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'(z)| < + \infty.$ In this paper, we consider the following complex integral operator, namely the $\beta$-Ces\`{ a}ro operator \begin{ equation} C_\beta(f)(z)=\int_{ 0}^{ z}\frac{ f(w)}{ w(1-w)^{ \beta}}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 $\beta$-Ces\`{ a}ro operators and their generalization. Also we calculate the essential norm and spectrum of these operators.
Categories: math.FA