# A Hankel matrix acting on spaces of analytic functions

## TL;DR

This paper studies Hankel matrices defined by measures on [0,1) and their induced operators on analytic function spaces, extending classical Hilbert operator results to Hardy and Möbius invariant spaces.

## Contribution

It generalizes the classical Hilbert operator by analyzing Hankel matrices from measures and explores their action on Hardy and Möbius invariant spaces, improving recent results.

## Key findings

- Extended the understanding of Hankel operators on Hardy spaces.
- Provided new bounds and conditions for boundedness of these operators.
- Enhanced the theoretical framework for operators acting on analytic 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 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 improve the results obtained in some recent papers concerning the action of the operators $H_\mu $ on Hardy spaces and on M\"obius invariant spaces.