The concept of partial isometry can be defined in other equivalent ways. If U is an isometric map defined on a closed subset H₁ of a Hilbert space H then we can define an extension W of U to all of H by the condition that W be zero on the orthogonal complement of H₁. Thus a partial isometry is also sometimes defined as a closed partially defined isometric map.

Partial isometries (and projections) can be defined in the more abstract setting of a semigroup with involution; the definition coincides with the one herein.

In finite-dimensional vector spaces, a matrix ${\displaystyle A}$  is a partial isometry if and only if ${\displaystyle A^{*}A}$  is the projection onto its support. Contrast this with the more demanding definition of isometry: a matrix ${\displaystyle V}$  is an isometry if and only if ${\displaystyle V^{*}V=I}$ . In other words, an isometry is an injective partial isometry.

Any finite-dimensional partial isometry can be represented, in some choice of basis, as a matrix of the form ${\displaystyle A={\begin{pmatrix}V&0\end{pmatrix}}}$ , that is, as a matrix whose first ${\displaystyle \operatorname {rank} (A)}$  columns form an isometry, while all the other columns are identically 0.

Note that for any isometry ${\displaystyle V}$ , the Hermitian conjugate ${\displaystyle V^{*}}$  is a partial isometry, although not every partial isometry has this form, as shown explicitly in the given examples.

For operator algebras, one introduces the initial and final subspaces:

${\displaystyle {\mathcal {I}}W:={\mathcal {R}}W^{*}W,\,{\mathcal {F}}W:={\mathcal {R}}WW^{*}}$ 

For C*-algebras, one has the chain of equivalences due to the C*-property:

${\displaystyle (W^{*}W)^{2}=W^{*}W\iff WW^{*}W=W\iff W^{*}WW^{*}=W^{*}\iff (WW^{*})^{2}=WW^{*}}$ 

So one defines partial isometries by either of the above and declares the initial resp. final projection to be W*W resp. WW*.

A pair of projections are partitioned by the equivalence relation:

${\displaystyle P=W^{*}W,\,Q=WW^{*}}$ 

It plays an important role in K-theory for C*-algebras and in the Murray-von Neumann theory of projections in a von Neumann algebra.

Any orthogonal projection is one with common initial and final subspace:

${\displaystyle P:{\mathcal {H}}\rightarrow {\mathcal {H}}:\quad {\mathcal {I}}P={\mathcal {F}}P}$ 

Any isometric embedding is one with full initial subspace:

${\displaystyle J:{\mathcal {H}}\hookrightarrow {\mathcal {K}}:\quad {\mathcal {I}}J={\mathcal {H}}}$ 

Any unitary operator is one with full initial and final subspace:

${\displaystyle U:{\mathcal {H}}\leftrightarrow {\mathcal {K}}:\quad {\mathcal {I}}U={\mathcal {H}},\,{\mathcal {F}}U={\mathcal {K}}}$ 

(Apart from these there are far more partial isometries.)

On the two-dimensional complex Hilbert space the matrix

${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}}$ 

is a partial isometry with initial subspace

${\displaystyle \{0\}\oplus \mathbb {C} }$ 

and final subspace

${\displaystyle \mathbb {C} \oplus \{0\}.}$ 

Other possible examples in finite dimensions are $${\displaystyle A\equiv {\begin{pmatrix}1&0&0\\0&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\0&0&0\end{pmatrix}}.}$$  This is clearly not an isometry, because the columns are not orthonormal. However, its support is the span of ${\displaystyle \mathbf {e} _{1}\equiv (1,0,0)}$  and ${\displaystyle {\frac {1}{\sqrt {2}}}(\mathbf {e} _{2}+\mathbf {e} _{3})\equiv (0,1/{\sqrt {2}},1/{\sqrt {2}})}$ , and restricting the action of ${\displaystyle A}$  on this space, it becomes an isometry (and in particular, a unitary). One can similarly verify that ${\displaystyle A^{*}A=\Pi _{\operatorname {supp} (A)}}$ , that is, that ${\displaystyle A^{*}A}$  is the projection onto its support.

Partial isometries do not necessarily correspond to squared matrices. Consider for example, $${\displaystyle A\equiv {\begin{pmatrix}1&0&0\\0&{\frac {1}{2}}&{\frac {1}{2}}\\0&0&0\\0&{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}.}$$ This matrix has support the span of ${\displaystyle \mathbf {e} _{1}\equiv (1,0,0)}$  and ${\displaystyle \mathbf {e} _{2}+\mathbf {e} _{3}\equiv (0,1,1)}$ , and acts as an isometry (and in particular, as the identity) on this space.

Yet another example, in which this time ${\displaystyle A}$  acts like a non-trivial isometry on its support, is$${\displaystyle A={\begin{pmatrix}0&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\1&0&0\\0&0&0\end{pmatrix}}.}$$ One can readily verify that ${\displaystyle A\mathbf {e} _{1}=\mathbf {e} _{2}}$ , and ${\displaystyle A\left({\frac {\mathbf {e} _{2}+\mathbf {e} _{3}}{\sqrt {2}}}\right)=\mathbf {e} _{1}}$ , showing the isometric behavior of ${\displaystyle A}$  between its support ${\displaystyle \operatorname {span} (\{\mathbf {e} _{1},\mathbf {e} _{2}+\mathbf {e} _{3}\})}$  and its range ${\displaystyle \operatorname {span} (\{\mathbf {e} _{1},\mathbf {e} _{2}\})}$ .

On the square summable sequences, the operators

${\displaystyle R:\ell ^{2}(\mathbb {N} )\to \ell ^{2}(\mathbb {N} ):(x_{1},x_{2},\ldots )\mapsto (0,x_{1},x_{2},\ldots )}$ 

${\displaystyle L:\ell ^{2}(\mathbb {N} )\to \ell ^{2}(\mathbb {N} ):(x_{1},x_{2},\ldots )\mapsto (x_{2},x_{3},\ldots )}$ 

which are related by

${\displaystyle R^{*}=L}$ 

are partial isometries with initial subspace

${\displaystyle LR(x_{1},x_{2},\ldots )=(x_{1},x_{2},\ldots )}$ 

and final subspace:

${\displaystyle RL(x_{1},x_{2},\ldots )=(0,x_{2},\ldots )}$ .

• John B. Conway (1999). "A course in operator theory", AMS Bookstore, ISBN 0-8218-2065-6
• Carey, R. W.; Pincus, J. D. (May 1974). "An Invariant for Certain Operator Algebras". Proceedings of the National Academy of Sciences. 71 (5): 1952–1956. Bibcode:1974PNAS...71.1952C. doi:10.1073/pnas.71.5.1952. PMC 388361. PMID 16592156.
• Alan L. T. Paterson (1999). "Groupoids, inverse semigroups, and their operator algebras", Springer, ISBN 0-8176-4051-7
• Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". World Scientific ISBN 981-02-3316-7
• Stephan Ramon Garcia; Matthew Okubo Patterson; Ross, William T. (2019). "Partially isometric matrices: A brief and selective survey". arXiv:1903.11648 [math.FA].

• Important properties and proofs
• Alternative proofs