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

Abstract

Let $\mu$ be a positive Borel measure on the interval [0,1). For $\beta >0$, The generalized Hankel matrix ${\mathcal{H}}_{\mu ,\beta }=({\mu }_{n,k,\beta }{)}_{n,k\ge 0}$ 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.

arXiv admin note: substantial text overlap with arXiv:2206.12024