Finding difficulty in giving an example:
An operator in B(H) such that $\|T\| \neq \sup_{\|h\|=1} |\langle Th,h\rangle |$.
2026-03-25 07:49:26.1774424966
$\|T\| \neq \sup_{\|h\|=1} |\langle Th,h\rangle |$
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The equality holds for selfadjoints. The typical candidate for a counterexample is then the canonical non-normal operator, $$T=\begin{bmatrix} 0&1\\0&0\end{bmatrix} .$$ It is easy to check that $\|T\|=1$, but $|\langle Th,h\rangle|\leq1/2$ for all $h$ with $\|h\|=1$.