单圆盘上压缩移位算子的基本性质，如：谱和不变子空间等的刻画分别依赖于内函数在圆盘内部的零点，边界的奇异点和因子。双圆盘内函数结构的复杂性决定了双圆盘Hardy空间上压缩移位算子的结构要更复杂。本项目将利用Nagy-Foias算子模型理论，通过建立双圆盘内函数的结构与双圆盘压缩移位算子的关系，研究如下问题：1.压缩移位算子何时是可约；如果可约，约化子空间如何描述？ 2. 压缩移位算子的不变子空间问题 3. 压缩移位算子的谱等算子结构和代数性质。这些研究会推动对双圆盘内函数结构的深入刻画，理清双圆盘压缩移位算子的结构以及和内函数之间的关系，为一般C_{\cdot 0}算子结构的研究提供有价值的理论依据，对不变子空间问题的解决会有推动作用。

Described properties of compressed shifts on disk such as the spectrum and invariant subspaces depend on zeros of inner on the open disk, the singular point on the boudedary and the inner factor. However, the complexity for the structure of the inner function on bidisk yields that the structure of compressed shifts on the Hardy space over the bidisk will more complexity than compressed shifts on the Hardy space over the disk. In this project, we will establish the relationship the of structure between the inner function and compressed shifts over the bidisk by Nagy-foias theory,and study the problem as follows:1.The reducibility of compressed shifts; described the reducing subspace.2. Described the invariant subspace of compressed shifts. 3. Described the spectrum and algebra of compressed shifts.These studies promote the in-depth description of the structure of the inner function on the bidisk, clarify the relationship between the structure of compression shifts and the inner function on the bidisk, provide a valuable theoretical basis for study the structure of the general C_{\cdot 0} operator, and promote to solve the invariant subspace problem.