The orthogonal complement of its kernel is called the initial subspace and its range is called the final subspace.
Partial isometries appear in the polar decomposition.
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 is a partial isometry if and only if is the projection onto its support. Equivalently, any finite-dimensional partial isometry can be represented, in some choice of basis, as a matrix of the form , that is, as a matrix whose first columns form an isometry, while all the other columns are identically 0.
Yet another general way to characterize finite-dimensional partial isometries is to observe that partial isometries coincide with the Hermitian conjugates of isometries, meaning that a given is a partial isometry if and only if is an isometry. More precisely, if is a partial isometry, then is an isometry with support the range of , and if is some isometry, then is a partial isometry with support the range of .
For operator algebras one introduces the initial and final subspaces:
For C*-algebras one has the chain of equivalences due to the C*-property:
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:
Any orthogonal projection is one with common initial and final subspace:
Any isometric embedding is one with full initial subspace:
Any unitary operator is one with full initial and final subspace:
(Apart from these there are far more partial isometries.)
On the two-dimensional complex Hilbert space the matrix
is a partial isometry with initial subspace
and final subspace
Generic finite-dimensional examples
Other possible examples in finite dimensions are
Partial isometries need not correspond to squared matrices. Consider for example,
Yet another example, in which this time acts like a non-trivial isometry on its support, is
Leftshift and Rightshift
On the square summable sequences the operators
which are related by
are partial isometries with initial subspace
and final subspace:
- 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].