# Compression (functional analysis)

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In [functional analysis](/source/functional_analysis), the '''compression''' of a [linear operator](/source/linear_operator) ''T'' on a [Hilbert space](/source/Hilbert_space) to a [subspace](/source/Linear_subspace) ''K'' is the operator

:<math>P_K T \vert_K : K \rightarrow K </math>,

where <math>P_K : H \rightarrow K</math> is the [orthogonal projection](/source/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](/source/invariant_subspace) for ''T'', then the compression of ''T'' to ''K'' is the [restricted](/source/restriction_(mathematics)) operator ''K&rarr;K'' sending ''k'' to ''Tk''.

More generally, for a linear operator ''T'' on a Hilbert space <math>H</math> and an [isometry](/source/isometry) ''V'' on a subspace <math>W</math> of <math>H</math>, define the '''compression''' of ''T'' to <math>W</math> by

:<math>T_W = V^*TV : W \rightarrow W</math>,

where <math>V^*</math> is the [adjoint](/source/hermitian_adjoint) of ''V''. If ''T'' is a [self-adjoint operator](/source/self-adjoint_operator), then the compression <math>T_W</math> is also self-adjoint.
When ''V'' is replaced by the [inclusion map](/source/inclusion_map) <math>I: W \to H</math>, <math>V^* = I^*=P_K : H \to W</math>, and we acquire the special definition above.

==See also==
* [Dilation (operator theory)](/source/Dilation_(operator_theory))

==References==
* P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.

Category:Functional analysis

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