The model operator S(u) have played an important role in developing both function and operator theory over the past century. In particular, the lattice of invariant subspaces and the spectrum of S(u) have been very well understood. This success led people to study the Hardy space over the bidisk. We are interested in the independent behavior of compressed shift operator Sz on the Beurling quotient modules. In this talk, we study the invariant subspace, reducibility and spectrum of Sz on on the Beurling quotient modules. This is joint work with Yufeng Lu and Senhua Zhu.

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