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

Abstract

For each $\alpha >0$, the $\alpha$-Bloch space consists of all analytic functions $f$ on the unit disk satisfying $\underset{\left|z\right|<1}{sup}{\left(1-|z{|}^2\right)}^{\alpha }\left|f^{\prime }\left(z\right)\right|<+\infty$. We consider the following complex integral operators, namely the $\beta$-Cesàro operator

$$C_{\beta }\left(f\right)\left(z\right)={\int }_0^z\frac{f\left(w\right)}{w{\left(1-w\right)}^{\beta }}dw$$

and its generalization, acting from the $\alpha$-Bloch space to itself, where $f\left(0\right)=0$ and $\beta \in ℝ$. We investigate the boundedness and compactness of the $\beta$-Cesàro operators and their generalizations. Also we calculate the essential norm and spectrum of these operators.