Remark 2

— The relation $u^*u = 1_E$ does not characterize all the automorphisms of the hilbertian space $E$. For example, let $E = \ell^2(\mathbf{N})$ and let $u$ be defined by $u(x_n) = x_{n-1}$ for $n \geq 1$ and $u(x)_0 = 0$. We have $\|u(x)\| = \|x\|$ for all $x \in E$, that is, $u^*u = 1_E$, but $u$ is not surjective.