I am trying to solve this problem but I fail in the converse part…
Show that the product of two self-adjoint operators is self-adjoint if and only
if the two operators commute.
($\Rightarrow$) If we suppose that $T,U,V$ are self-adjoint operators such that $T=UV$ and $U^*=U,V^*=V,T^*=T$ we have by Theorem that:
$$(UV)^*=V^*U^*=VU$$
but we know, by hypothesis, that: $$(UV)^*=T^*=T=UV$$.We already know that the adjoint operator is unique, so we hav that $UV=VU$ and we are done with this part.
So my troubles are this the converse, can anyone help please.
Best Answer
Assume $UV = VU$, and $U,V$ are self-adjoint. $$(UV)^*=V^* U^* = V U = UV \implies (UV)^* = UV$$ i.e, $UV$ is self adjoint. Hopefully that helps.