On the Expression of Generalized Inverses of Perturbed Bounded Linear Operators

Abstract

Let $X$ and $Y$ be two Hilbert spaces or Banach spaces, and let $T \colon X \rightarrow Y$ be a bounded linear operator with closed range. Let $\tilde{T} = T + \delta T$ with $\|\delta T\| \|T^{\dagger}\| < 1$. We give some equivalent conditions for the generalized inverse of $\tilde{T}$ to have the simplest expression $\tilde{T}^{\dagger} = (I + T^{\dagger}\delta T)^{-1} T^{\dagger} = T^{\dagger} (I + \delta T T^{\dagger})^{-1}$.