TITLE: Norm of operator in a Hilbert space
QUESTION [8 upvotes]: Consider a complex Hilbert space $H$ and an operator $T\in\mathcal{L}(H,H)$. Define
$$\|T\|=\sup_{\|x\|=\|y\|=1}\lvert\langle Tx, y\rangle\rvert$$, $$\||T|\|=\sup_{\|x\|=1}\lvert\langle Tx,x\rangle\rvert$$
Can you find an example of a complex Hilbert space $H$ and an operator $T$ such that $\|T\|=2\||T|\|$. I've been thinking about it and I can't find it.

REPLY [5 votes]: Take, for example, 
$$
T=\begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}.
$$
Then
$$
\sup_{\|x\|=1}|x^TTx|=\sup_{\|x\|=1}|x_1x_2|=\sup_{t}|\cos t\sin t|=\frac12\sup_t|\sin 2t|=\frac12,
$$
but
$$
\sup_{\|x\|=\|y\|=1}|y^TTx|=\sup_{\|x\|=\|y\|=1}|y_1x_2|=1.
$$