Mean Lipschitz spaces and a generalized Hilbert operator

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

$$\begin{aligned}\mathcal {H}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n,k}{a_k}\right) z^n\end{aligned}$$

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.