The expression of Moore--Penrose inverse of $A-XY^*$

TL;DR

This paper derives new explicit formulas for the Moore--Penrose inverse of the operator $A-XY^*$ in Hilbert spaces, expanding understanding of its structure under specific conditions.

Contribution

It provides novel expressions for the Moore--Penrose inverse of $A-XY^*$ when certain range conditions are satisfied, extending previous results.

Findings

New formulas for $(A-XY^*)^+$ under specific conditions

Conditions involving the ranges of $X$, $Y$, and $A$

Enhanced understanding of Moore--Penrose inverse in operator theory

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 $R(X)\subseteq R(A),R(Y)\subseteq R(A^*)$. In this short note, we give some new expressions of the Moore--Penrose inverse $(A-XY^*)^+$ of $A-XY^*$ under certain suitable conditions.