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Let $H$ be a hilbert space over the complex numbers. Let $T:H\to H$ be a linear bounded operator prove that $$\| T\|= \sup_{\| f\|=1} |\langle Tf,f \rangle|.$$

Obviously $\sup_{\| f\|=1} |\langle Tf,f\rangle| \le \| T\|$ by C-S inequality. But how about the other inequality?

ViktorStein
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Agumon
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    this is true for selfadjoint (and also normal) operators, not for all bounded ones – user8268 May 28 '14 at 08:34
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    try $\begin{pmatrix} 0&1\0&0\end{pmatrix}$ to see that it's wrong even in $\mathbb C^2$ – user8268 May 28 '14 at 08:43
  • The right hand side is $|\langle M (x_1, x_2)^T, (x_1, x_2)^T\rangle| = |x_2 \overline{x_1}| \leq \frac{1}{2}(|x_2^2| + |x_1^2|) \leq \frac{1}{2} < 1$. Here I used $xy \leq \frac{1}{2}(x^2 + y^2)$ for real $x,y$ and $M$ as the above matrix. In my first comment I confused the statements regarding the norm and the statement that $\langle Tf, f\rangle = 0$ for all $f$ implies $T = 0$, sorry for that. – PhoemueX May 28 '14 at 08:55

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As user8268 points out, this is wrong even in complex hilbert spaces.

But there is at least the following substitute:

Define $B(u,v) := \langle Tu, v\rangle$. Then $B$ is sesquilinear (i.e. linear in the first argument and antiliear in the second). Define $Q(u) := B(u,u)$. The sesquilinearity of $B$ implies the polarization identity

$$B(u,v) = \frac{1}{4} \sum_{j=0}^3 i^j Q(u + i^j v),$$

where $i = \sqrt{-1}$.

We thus derive

$$|\langle Tu, v\rangle| = |B(u,v)| \leq \frac{1}{4} \sum_{j=0}^3 |Q(u + i^j v)|.$$

The right hand side can be estimated by $4 \cdot \sup_{\Vert f \Vert = 1}|\langle Tf, f\rangle|$, because $\Vert u + i^j v\Vert \leq 2$ if $\Vert u \Vert, \Vert v \Vert \leq 1$.

Then use $\Vert Tu \Vert = \sup_{\Vert v \Vert \leq 1} |\langle Tu, v\rangle|$.

PhoemueX
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    "The right hand side can be estimated by $4 \cdot \sup_{\Vert f \Vert = 1}|\langle Tf, f\rangle|$" Yes, and a better bound $2 \cdot \sup_{\Vert f \Vert = 1}|\langle Tf, f\rangle|$ is proved in this post. This bound is achieved by $\begin{pmatrix} 0&1\0&0\end{pmatrix}$. – hbghlyj Jun 07 '24 at 22:31