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Thread: Inverse of Mapping from Hilbert Space to Hilbert Space exists

  1. #1
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    Inverse of Mapping from Hilbert Space to Hilbert Space exists

    Let S = I + T^{\ast}T : H \longrightarrow H, where T is linear and bounded.

    Show that S^{-1} S(H) \longrightarrow H exists.

    (I is the Identity operator)
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  2. #2
    Super Member Rebesques's Avatar
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    Hint.

    $\displaystyle (1+x)(1-x+x^2-x^3+...) = 1$
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    I'm lost as to how to use this to find a solution for the question.
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  4. #4
    Super Member Rebesques's Avatar
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    Let $\displaystyle K=T^*\circ T, \ K^2=K\circ K,
    K^3=K\circ K\circ K$ etc

    and consider

    $\displaystyle I-K+K^2-K^3+...$

    You 'll also need to guarantee this converges.
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