Reducibility Of Compressed Shift Operators On The Homogeneous Quotient Module

In this paper,we show that the compressed shift operator SzK has at least NK distinct non-trivial minimal reducing subspaces on the quotient module N=H2(T2)(?)[z~N-wN].Moreover,there is a unique non-trivial minimal reducing subspace M of S_φ(zH)on N such that S_φ(ZN)|M is unitary equivalent to Bergman shift Mz,where φ is a finite Blaschke product.The Beurling type theorem related to operator FwN on defect space[ZN-wN](?)z[z~N-wN]is also studied.