Hilbert空间上线性算子广义逆A(2)T,S的存在性及其表示式

设H₁和H₂是两个Hilbert空间, B(H₁,H₂)表示从H₁到H₂的所有有界线性算子的集合, T和S分别是H₁和H₂的两个闭子空间. 如果存在线性算子X ∈ B(H₂,H₁)满足XAX=X, R(X)=T, N(X)=S,则称X为线性算子$A$的具有指定像空间T和零空间S的外逆,记为A⁽²⁾_T,S. 该文进一步研究了线性算子广义逆A⁽²⁾_T,S存在的若干等价条件及其性质,建立了算子广义逆A⁽²⁾_T,S的表示形式.

The Existence and Expressions for the Generalized Inverse A(2)T,S of Linear Operator in Hilbert Space

Let H₁ and H₂ be Hilbert spaces. In this paper, the authors present some equivalent conditions for the existence of generalized inverse A⁽²⁾_T,S of a bounded linear operator A ∈ B(H₁,H₂), which is a bounded linear operator X ∈ B(H₂,H₁) satisfyingXAX=X, R(X)=T, N(X)=S.The basic properties and some expressions of the generalized inverse A⁽²⁾_T,S are also investigated.