Properties of $\beta$-Cesàro operators on $\alpha$-Bloch space

Abstract

For each $\alpha >0$, the $\alpha$-Bloch space is consisting of all analytic functions $f$ on the unit disk satisfying ${\sup }_{|z|<1}(1-|z{|}^2{)}^{\alpha }|f^{\prime }(z)|<+\infty .$ In this paper, we consider the following complex integral operator, namely the $\beta$-Cesàro operator \begin{equation} C_β(f)(z)=\int_{0}^{z}\frac{f(w)}{w(1-w)^β}dw \nonumber \end{equation} and its generalization, acting from the $\alpha$-Bloch space to itself, where $f(0)=0$ and $\beta \in \mathbb{R}$. We investigate the boundedness and compactness of the $\beta$-Cesàro operators and their generalization. Also we calculate the essential norm and spectrum of these operators.

24 pages, Rocky Mountain Journal of Mathematics (to appear)