Compression (functional analysis)
Encyclopedia
In functional analysis
, the compression of a linear operator T on a Hilbert space
to a subspace
K is the operator
where
is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space. If K is an invariant subspace
for T, then the compression of T to K is the restricted
operator K→K sending k to Tk. General, let
be isometry
on Hilbert space
, subspace of Hilbert space 
(T on 
). We define compression 
of a linear operator T on a Hilbert space
to a subspace
W as linear operator
. If T is hermitian operator, then compression is hermitian too.
When replace V with identity
(that implies 
),
we encounter special definition.
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the compression of a linear operator T on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
to a subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
K is the operator
where
is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space. If K is an invariant subspaceInvariant subspace
In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
for T, then the compression of T to K is the restricted
Restriction
Restriction may refer to:* Restriction , an aspect of a mathematical function* Restrictions , an album by Cactus* Restriction enzyme, a type of enzyme that cleaves genetic material...
operator K→K sending k to Tk. General, let
be isometryIsometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
on Hilbert space
, subspace of Hilbert space (T on ). We define compression of a linear operator T on a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
to a subspaceSubspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
W as linear operator
. If T is hermitian operator, then compression is hermitian too.When replace V with identity
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
(that implies ),we encounter special definition.