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Proof that HK is hermitian operator only if HK=KH

  1. May 2, 2013 #1
    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? o_O)

    2.) Now lets 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.
     
  2. jcsd
  3. May 2, 2013 #2

    micromass

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    That's completely correct!!

    And yes, direction is the right word.
     
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