Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast\|$? Let $H$ be a Hilbert space and $u \in B(H)$ and let $u^\ast$ denote its adjoint. I know that $\|u\|=\|u^\ast\|$. But now I am wondering:

Does $\|u\|=\|u^\ast\|$ imply $\|uh\| = \|u^\ast h\|$ for all $h\in
 H$?

At first I thought that yes but on second thought I can't argue why it should be true.

Consider $H=\mathbb{R}^2$, and $u\in B(H)$ is given by the matrix $\begin{bmatrix} 1& 1\\
0 &1\end{bmatrix}$, then $u^*$ is the transpose $\begin{bmatrix} 1& 0\\
1 &1\end{bmatrix}$. Let $h=\begin{bmatrix} 1\\
2 \end{bmatrix}$, then
$$uh=\begin{bmatrix} 3\\
2 \end{bmatrix},\quad u^*h=\begin{bmatrix} 1\\
3 \end{bmatrix}$$
clearly $\Vert uh\Vert\neq\Vert u^*h\Vert$.
In fact, it is not hard to show that $\Vert uh\Vert=\Vert u^*h\Vert,\forall h\in H$ if and only if $u$ is normal, i.e. $uu^*=u^*u$. 
