- #1
skrat
- 740
- 8
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 let's 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.
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 let's 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.
