Generalized Ces\`{a}ro-like operator from a class of analytic function spaces to analytic Besov spaces

Abstract

Let $\mu$ be a finite positive Borel measure on $[0,1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. For $0<\alpha<\infty$, the generalized Ces\`aro-like operator $\mathcal{C}_{\mu,\alpha}$ is defined by $$ \mathcal {C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $\mu_n$ 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})$ with $\Lambda^{s}_{\frac{1}{s}}\subset X\subset\mathcal {B}$. In this paper, for $1\leq p <\infty$, we characterize the measure $\mu$ for which $\mathcal{C}_{\mu,\alpha}$ is bounded(or compact) from $X$ into analytic Besov space $B_{p}$.