Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 355-378.

On multiplication operators on the Bergman space: Similarity, unitary equivalence and reducing subspaces

Authors: Kunyu Guo

$1$ and Hansong Huang

$2$
Author institution:

$1$ School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

$2$ Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

Summary: In this paper, we study similarity, unitary equivalence and reducing subspace problems of multiplication operators with symbols of finite Blaschke products on the Bergman space

$L^2_a(\mathbb{D})$ . By using Rudin's method, we establish a representation theorem of

$L^2_a$ -functions related to a given finite Blaschke product. As an immediate consequence, one sees that for two finite Blaschke products

$B_1,\,\,B_2$ ,

$M_{B_1}$ is similar to

$M_{B_2}$ if and only if

$\deg B_1=\deg B_2$ . By a different method, this similarity result also was independently obtained by Jiang and Li. Then we turn to the study of reducing subspaces of multiplication operators. It is shown that if

$B$ is a finite Blaschke product with

$\deg B\leqslant 6$ , then the number of minimal reducing subspaces of

$M_B$ is at most

$\deg B$ . The best previous known results were for the cases of

$\deg B=2,\, 3,\,4$ .

Keywords: Bergman space, finite Blaschke product, minimal reducing subspaces, unitary equivalence, similarity

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