[Math] Exercise about compact operator.

Question

In $X=\ell^p$, $p\in[1,\infty]$ we consider:
$$
T(x_1,x_2,x_3,\ldots)=(0,x_1,0,x_3,\ldots)
$$
Prove that $T$ isn't a compact operator and that $T^2$ is a compact operator.
I think I solved the second point of the exercise because $T^2 x= 0$ always considering $T^2=T(T(x))$ (but I'm not sure this is $T^2$…). Anyway I'd like to find a sequence $\{x_n\}$ such that $x_n\rightharpoonup x$ but $Tx_n\nrightarrow Tx$.

Answer 1

Consider the subspace $U=\{(x_1,0,x_3,0,\dots)\}\subset\ell^p$. The restriction $T_{|_U}$ is the right shift operator on $U$. But we know that this is not a compact operator (e.g. looking at the spectrum).