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

Abstract

For each α>0, the α-Bloch space consists of all analytic functions f on the unit disk satisfying sup|z|<1(1−|z|2)α|f′(z)|<+∞. We consider the following complex integral operators, namely the β-Cesàro operator C β ( f ) ( z ) = ∫ 0 z f ( w ) w ( 1 − w ) β d w and its generalization, acting from the α-Bloch space to itself, where f(0)=0 and β∈ℝ. We investigate the boundedness and compactness of the β-Cesàro operators and their generalizations. Also we calculate the essential norm and spectrum of these operators.