• January 27th 2010, 03:25 PM
putnam120
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 $\parallel T\parallel^2\le\parallel T^*T\parallel$
• January 27th 2010, 07:04 PM
Black
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, $\|T^*\|=\|T\|$.
• January 27th 2010, 07:41 PM
putnam120
Oh Wow how did I not see that. Thanks.