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 $\mathcal{H}_{\mu,\alpha}=(\mu_{n,k,\alpha})_{n,k\geq 0}$ 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 $\mathcal{H}_{\mu,\alpha}$ ($\alpha\geq 2$) is a bounded (resp., compact) operator from the Bloch type space $\mathscr{B}_{\beta}$ ($0<\beta<\infty$) into $\mathscr{B}_{\alpha-1}$. 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.