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PDE-Cayley Transform Of An Operator

  1. May 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that a unitary operator [itex]U[/itex] acting on a Hilbert space [itex]H[/itex] is the Cayley transform of some self-adjoint operator if and only if [itex]1[/itex] is not an eigenvalue of [itex]U[/itex].


    Hope someone will be able to help

    Thanks !


    2. Relevant equations

    3. The attempt at a solution
    At the first direction, I have tried assuming that 1 is an eigenvalue for the eigenfunction [itex]f[/itex]. This implies [itex]Uf=f[/itex] ,where [itex]U=(L-i)(L+i)^{-1}[/itex] for some [itex]L[/itex].
    But since [itex]U^* U=1[/itex] we get that also [itex]U^* f =f [/itex] , and thus:
    [itex]<Uf,f>=<Uf,U(U^*f)>=<f,Uf>[/itex], which implies [itex]U[/itex] is symmetric, and in particular [itex]U=U^*[/itex] and [itex]U^2=1[/itex]. I can't figure out how to get some contradiction out of this. Maybe we know that [itex]U[/itex] must be one-to-one or something?

    As for the other direction, I need some detailed guidance.


    Thanks in advance
     
  2. jcsd
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