounded (resp. compact) from the Dirichlet space $\mathcal{D}^{2}_{\alpha}$ to $\mathcal{D}^{2}_{\beta}$ for all possible indicators $\alpha,\beta \in \mathbb{R}$. We also study the action of the operator $\mathcal{H}_{g}$ on the space of bounded analytic functions $H^{\infty}$, which generalizes the known results for the classical Hilbert operator $\mathcal {H}$ acting on $H^{\infty}$. In particular, we consider the boundedness of the operator $\mathcal{H}_{g}$ with a symbol of non-negative Taylor coefficients, acting on logarithmic Bloch spaces and on Korenblum spaces. This work generalizes the corresponding results for the classical Hilbert operator.