Reducing Subspaces Of Some Multiplication Operators On The Bergman Space Over Polydisk

The operator theory on the functional space is inextricably bound up with the function theory and operator theory. At present, the structure of some representative linear operator on the functional space is one of the hot topics of operator theory, where the structure of the invariant subspace has always been one of the most basic research issue. So far, the invariant subspace problem on separable Hilbert space is still a famous open problem in the operator theory, that is whether every linear operator on separable Hilbert space has a non-trivial invariant subspace? In 1996, H. Hedenmalm, S. Richterand K. Seip[1] proved that the invariant subspace problem on the infinite dimension separable Hilbert space is equivalent to a invariant subspace problem of multiplication operator with symbol z on Bergman space.Thus it has been drawn many scholars’depict concerns. As a special kind of invariant subspace, the reducing subspace of multiplication operator on Bergman space also has important the oretical significance, so that many scholars have carried out extensive and in-depth study.In this paper, we mainly research the reducing subspaces of multiplication operator on unweighted and weighted Bergman space, and also completely characterize the minimal reducing subspaces.In section one, we discuss the structure of reducing subspaces and minimal reducing subspaces of multiplication operators on the unweighted and weighted Bergman spaces. By the orthogonal decompose of the polynomials according to the reducing subspaces, we characterize the minimal reducing subspaces of multiplication operator completely.In section two, we keep on the research of the reducing subspaces and minimal reducing subspaces of some multiplication operators over polydisk. On the weighted Bergman space, we get the similar conclusions with. On the weighted Bergman space we prove that every reducing subspace of multiplication operator is a direct sum of some minimal reducing subspaces, where the number of the minimal reducing subspaces are at most countable. Naturally, we consider how to characterize the minimal reducing subspaces. According to some examples, we find that the reducing subspaces of multiplication operators are more complicated than the reducing subspaces on A2(Dk) and A2α(D3). Then we start the research with some special cases, and get the characteristics of minimal reducing subspaces under certain conditions. By symmetrical rotation of algebra, we get a complete characterization of the reducing subspaces. Finally, we prove that every minimal reducing subspace is generated by a monomial when the weight coefficient is irrational. When the weight coefficient is rational, every minimal reducing subspace is generated by a polynomial which contains at most two non-zero coefficients.