- #1
skrat
- 740
- 8
Let [itex]H[/itex] and [itex]K[/itex] be hermitian operators on vector space [itex]U[/itex]. Show that operator [itex]HK[/itex] is hermitian if and only if [itex]HK=KH[/itex].
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 [itex]HK[/itex] is hermitian, than [itex]HK=(HK)^{*}=K^{*}H^{*}[/itex]. But [itex]H[/itex] and [itex]K[/itex] are also hermitian, therefore [itex]K^{*}H^{*}=KH[/itex] so [itex]HK=KH[/itex] 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 [itex]HK=KH[/itex]. Since [itex]H[/itex] and [itex]K[/itex] are hermitian: [itex]KH=K^{*}H^{*}=(HK)^{*}=HK[/itex] (last equality comes from the statement at the beginning thah [itex]KH=HK[/itex]). But if [itex](HK)^{*}=HK[/itex] than [itex]HK[/itex] 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 [itex]HK[/itex] is hermitian, than [itex]HK=(HK)^{*}=K^{*}H^{*}[/itex]. But [itex]H[/itex] and [itex]K[/itex] are also hermitian, therefore [itex]K^{*}H^{*}=KH[/itex] so [itex]HK=KH[/itex] 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 [itex]HK=KH[/itex]. Since [itex]H[/itex] and [itex]K[/itex] are hermitian: [itex]KH=K^{*}H^{*}=(HK)^{*}=HK[/itex] (last equality comes from the statement at the beginning thah [itex]KH=HK[/itex]). But if [itex](HK)^{*}=HK[/itex] than [itex]HK[/itex] is hermitian.
proof finished.