# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 34 (2005) pp.135-147

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### 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.

MSC(Primary) 30D55 multiplication operator, integration operator, Bloch space, boundedness,compactness