Register to reply

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

by CornMuffin
Tags: compact, implies, invertible, prove
Share this thread:
Mar31-12, 03:05 PM
P: 64
1. The problem statement, all variables and given/known data
[itex]X[/itex] is a Banach space
[itex]S\in B(X)[/itex] (Bounded linear transformation from X to X)
[itex]T\in K(X)[/itex] (Compact bounded linear transformation from X to X)

[itex]S(I-T)=I[/itex] if and only if [itex](I-T)S=I[/itex]

The question also asks to show that either of these equalities implies that [itex]I-(I-T)^{-1}[/itex] 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:
[itex]S(I-T)=I\Rightarrow S-ST=I \Rightarrow S=I+ST[/itex]
[itex]I-(I-T)^{-1} = I-S = I-(I+ST) = ST [/itex]
And ST is compact since T is compact
Phys.Org News Partner Science news on
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
Apr4-12, 12:01 PM
Sci Advisor
HW Helper
P: 2,020
Hint: Fredholm alternative.

Register to reply

Related Discussions
Left invertible mapping left inverse of matrix Calculus & Beyond Homework 5
If AB^2-A Invertible Prove that BA-A Invertible Calculus & Beyond Homework 5
Left invertible Linear & Abstract Algebra 10
Prove that asquare matrix A is invertible if nad only if A[sup]T[/sup]A is invertible Calculus & Beyond Homework 3
Two square invertible matrices, prove product is invertible Calculus & Beyond Homework 1