# Prove: For T Compact, left or right invertible implies invertible

by CornMuffin
Tags: compact, implies, invertible, prove
 P: 64 1. The problem statement, all variables and given/known data $X$ is a Banach space $S\in B(X)$ (Bounded linear transformation from X to X) $T\in K(X)$ (Compact bounded linear transformation from X to X) $S(I-T)=I$ if and only if $(I-T)S=I$ The question also asks to show that either of these equalities implies that $I-(I-T)^{-1}$ is compact. 2. Relevant equations 3. The attempt at a solution I have tried using the adjoint, cause S is invertible if and only if S* is invertible. but that didn't get me anywhere. If there happens to be a theorem that says ST = TS, then it would be easy, but i couldn't find anything like that. For the second part: $S(I-T)=I\Rightarrow S-ST=I \Rightarrow S=I+ST$ $I-(I-T)^{-1} = I-S = I-(I+ST) = ST$ And ST is compact since T is compact
 Sci Advisor HW Helper P: 2,020 Hint: Fredholm alternative.

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