Question

suppose s t ∈ l(v) are self-adjoint. Prove that st is self-adjoint if and only if st=ts

Answer 1

Since $s, t, st$ are self-adjoint, then $s^*=s,t^*=t$ and $(st)^*=st$ .

Then $st=(st)^∗=t^∗s^∗=ts .$

Since $s^*=s,t^*=t$ and $st=ts$ ,

we have $(st)^*=t^*s^*=ts=st$ .

So $st$ is self-adjoint