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

[CornMuffin]

Homework Statement

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.

Homework Equations

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

 

[morphism]

Hint: Fredholm alternative.