Hilbert空间上线性算子广义逆A_(T,S)^((2))的存在性及其表示式

The Existence and Expressions for the Generalized Inverse of Linear Operator in Hilbert Space

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AT,S(2)．该文进一步研究了线性算子广义逆AT,S(2)存在的若干等价条件及其性质,建立了算子广义逆AT,S(2)的表示形式．

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）satisfying. XAX=X,R（X）=T and N（X）=S The basic properties and some expressions of the generalized inverse A^（2） T,S are also investigated.