在数学不同分支中，Grassmannian的研究对象各有不同。传统的研究方法多基于微分几何、代数拓扑和抽象代数。在本项目中，Grassmannian具体的研究对象是Hilbert 空间中的闭子空间，或正交射影集合。我们将以算子论和算子代数为基础，以算子分块矩阵技巧，算子谱理论和解算子方程这一全新的视角为切入点，对正交算子对和与之密切相关的多个算子之间的几何结构，谱结构进行深入研究，给出所涉及到的算子的分块算子矩阵具体表示。在此基础上对一些受限的Grassmannian进行不同角度的研究。例如，借助于算子分块矩阵，分别给出Sato-Grassmannian和射影对的交织算子的完全刻画。进而，在梳理Grassmannian的传统研究方法和本项目提供的新思路对接中，进一步探索算子论和算子代数，以及Grassmannian研究的新方向和新问题。

In different branches of mathematics, subjects in the studying of Grassmannians are different. Traditional methods are based on differential geometry, algebraic topology and algebra. In the project，the Grassmannian is as a set of closed subspaces of a Hilbert space. We also consider subjects as a set of orthogonal projections on closed subspaces of a Hilbert space in the studying of Grassmannians since there exists an one to one corresponding between the set of all closed subspaces in a Hilbert space and the set of orthogonal projections on closed subspaces of a Hilbert space. The project will be divided into three levels. First, based on operator theory and operator algebra and by new perspective of block operator matrix technique, spectral theory of operators and solving operator equation, geometry and spectral constructions of pairs of orthogonal projections and closely related operators will be deeply studied and block operator matrix representations of operators involved in studying will be given. For example, general explicit expressions direct rotations of two orthogonal projections will be obtained. Second, in view of the first step, some special Grassmannian will be discussed. For example, by means of that spectral characterizations and operator matrix representations of Sato- Grassmannian, spectral characterizations and operator matrix representations of Hopf-Rinow theorem concerning geodesic will be got. Third, comparing the traditional research methods with new idea used in this project， new directions and new problems in the studying of operator theory, operator algebra and Grassmannian will be explored and discovered in the further.