Let $T$ be a continuous and bounded self-adjoint compact operator on a Hilbert space H.
I want to prove that if $T^2=0$, then $T=0$.
Is there any thing wrong with the following:
$T^2$ = $TT^*=0$ impiles that all of T's eigenvalues are zero, and so as T is a compact operator and therefore has finite rank, all of T's eigenvalues are zero, and so $T=0$.