Cesàro-like operators acting on a class of analytic function spaces

Abstract

Let $\mu$ be a finite positive Borel measure on [0, 1) and let $H(\mathbb {D})$ be the space of all analytic function in the unit disc $\mathbb {D}$. The Cesàro-like operator $\mathcal {C}_\mu$ is defined in $H(\mathbb {D})$ as follows: If $f \in H(\mathbb {D})$, $f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} (z\in \mathbb {D})$, then

$$\begin{aligned} \mathcal {C}_\mu (f)(z)=\sum ^\infty _{n=0}\left( \mu _n\sum ^n_{k=0}a_k\right) z^n,\quad z\in \mathbb {D}, \end{aligned}$$

where for $n\ge 0$, $\mu _n$ denotes the n-th moment of the measure $\mu$, that is, $\mu _n=\int _{0}^{1} t^{n}d\mu (t)$. For $s > 1$, let X be a Banach subspace of $H(\mathbb {D})$ lying between the mean Lipschtz space $\Lambda ^{s}_{\frac{1}{s}}$ and the Bloch space $\mathcal {B}$. In this paper we characterize the measures $\mu$ as above for which $\mathcal {C}_\mu$ is bounded (compact) from X into any of the Hardy spaces $H^{p} ~ (1\le p\le \infty )$.