A generalized Hilbert operator acting on conformally invariant spaces

TL;DR

This paper investigates a generalized Hilbert operator defined via a Hankel matrix with moments of a measure, focusing on its action on conformally invariant spaces like Bloch, BMOA, Besov, and Q_s spaces.

Contribution

It extends the study of Hilbert-type operators to conformally invariant spaces, analyzing their boundedness and behavior beyond Hardy spaces.

Findings

Operators are bounded on certain conformally invariant spaces.

Characterization of measures for boundedness on these spaces.

New insights into the operator's behavior on complex function spaces.

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 orden $n$ 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 $\D $. This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu }$ on Hardy spaces has been recently studied. This paper is devoted to study the operators $H_\mu $ acting on certain conformally invariant spaces of analytic functions on the disc such as the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces.