Generalized Cesàro-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 Cesàro-like operator ${\mathcal{C}}_{\mu ,\alpha }$ is defined by $${\mathcal{C}}_{\mu ,\alpha }(f)(z)=\sum _{n=0}^{\infty }\left({\mu }_n\sum _{k=0}^n\frac{\Gamma (n-k+\alpha )}{\Gamma (\alpha )(n-k)!}{a}_k\right)z^n,z\in \mathbb{D},$$ 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 $H(\mathbb{D})$ with ${\Lambda }_{\frac{1}{s}}^s\subset X\subset \mathcal{B}$. In this paper, for $1\le p<\infty$, we characterize the measure $\mu$ for which ${\mathcal{C}}_{\mu ,\alpha }$ is bounded(or compact) from $X$ into analytic Besov space $B_p$.