On Quasi-similarity And Reducing Subspaces Of Multiplication Operator On A Certain Closed Subspace

The invariant subspace and reducing subspace problems are interesting and impor-tant themes in operator theory. It is conjectured that every bounded linear operator does have a non-trivial closed invariant subspace. The commutant of operator is an important concept in characterizing the reducing subspaces of the operator. From the information of the commutant, people can research the similar equivalence and unitary equivalence of the operators. The invariant subspace and reducing subspace problems on the Hardy and Bergman space have been studied extensively in the literature. In this article, we gener-alized some results of Bergman space of unit disk to a certain subspace of the weighted Bergman space of the unit ball.Let A2a(Bn)(a> -1) denote the weighted Bergman space of the unit ball which consists of all holomorphic functions/in L2(Bn, dva). A certain subspace of the weight-ed Bergman space of the unit ball is (si≥1,i= 1,2,…,n). In this paper, first we prove that the multiplication operator Mz(ms1,…,msn) is quasi-similar to on the certain closed subspace. Then we characterize the reducing sub-spaces of Mz(ms1,…,msn),by applying the technique of the operator theory, it is shown that Mz(ms1,…,msn)has exactly 2m reducing subspaces on this closed subspace.The main structure of this article is:In the first section, we give out some concepts. We introduce the concept of the weighted Bergman space of the unit ball, quasi-similarity and reducing subspaces. The concept of projection operator and its relationship with reducing subspace. In addition, we list Stirling’s formula which is needed in the process.In the second section, we prove that the multiplication operator Mz(ms1,…,msn) is quasi-similar to on the certain closed subspace.In the third section, with the help of projection operators, we characterize the re-ducing subspaces ofMz(ms1,…,msn) , it is shown that Mz(ms1,…,msn) has exactly 2m reducing subspaces on the certain subspace.