Reducing Subspace Of The Weighted Bergman Space

Structures of Operator are the main problem that the operator theorists study,one of the most problems is the invariant subspace problem:whether every bounded operator on a separable infinite dimension Hilbert space has a nontrivial invariant sub-space?There is not an anticipated way for resolving the problem until now.Bercovici,Foia and Pearcy [2] give a sufficient condition by the Bergman shift M_z,or that's to say,the existence of the invariant subspace problem for Hilbert space operator is equivalent to whetherLatM_z is saturated,i.e.,for any M, N∈LatM_z,M(?)N,with dim(M(?)N) = oo,whether there always exists someΩ∈LatM_z such that M(?)Ω(?) N. Thus how to figure out the structure or to get some deeper information about LatM_z is definitely interesting and important.Instead of describing LatM_z itself, some efforts have been made to describe the structure of the reducing subspace of T_φon the Bergman space withφa finite Blaschke product. In this paper,we lift the weighted Bergman space as a subspace of Hardy space on poly-disc.The Toeplitz operator on the weighted Bergman space can be studied via the analytic Toeplitz operator on the Hardy space on poly-disc. We obtain that every analytic Toeplitz operator induced by a finite blaschke productφon the weighted Bergman space has an nontrivial reducing subspace M such that the restriction of T_φon M is unitary equivalent to weighted Bergman shift.At the end of the paper,we mainly discucc the property of invaiant subspace of the weighted Hardy space,and get some interesting structure theorems about invariant subspace of the weighted Hardy space.