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Math Help - square root lemma

  1. #1
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    square root lemma

    Hello,

    in the book functional analysis by reed, there is a lemma which says:
    If A \in L(H) is a bounded linear operator on a Hilbert Space, and  A\ge0. Then there is a unique \( B \in L(H) \) B\ge0 s.t. B^2 = A.


    I have a question about the proof. The author argues:
    "It is sufficient to consider the case where  \|A\| \ge 1."

    Why is it sufficient to proof just this case?

    I couldn't find the right argument.

    Regards
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  2. #2
    Super Member girdav's Avatar
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    Re: square root lemma

    If the result has been shown for the operators which has a norm \geq 1, let T\in\mathcal{B}(H), T\geq 0. If T=0 take B=0 and if \lVert T\rVert \neq 0 then S:=\frac{T}{\lVert T\rVert}\in\mathcal{B}(H) and S\geq 0. We can find an unique B\geq 0 such that B^2=S i.e. \lVert T\rVert^2B^2 =T. We take B' := \lVert T\rVert B.
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  3. #3
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    Re: square root lemma

    Thank you very much!
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