A Hankel matrix acting on spaces of analytic functions

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 $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$, 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 Möbius invariant spaces.

arXiv admin note: text overlap with arXiv:1612.08304