In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
a partial isometry is a linear map between Hilbert spaces such that it is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
on the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of its
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
.
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
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
.
General
The concept of partial isometry can be defined in other equivalent ways. If ''U'' is an isometric map defined on a closed subset ''H''
1 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 ''H''
1. 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
.
Operator Algebras
For
operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of ...
s one introduces the initial and final subspaces:
:
C*-Algebras
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:
:
It plays an important role in
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
for C*-algebras and in the
Murray-
von Neumann Von Neumann may refer to:
* John von Neumann (1903–1957), a Hungarian American mathematician
* Von Neumann family
* Von Neumann (surname), a German surname
* Von Neumann (crater), a lunar impact crater
See also
* Von Neumann algebra
* Von Ne ...
theory of projections in a
von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algebra ...
.
Special Classes
Projections
Any orthogonal projection is one with common initial and final subspace:
:
Embeddings
Any isometric embedding is one with full initial subspace:
:
Unitaries
Any
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
is one with full initial and final subspace:
:
''(Apart from these there are far more partial isometries.)''
Examples
Nilpotents
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
This is clearly not an isometry, because the columns are not orthonormal. However, its support is the span of
and
, and restricting the action of
on this space, it becomes an isometry (and in particular a unitary). One can similarly verify that
, that is, that
is the projection onto its support.
Partial isometries need not correspond to squared matrices. Consider for example,
This matrix has support the span of
and
, and acts as an isometry (and in particular, as the identity) on this space.
Yet another example, in which this time
acts like a non-trivial isometry on its support, is
One can readily verify that
, and
, showing the isometric behavior of
between its support
and its range
.
Leftshift and Rightshift
On the square summable sequences the operators
:
:
which are related by
:
are partial isometries with initial subspace
:
and final subspace:
:
.
References
*John B. Conway (1999). "A course in operator theory", AMS Bookstore,
*
*Alan L. T. Paterson (1999).
Groupoids, inverse semigroups, and their operator algebras, Springer,
*Mark V. Lawson (1998).
Inverse semigroups: the theory of partial symmetries.
World Scientific
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various ...
*
External links
Important properties and proofsAlternative proofs
{{DEFAULTSORT:Partial Isometry
Operator theory
C*-algebras
Semigroup theory