Norm of adjoint

**putnam120** · January 27th 2010, 02:25 PM

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 $\parallel T\parallel^2\le\parallel T^*T\parallel$

**Black** · January 27th 2010, 06:04 PM

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, $\|T^*\|=\|T\|$ .

**putnam120** · January 27th 2010, 06:41 PM

Oh Wow how did I not see that. Thanks.

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