Expressions and perturbations for the Moore--Penrose inverse of bounded linear operators in Hilbert spaces

Let $A,X,Y$ be bounded linear operators. In this paper, we present the explicit expression for the Moore--Penrose inverse of $A-XY$. In virtue of the expression of $(A+X)^\dag$, we get the upper bounds of $\|(A+X)^\dag\|$ and $\|(A+X)^\dag-A^\dag\|$.