Originally Posted by

**Dinkydoe** I'm struggeling with the following questions:

Prove or disprove the following statements:

(a) If $\displaystyle \mathcal{H}$ is a complex Hilbert-space and $\displaystyle T\in B(\mathcal{H})$ be a compact operator such that $\displaystyle \sigma(T) = \left\{0\right\}$ (the spectrum) then $\displaystyle T= 0$

(b) If $\displaystyle \mathcal{H}$ is a complex Hilbert-space and $\displaystyle T\in B(\mathcal{H})$ such that $\displaystyle T^2$ is compact, then the set $\displaystyle \left\{\lambda\in\sigma(T):|\lambda|>\delta \right\}$ is finite for each $\displaystyle \delta>0$.

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 $\displaystyle T$ such that $\displaystyle \sigma(T)$ is (not countable) infinite, this should be a counter example for (b). However I'm not even sure such T exists.