Partial isometry - Knowlpedia

Let $H$ and $K$ be Hilbert spaces. A bounded operator $V:H\to K$ is a partial isometry if its restriction to $(\ker V)^\perp$ preserves norms. The closed subspace $(\ker V)^\perp$ is the initial space, and $\operatorname{Ran}V$ is the final space; the final space is closed because this restriction is an isometry. Equivalently, using the adjoint, $V^*V$ is the orthogonal projection onto the initial space. Then $VV^*$ is the orthogonal projection onto the final space.

Polar decomposition

Every bounded operator $T:H\to K$ has a polar decomposition

T=V|T|,\qquad |T|=(T^*T)^{1/2},

where $V$ is a partial isometry with initial space $\overline{\operatorname{Ran}|T|} =\overline{\operatorname{Ran}T^*}$ and final space $\overline{\operatorname{Ran}T}$ . Requiring $\ker V=\ker T$ makes this partial isometry unique. Thus partial isometries are the correct polar factors even when $T$ is neither injective nor surjective.

Operator-algebra formulation

An element $v$ of a $C^*$ -algebra is called a partial isometry when $v^*v$ is a projection; this implies that $vv^*$ is also a projection. In a concrete operator algebra the two projections record the initial and final spaces. Partial isometries therefore implement Murray–von Neumann equivalence between projections.

Examples and non-examples

Every orthogonal projection, unitary operator, and isometric embedding is a partial isometry. The unilateral shift is a partial isometry whose initial space is all of $\ell^2(\mathbb N)$ and whose final space has codimension one. The zero operator also qualifies, with zero initial and final spaces. By contrast, $2I$ on a nonzero Hilbert space is not a partial isometry because it does not preserve norms on the orthogonal complement of its kernel.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. DOI record. Relevant: Chapter II on adjoints and polar decomposition.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. DOI record. Relevant: §2.5 on partial isometries and projections.