The expression of Moore--Penrose inverse of $A-X{Y}^{\ast }$

Abstract

Let $K,H$ be Hilbert spaces and let $L(K,H)$ denote the set of all bounded linear operators from $K$ to $H$. Let $A\in L(H)\triangleq L(H,H)$ with $R(A)$ closed and $X,Y\in L(K,H)$ with $R(X)\subseteq R(A),R(Y)\subseteq R(A^{\ast })$. In this short note, we give some new expressions of the Moore--Penrose inverse $(A-X{Y}^{\ast }{)}^{+}$ of $A-X{Y}^{\ast }$ under certain suitable conditions.

arXiv comment6 pages