正交投影算子序列积的相关性质

On the Related Properties of the Sequential Product of Orthogonal Projection Operators

摘要: 本文以复可分Hilbert空间为研究背景，聚焦正交投影算子序列积的相关性质展开探究。记$\epsilon \left(H\right)$ 为该空间全体正压缩算子的集合，为全体正交投影算子的集合。对于$A\in \epsilon \left(H\right)$ ，$B\in \epsilon \left(H\right)$ ，定义$A\circ B=A^{\frac{1}{2}}B{A}^{\frac{1}{2}}$ 为$A$ 和$B$ 的序列积。研究借助空间分解的方法，推导并证明了正交投影算子序列积在幂运算、广义逆存在性等方面的核心定理。

Abstract: This paper takes the complex separable Hilbert space as the research background and focuses on exploring the relevant properties of the sequential product of orthogonal projection operators. Let $\epsilon \left(H\right)$ denote the set of all positive contraction operators on this space, and denote the set of all orthogonal projection operators. For $A\in \epsilon \left(H\right)$ , $B\in \epsilon \left(H\right)$ , the sequential product of A and B is defined as $A\circ B=A^{\frac{1}{2}}B{A}^{\frac{1}{2}}$ . By means of the space decomposition method, this study deduces and proves the core theorems of the sequential product of orthogonal projection operators in terms of power operations and the existence of generalized inverses.