模型空间上的截断Toeplitz算子因其与复分析、算子理论、数学物理等多个重要分支的深刻联系近年来受到广泛的关注。约化子空间问题是算子理论研究的重要课题之一。本项目将以Sz.-Nagy和Foias模型理论、换位提升定理和模型空间理论等为工具研究下面三个问题。1.模型空间上以内函数（特别是有限Blaschke积）为符号的截断Toeplitz算子的可约性问题。2.模型空间上以内函数（特别是有限Blaschke积）为符号的截断Toeplitz算子对应的换位von Neumann代数结构。3.Cowen-Thomson定理在模型空间上的版本。本项目的目标是一方面更好的理解截断Toeplitz算子的结构，另一方面通过研究截断Toeplitz算子的可约性进而探讨一般压缩算子可约性问题的内在规律。

The truncated Toeplitz operators on model spaces have been attracted much attention in recent years due to its deep connections with complex branches such as complex analysis, operator theory, and mathematical physics. The problem of reducing subspace is one of the important topics in the study of operator theory. This project will use tools such as Sz.-Nagy and Foias model theory, the commutant lifting theorem, and model space theory to study the following three issues. 1. The reducibility of truncated Toeplitz operators with inner functions (especially the finite Blaschke products) on model spaces. 2. The structures of von Neumann algebras of truncated Toeplitz operators whose symbols are inner functions (especially a finite Blaschke products) on model spaces. 3. The version of Cowen-Thomson theorem on model spaces. The goal of this project is to better understand the structures of truncated Toeplitz operators on the one hand, and on the other hand, to discuss the inherent laws of the reducibility of general contraction operators by studying the reducibility of truncated Toeplitz operators.