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Homework Help: Two Functional Analysis Questions

  1. May 27, 2012 #1
    1. The problem statement, all variables and given/known data
    1. Given an operator [tex]H[/tex] , and a sequence [tex]\{ H_n \} _{n\geq 1 } [/tex] in an arbitrary Hilbert Space , such that both [tex]H[/tex] and [tex] H_n [/tex] are self-adjoint .

    How can I prove that if [tex]||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0 [/tex] and if [tex]H[/tex] has an isolated eigenvalue [tex]\lambda[/tex] of multiplicity one, then for large enough [tex]n[/tex], [tex]H_n[/tex] also have isolated eigenvalues [tex]\lambda _ n[/tex] of multiplicty one that converge to [tex]\lambda[/tex].

    2. Given an operator [tex]H[/tex] , and a sequence [tex]\{ H_n \} _{n\geq 1 } [/tex] in an arbitrary Hilbert Space , such that both [tex]H[/tex] and [tex] H_n [/tex] are self-adjoint and non-negative.

    How can I prove that [tex]||(H_n+1)^{-1} - (H+1) ^ {-1} || \to 0 [/tex] is equivalent to
    [tex]||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0 [/tex] ?



    2. Relevant equations

    3. The attempt at a solution
    I really have no idea about it... I assume it has something to do with the self-adjointness...maybe some estimations on the inner-product...but can't figure out what

    Thanks in advance
     
  2. jcsd
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