I'm struggeling with the following questions:
Prove or disprove the following statements:
(a) If is a complex Hilbert-space and be a compact operator such that (the spectrum) then
(b) If is a complex Hilbert-space and such that is compact, then the set is finite for each .
I suspect (a) is false: Taking a Hilbert-space of finite dimension, and a nilpotent matrix should be a counterexample.
I suspect (b) is false. If we find a compact operator such that is (not countable) infinite, this should be a counter example for (b). However I'm not even sure such T exists.
Any help would be appreciated!
So if I'm right, and please correct me where I'm wrong:
There's a Theorem in my book that more or less states: For compact operators we have (Pointspectrum = Spectrum). And another Theorem states that (*)for any t > 0 the set of all distinct eigenvalues of with is finite.
Thus given your hint: if is such that is invertible iff are invertible
Thus . And since is compact we have (*), thus since should follow that for any , we have is finite.