Isometry coisometry and partial isometry

Assume Countable Choice and let $H,K$ be nonzero complex Hilbert spaces with $U\in \mathcal{B}(H,K)$ a bounded linear operator (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

• $U$ is an isometry when $\mathrm{\parallel }Ux\mathrm{\parallel }=\mathrm{\parallel }x\mathrm{\parallel }$ for every $x\in H$; equivalently $U^{\ast }U=I_H$.
• $U$ is a coisometry when $U^{\ast }$ is an isometry; equivalently $U{U}^{\ast }=I_K$.
• $U$ is a partial isometry when $U$ vanishes on its kernel and is isometric on the orthogonal complement of its kernel: $x\in \ker U⟹Ux=0,x\in (\ker U{)}^{\perp }⟹\mathrm{\parallel }Ux\mathrm{\parallel }=\mathrm{\parallel }x\mathrm{\parallel }\mathrm{.}$ The closed subspace $(\ker U{)}^{\perp }$ is the initial space of $U$, and the closed subspace $\mathrm{ran}U$ is its final space.

The equivalence in the isometry clause. If $U^{\ast }U=I$ then $\mathrm{\parallel }Ux{\mathrm{\parallel }}^2=⟨U^{\ast }Ux,x⟩=⟨x,x⟩$. Conversely, if $\mathrm{\parallel }Ux\mathrm{\parallel }=\mathrm{\parallel }x\mathrm{\parallel }$ for every $x$, then $S:=U^{\ast }U-I$ is self-adjoint and $⟨Sx,x⟩=\mathrm{\parallel }Ux{\mathrm{\parallel }}^2-\mathrm{\parallel }x{\mathrm{\parallel }}^2=0$ for every $x$; the four-term expansion of the sesquilinear form $(x,y)\mapsto ⟨Sx,y⟩$ applied to the vanishing diagonal values gives $⟨Sx,y⟩=0$ for all $x,y$, hence $S=0$, that is $U^{\ast }U=I$ (Hilbert-adjoint identities for the adjoint identities).

Well-definedness of the subspaces. The kernel $\ker U$ is a closed linear subspace because $U$ is bounded and linear, so its orthogonal complement is a closed subspace and the orthogonal-decomposition theorem gives $H=\ker U\oplus (\ker U{)}^{\perp }$ (Orthogonal decomposition by a closed subspace, Orthogonality and the orthogonal complement). The range of a partial isometry is closed: $U$ is isometric on the closed subspace $(\ker U{)}^{\perp }$ and vanishes on its orthogonal complement, so $U$ carries the unit sphere of $(\ker U{)}^{\perp }$ to a closed set and $\mathrm{ran}U=U[(\ker U{)}^{\perp }]$ is closed. In the terminology of The Hilbert orthogonal projection onto a closed subspace, the orthogonal projection onto the initial space is an orthogonal projection in the sense of that item, and the partial isometry restricted to it is an isometry onto $\mathrm{ran}U$.

Immediate cases and conventions. Every isometry and every coisometry is a partial isometry: an isometry has $\ker U=\{0\}$ and is isometric on $H=(\ker U{)}^{\perp }$, and a coisometry has $(\ker U{)}^{\perp }=\mathrm{ran}{U}^{\ast }$ on which it is isometric (Hilbert-adjoint identities). The zero operator is a partial isometry, with $\ker U=H$ and $\mathrm{ran}U=\{0\}$ (Hilbert space). A partial isometry need not be an isometry and need not be unitary; the unilateral shift on the companion page is an isometry that is not a coisometry.