I'm searching for an example of a Banachspace $X$ and an operator $T\in K(X)/\{0\}$ with spectrum $\sigma(T)=\{0\}$ and one of the following properties:
i) $\sigma_p(T)=\{0\}$
ii) $\sigma_c(T)=\{0\}$
iii) $\sigma_r(T)=\{0\}$
The Banachspaces can be different for every operator which fulfills i), ii) or iii)
For i) I thought about an operator with a Banachspace $X$ with $dim X<\infty$. Because for these Banachspaces $\sigma(T)=\sigma_p(T)$ So for i) I just need an operator T with $\sigma(T)=\{0\}$
Can someone give me an example for the other two problems? Or are there any operators which fulfill these properties? Thanks in advance.
If any definitions aren't clear, I can add them.
Volterra! That should do it!!
https://en.wikipedia.org/wiki/Volterra_operator