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

TL;DR

This paper characterizes when a generalized Cesàro-like operator, defined via a measure and a parameter, acts boundedly or compactly from certain analytic function spaces to Besov spaces.

Contribution

It provides a characterization of measures for which the generalized Cesàro-like operator is bounded or compact between specific analytic function spaces and Besov spaces.

Findings

Characterization of measures for boundedness of the operator.

Conditions for compactness of the operator.

Extension of Cesàro operator theory 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 $B_{p}$.