Hilbert-Type Operators Acting on Bergman 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 \(n\,=\,0, 1, 2, \ldots \), \(\mu _n\) denotes the moment of order n of \(\mu \). This matrix formally induces an operator, called also \({\mathcal {H}}_\mu \), on the space of all analytic functions in the unit disc \({\mathbb {D}}\) as follows: If f is an analytic function in \({\mathbb {D}}\), \(f(z)=\sum _{k=0}^\infty a_kz^k\), \(z\in {{\mathbb {D}}}\), \({\mathcal {H}}_\mu (f)\) is formally defined by

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

This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators \(H_\mu \) acting on the Bergman spaces \(A^p\), \(1\le p<\infty \). Among other results, we give a complete characterization of those \(\mu \) for which \({\mathcal {H}}_\mu \) is bounded or compact on the space \(A^p\) when p is either 1 or greater than 2. We also give a number of results concerning the boundedness and compactness of \(\mathcal H_\mu \) on \(A^p\) for the other values of p, as well as on its membership in the Schatten classes \({\mathcal {S}}_p(A^2)\).