Isometry Operator

Definition

An operator $T:\mathcal{H}\to \mathcal{H}$ is an isometry if $\Vert Tf\Vert =\Vert f\Vert$ for all $f\in \mathcal{H}$.

Proposition

1. If $T$ is an isometry, then $⟨Tf,Tg⟩=⟨f,g⟩$ for every $f,g\in \mathcal{H}$. $T^{\ast }T=I$.
2. . If $T$ is an isometry and $T$ is surjective, then $T$ is unitary and $T{T}^{\ast }=I$.
3. There exist isometry operator that is not unitary.
4. Show that if $T^{\ast }T$ is unitary then $T$ is an isometry.