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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