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Math Help - 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.

    (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 K=T^*\circ T, \ K^2=K\circ K, <br />
 K^3=K\circ K\circ K etc

    and consider

    I-K+K^2-K^3+...

    You 'll also need to guarantee this converges.
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