Generalized Hilbert Operator Acting on Weighted Bergman Spaces and on Dirichlet Spaces

TL;DR

This paper characterizes measures for which a generalized Hilbert operator acts boundedly or compactly on weighted Bergman and Dirichlet spaces, extending classical operator theory in complex analysis.

Contribution

It provides a characterization of measures inducing bounded and compact generalized Hilbert operators on weighted Bergman and Dirichlet spaces, generalizing previous results.

Findings

Characterization of measures for boundedness

Criteria for compactness of the operator

Extension to Dirichlet spaces

Abstract

Let $\mu$ be a positive Borel measure on the interval [0,1). For $\beta > 0$, The generalized Hankel matrix $\mathcal{H}_{\mu,\beta}= (\mu_{n,k,\beta})_{n,k\geq0}$ with entries $\mu_{n,k,\beta}= \int_{[0.1)}\frac{\Gamma(n+\beta)}{n!\Gamma(\beta)} t^{n+k}d\mu(t)$, induces formally the operator $$\mathcal{H}_{\mu,\beta}(f)(z)=\sum_{n=0}^\infty \left(\sum_{k=0}^\infty \mu_{n,k,\beta}a_k\right)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. In this paper, we characterize those positive Borel measures on $[0,1)$ such that $\mathcal{H}_{\mu,\beta}(f)(z)= \int_{[0,1)} \frac{f(t)}{{(1-tz)^\beta}} d\mu(t)$ for all in weighted Bergman Spaces $A_{\alpha}^p(0<p<\infty,\; \alpha>-1)$, and among them we describe those for which $\mathcal{H}_{\mu,\beta}(\beta>0)$ is a bounded(resp.,compact) operator on weighted Bergman spaces and Dirichlet spaces.