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Is this operator Hermitian?

  1. Oct 20, 2010 #1
    1. Let G be an operator on H (Hilbert Space). Show that:
    (a) H = 1/2 (G + G[tex]^{\dagger}[/tex]) is Hermitian.
    (b) K = -1/2 (G - G[tex]^{\dagger}[/tex]) is Hermitian.
    (c) G = H + iK
    .

    2. Relevant equations
    ...

    3. The attempt at a solution:

    (a) Since the adjoint of the sum of two operators does not change their position (addition of operators is commutative), it is very straight forward.

    (b) This is where I run into trouble, and I think it is because the problem is wrong. The operator given appears to be anti-hermitian (skew-hermitian), however I would like confirmation. This result makes (c) very difficult, as it uses an incorrect answer. My suspicion is that the intended question includes an i (as in, H = -i/2 (...)):

    K[tex]^{\dagger}[/tex] = -1/2 (G[tex]^{\dagger}[/tex] - G) = 1/2 (G - G[tex]^{\dagger}[/tex]) = -K.

    (c) Well if I am correct about (b), (c) is wrong. I was just hoping someone could confirm I am right, or show me how K is Hermitian. I can take it from there.
     
  2. jcsd
  3. Oct 20, 2010 #2
    If the problem should be as stated this will give a condition for G that allows to solve all equations.
    If G is arbitrary and the equations should be satisfied for general G there has to be a change in definition.
    K= -i/2(...)
     
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