Generalized Hilbert Operator Acting on Bloch Type Spaces

Abstract

Let $\mu$ be a positive Borel measure on the interval [0,1). For $\alpha >0$, the Hankel matrix ${\mathcal{H}}_{\mu ,\alpha }=({\mu }_{n,k,\alpha }{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k,\alpha }={\int }_{[0,1)}\frac{\Gamma (n+\alpha )}{n!\Gamma (\alpha )}{t}^{n+k}d\mu (t)$ formally induces the operator $${\mathcal{H}}_{\mu ,\alpha }(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k,\alpha }{a}_k\right)z^n$$ on the space of all analytic functions $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ in the unit disc $\mathbb{D}$. In this paper, we characterize the measures $\mu$ for which ${\mathcal{H}}_{\mu ,\alpha }$ ($\alpha \ge 2$) is a bounded (resp., compact) operator from the Bloch type space ${\mathcal{B}}_{\beta }$ ($0<\beta <\infty$) into ${\mathcal{B}}_{\alpha -1}$. We also give a necessary condition for which ${\mathcal{H}}_{\mu ,\alpha }$ is a bounded operator by acting on Bloch type spaces for general cases.