A generalized Hilbert operator acting on conformally invariant 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 formally induces the operator

$${\mathcal{H}}_{\mu }\left(f\right)\left(z\right)={\sum }_{n=0}^{\infty }\left({\sum }_{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)z^n$$ on the space of all analytic functions $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{z}^k$, in the unit disk $\mathbb{D}$. 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 article is devoted to a study of the operators $H_{\mu }$ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the $Q_s$-spaces.