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Homework Help: Linear Operator formulation

  1. Sep 7, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that any linear operator [itex]\hat{L}[/itex] can be written as [itex]\hat{L} = \hat{A} + i\hat{B}[/itex], where [itex]\hat{A}[/itex] and [itex]\hat{B}[/itex] are Hermitian operators.

    2. Relevant equations
    The properties of hermitian operators.

    3. The attempt at a solution
    I am not sure where to start with this one. For example, we know that if an operator, A is hermitian, then [itex]\langle g\mid A f \rangle = \langle f\mid A g\rangle^*[/itex]. But I do not see how to break up L into any combination of other operators. Any help would be appreciated, perhaps a nudge in the right direction.
  2. jcsd
  3. Sep 7, 2014 #2


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    Science Advisor

    What would ##\hat L^\dagger## look like?
  4. Sep 7, 2014 #3


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    Gold Member

    If ##\hat{A},\hat{B}## are Hermitian then ##i\hat{B}## is anti-Hermitian. So the problem is really just asking you to prove that any operator is a sum of a Hermitian part and an anti-Hermitian part. This is very similar that any real linear operator is a sum of a symmetric part and an anti-symmetric part. Do you know how that property is proved?
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