# Proof that HK is hermitian operator only if HK=KH

1. May 2, 2013

### skrat

Let $H$ and $K$ be hermitian operators on vector space $U$. Show that operator $HK$ is hermitian if and only if $HK=KH$.

I tried some things but I don't know if it is ok. Can somebody please check? I got a hint on this forum that statements type "if only if" require proof in both directions, so here is how it goes:

1.)Lets say that $HK$ is hermitian, than $HK=(HK)^{*}=K^{*}H^{*}$. But $H$ and $K$ are also hermitian, therefore $K^{*}H^{*}=KH$ so $HK=KH$ proof finished in one direction. (do you say direction or do you say way or what do you say in english? )

2.) Now lets say that $HK=KH$. Since $H$ and $K$ are hermitian: $KH=K^{*}H^{*}=(HK)^{*}=HK$ (last equality comes from the statement at the beginning thah $KH=HK$). But if $(HK)^{*}=HK$ than $HK$ is hermitian.

proof finished.

2. May 2, 2013

### micromass

That's completely correct!!

And yes, direction is the right word.