$T \in B(H)$, and $T = S^2$ for some self adjoint operator $S \in B(H)$. I need to prove that T is compact if and only if S is compact.
If S is compact, it is easy to show that T is compact since S is also bounded and the product of compact operator and bounded linear operator is compact. But how can I prove the converse of it?
Also, is self-adjoint necessary here, if we drop the self-ajoint property, does the statement still hold?