In functional analysis, the '''compression''' of a linear operator ''T'' on a Hilbert space to a 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 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''.
More generally, for a linear operator ''T'' on a Hilbert space <math>H</math> and an 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 of ''V''. If ''T'' is a self-adjoint operator, then the compression <math>T_W</math> is also self-adjoint. When ''V'' is replaced by the 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)
==References== * P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.
Category:Functional analysis
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