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 ...

, the compression of a linear operator ''T'' on a Hilbert space to a subspace ''K'' is the operator :

P_K T \vert_K : K \rightarrow K

, where

P_K : H \rightarrow K

is the

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

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 In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General descri ...

for ''T'', then the compression of ''T'' to ''K'' is the restricted operator ''K→K'' sending ''k'' to ''Tk''. More generally, for a linear operator ''T'' on a Hilbert space

H

and 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 ...

''V'' on a subspace

W

H

, define the compression of ''T'' to

W

by :

T_W = V^*TV : W \rightarrow W

, where

V^*

is the

adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...

of ''V''. If ''T'' is a

self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...

, then the compression

T_W

is also self-adjoint. When ''V'' is replaced by the

inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota ...

I: W \to H

V^* = I^*=P_K : H \to W

, and we acquire the special definition above.

References

* P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982. Functional analysis {{mathanalysis-stub

See also

References