If $A$ is a linear operator on a normed vector space $X$ and $A^k$ is compact for some positive integer $k$, is it possible that $A$ is unbounded?
If not, how to prove that $A$ is bounded?
If $A$ is a linear operator on a normed vector space $X$ and $A^k$ is compact for some positive integer $k$, is it possible that $A$ is unbounded?
If not, how to prove that $A$ is bounded?
Of course, if $k=1$ then $A$ is bounded. Otherwise, it doesn't need to be true. Let $H$ a separable Hilbert space with $\{e_n\}$ a Hilbert basis and define $Ae_{kj+l}=\begin{cases} je_{kj+l+1}&\mbox{ if }0\leq l
This is almost the same as Davide's answer:
Let $S:\ell_2\rightarrow\ell_2$ be any unbounded operator. Define $T:\ell_2\oplus_2\ell_2\rightarrow\ell_2\oplus_2\ell_2$ via $T(x,y) = \bigl(0,S(x)\bigr)$. Then $T^k=0$ for any $k>1$ and thus compact, but $T$ is unbounded.