On The Strong Irreducibility Of Toeplitz Operator And The Reducing Subspaces Of A Class Of Multiplication Operator On The Weighted Bergman Space

Characterization of commutanting of an operator on a Hilbert space, can make people better understand the structure of the given operator. An operator is strongly irreducible, if any nontrivial idempotent operator does not commute with this operator. Studying reducing subspace problem of an operator is searching a projection operator which can commutate with the operator, so it is an important problem.In this paper, we firstly give the sufficient condition of a class of analytic Toeplitz operator which is strongly irreducible on the weighted Bergman space, and then give the reducing sub-spaces of multiplication operators with matrix method. At last, we give the winding number of a class of function.The main structure of the content of this paper is arranged as follows:In chapter1, we introduce the definition of weighted Bergman space, the representation of inner product and the definition of multiplication operator, Toeplitz operator、winding number. In addition,we construct an orthonormal basis of Aα2(D) by the normalized reproducing kernel.In chapter2, we find the sufficient condition of a class of analytic Toeplitz operator which is strongly irreducible on the weighted Bergman space Aα2(D).In chapter3, we briefly prove that multiplication operator Mzn is similar to (?) Mz in terms of Stirling formula on the weighted Bergman space Aα2(D).In chapter4, we study that the multiplication operator Mzn has2n reducing subspaces, and give the minimal reducing subspaces on the weighted Bergman space Aα2(D).In chapter5, we prove the multiplication operator MB(z) where B(z)=(z-α/1-αz) n(0<|α|<1) has2n reducing subspaces on the weighted Bergman space Aα2(D). we also compute the winding number of a class of function.