Quasi-Wandering Subspace In The Hardy Space On The Bidisc

This paper concerns with quasi-wandering subspaces in the Hardy space on the bidisc, and we focus on the relation between invariant sub-spaces and quasi-wandering subspaces under some condition. Moreover, we also study the relation between backward shift invariant subspaces and P_NT_z^*M+P_NT_w^*M under some condition. The principal tool is Beurling Theorem in the paper.In Chapter 1, we give some basic definitions and symbols. At last, we discuss some related research ground and research significance.Chapter 2 is concerned with invariant subspaces, wandering subspaces and quasi-wandering subspaces in the one variable Hardy space and Bergman space. Moreover, we give the relation among the three subspaces.Chapter 3 is the main part of this paper. It is proved that let M be a nontrivial invariant subspace in the Hardy space on the bidisc, and M(?)N = H², if 0∈Z(M∩H²(z)) or 0∈Z(M∩H²(w)), then [P_MT_zN + P_MT_wN]_H² = M. It is proved that let M = q₁H²+q₂H², q₁ = q₁(z) and q₂ = q₁(w) are one variable inner fuctions, then [P_MT_zN+P_mT_wN]_H² = M. Moreover,we give some examples.In Chapter 4, we show that backward shift invariant subspace N could be generated by P_NT_z^*M+P_NT_w^*M under some condition. Moreover, we give some examples.