Are $T,T^2$ compact operators?

Question

$T:l_2\to l_2$ is defined by $T(x_1,x_2,\dots)=(0,x_1,0,x_3,0,x_5,\dots)$ we need to find whether $T, T^2$ are compact or not. I see here the definition of compact operator but I'm not able to apply. Could anyone give me hints?

$T^2(x_1,x_2,\dots)=T(0,x_1,0,x_3,0\dots)=(0,0,0,\dots)$.

Answer 1

Certainly $T^2$ is compact. But $T$ isn't, because the range of the unit ball contains $S:=\{e_{2k+1},k\geqslant 0\}$ (where $e_j$ are classical canonical Hilbert basis). The set $S$ is not compact for the same reason as the unit ball isn't.