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Math Help - Norm of adjoint

  1. #1
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    Norm of adjoint

    Suppose T\in L(H,H) where H is a Hilbert space. How do I go about showing \parallel T\parallel =\parallel T^*\parallel?
    So far all I am able to get to is that \parallel T\parallel^2\le\parallel T^*T\parallel
    Last edited by putnam120; January 27th 2010 at 07:45 PM.
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  2. #2
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    For x \in H

    \|T^{*}x\|^2=|\langle T^{*}x,T^{*}x\rangle|=|\langle x,TT^{*}x \rangle| \le \|x\|\|TT^{*}x\| \le \|x\|\|T\|\|T^*x\|.

    Divide by \|T^{*}x\| to get \|T^{*}x\| \le \|T\|\|x\| \Longrightarrow \|T^*\| \le \|T\|.

    Since T^{**}=T, we have \|T\| \le \|T^*\|. Therefore, \|T^*\|=\|T\|.
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  3. #3
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    Oh Wow how did I not see that. Thanks.
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