Mean Lipschitz spaces and a generalized Hilbert operator

TL;DR

This paper investigates how a generalized Hilbert operator, defined via a Hankel matrix associated with a measure, acts on mean Lipschitz spaces of analytic functions in the unit disk.

Contribution

It introduces a broad class of operators extending the classical Hilbert operator and analyzes their behavior on mean Lipschitz spaces, expanding understanding of operator theory in complex analysis.

Findings

Characterization of the boundedness of al H_ on mean Lipschitz spaces.

Conditions under which al H_ is compact or bounded.

Extension of classical results to a more general operator class.

Abstract

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of order $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb{D} $. This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators $\mathcal H_\mu $ on mean Lipschitz spaces of analytic functions.