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Math Help - Real eigenvalues of linear operator

  1. #1
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    Real eigenvalues of linear operator

    Let L : D--->H be a linear operator defined on a domain D in a complex Hilbert space H. Given that L is self-adjoint, show that its eigenvalues are real.

    I know Lf=af where a is the eigen value of L and f is the eigenfunction. L is self-adjoint <Lf,g>=<f,Lg>
    How do i show eigenvalues are real? many thanks
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Let L : D--->H be a linear operator defined on a domain D in a complex Hilbert space H. Given that L is self-adjoint, show that its eigenvalues are real.

    I know Lf=af where a is the eigen value of L and f is the eigenfunction. L is self-adjoint <Lf,g>=<f,Lg>
    How do i show eigenvalues are real? many thanks


    When working with orthonormal basis you know that L^{*}=\overline{L^t} , and if \lambda is an eignevalue of L then \overline{\lambda} is an eigenvalue of L^{*} , so...

    Tonio
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  3. #3
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    Also, suppose \lambda is an eigenvalue for self adjoint operator L. Then Lv= \lambda v for some v with norm 1.
    Now \lambda= \lambda<v, v>= <\lambda v, v>= <Lv, v>. Since L is self adjoint, <Lv, v>= <v, Lv>= <v, \lambda v>= \overline{<\lambda v, v>}= \overline{\lambda}<v, v>= \overline{\lambda}.
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