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Linearity of A Hermitian Operator

  1. Aug 31, 2007 #1
    Can anybody give me a hint about how can i show that if an operator is linear then it's hermitian conjugate is linear. Thanks for your help from now.
     
  2. jcsd
  3. Aug 31, 2007 #2

    CompuChip

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    I don't know the context, but if you have an inner product perhaps you can try to show that
    [tex]\langle \psi | T^* (\alpha \phi + \beta \psi) \rangle = \langle \psi | \alpha T^* \phi + \beta T^* \psi \rangle [/tex]
    for any [itex]\psi, \phi, \chi \in \operatorname{domain} T[/itex], which would prove the linearity of [itex]T^*[/itex]?
     
  4. Aug 31, 2007 #3
    i want to say that how can i show if [tex]A[/tex] is linear , [tex]A^\dagger[/tex] is linear.i dont know where can i start.please help.
     
  5. Aug 31, 2007 #4

    dextercioby

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    2 questions for the OP:
    * How do you define the domain of definition of the adjoint of a (possibly unbounded) densly defined linear operator in a Hilbert space ?
    * What is the definition of linearity for an unbounded operator in a Hilbert space ?

    I asked these 2 qtns because we want to the give the proof in the most general case, namely when the operator is unbounded but linear and densly defined.

    And btw, this is a purely mathematical problem, it has nothing to do with quantum mechanics.
     
    Last edited: Aug 31, 2007
  6. Aug 31, 2007 #5
    Explaining

    if A is lineer operator

    A[af(x)+bg(x)]=aAf(x) + bAg(x)

    x is the parameter of functions in Hilbert space
     
    Last edited: Aug 31, 2007
  7. Aug 31, 2007 #6

    dextercioby

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    Who's "x" ?
     
  8. Sep 1, 2007 #7

    CompuChip

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    And how is [itex]A^\dagger[/itex] defined?
     
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