A Generalized Hilbert Operator on $\textit{Bloch}$ Space and $\textit{BMOA}$ Spaces

Abstract

We consider the generalized Hilbert operators $\mathcal {H}_{a,b}$,

$$\mathcal {H}_{a,b}(f)(z)=\sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty }\frac{(b-a)_n\mu _{n,k}}{n!}a_k\right) z^n,$$

where $a,b\in \mathbb {C}$, $f(z)=\sum _{k=0}^{\infty }a_kz^k$ analytic on the unit disc $\mathbb {D}$, $\mu$ be a positive Borel measure on the interval [0, 1) and $\mu _n$ denote the moment of order n of $\mu$, that is, $\mu _n=\int _{[0,1)}t^n\mathrm{d}\mu (t)$ with $\mu _{n,k}=\mu _{n+k}$. This is one of the generalization of the classical Hilbert operator. In this paper, we characterize the measures $\mu$ and find the conditions on a, b such that $\mathcal {H}_{a,b}$ is bounded and compact on Bloch space and BMOA spaces.