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.

(Correct.)
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.