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Math Help - inner product

  1. #1
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    inner product

    Can you help me with this. I donít know what to do
    Let R, S, T are linear operators, where V is a complex inner product space.
    (i) Suppose that S is an isometry and R is a positive operator such that T=SR. Prove that R=square root of (T*T)
    (ii) Let σ denote the smallest singular value of T, and let σ*denote the largest singular value of T. Prove that σ<=|| T(v)/||v|| ||<= σ* for every nonzero v in V.

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  2. #2
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    Quote Originally Posted by bamby View Post
    Let R, S, T be linear operators, where V is a complex inner product space.
    (i) Suppose that S is an isometry and R is a positive operator such that T=SR. Prove that R=square root of (T*T)
    (ii) Let σ denote the smallest singular value of T, and let σ*denote the largest singular value of T. Prove that σ<=|| T(v)/||v|| ||<= σ* for every nonzero v in V.
    (i) If S is an isometry then S*S = I. Therefore T^*T = RS^*SR = R^2. But a positive operator has a unique positive square root, so R=(T^*T)^{1/2}.

    (ii) \|Tv\|^2 = \langle T^*Tv,v\rangle = \langle R^2v,v\rangle. But \sigma^2 I\leqslant R^2\leqslant \sigma^{*2}I. So \sigma^2\langle v,v\rangle \leqslant\langle R^2v,v\rangle\leqslant \sigma^{*2}\langle v,v\rangle, from which \sigma\|v\|\leqslant\|Tv\|\leqslant\sigma^*\|v\|.
    Last edited by Opalg; November 14th 2008 at 01:17 AM. Reason: corrected typo
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