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.