Multiplication operators, integration operatorsand companion operators on weighted Bloch space

Abstract

Let $g$ be an analytic function on the open unit disk $D$ in the complex plane $C$. We will study the following operator \[\ I_{g}(h) (z) := \int_0^z h'( \zeta )g( \zeta) d \zeta \ , \ J_g(h) (z) := \int_0^z h( \zeta )g'( \zeta) d \zeta\] on the Bloch space. In this paper, we will study the boundedness and compactness of $I_g$ on the $\alpha$-Bloch space , and the boundedness and compactness of products of $I_g$ and $J_g$ defined on the $\alpha$-Bloch space. And we will get the relationship of multiplication operator $M_g$ and the operators $I_g$, $J_g$ defined on the $\alpha$-Bloch space.