Assuming that $T^p$ is compact for some positive integer $p$. Prove that eigen values of $T: X \rightarrow X$ on a normed linear space is countable.
I have another problem. Assuming that $T^p$ is compact for some integer $p$ then nullspace of $T_\lambda(x)$ is finite dimensional where $T: X\rightarrow X$ is an operator on a normed linear space $X$.
What I know so far: I know that the result is true for $p =1$. But, I am not sure how to proceed for the case $T^p$. I think both the questions can be solved easily if I have something to relate $T^p$ to $T$. I am not sure, how.
Kindly provide me some hint or some resources where I can find more about this. Thanks in advance!
This is a consequence of spectra-mapping.
$Tx=\lambda x$ for some $x\neq 0$. Then $T^n=\lambda^n x$. This shows $\sigma_{p}(T) ^n=\{ \lambda^n:\lambda \in \sigma_p(T) \}\subseteq \sigma_p(T^n)$, which is countable. So $|\sigma_p(T)|\leq n|\sigma_p(T^n)|$ must be countable.