View source for Compression (functional analysis)

In [[Functional analysis|functional analysis]], the '''compression''' of a linear operator ''T'' on a [[Hilbert space]] to a [[Linear subspace|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|invariant subspace]] for ''T'', then the compression of ''T'' to ''K'' is the [[Restriction (mathematics)|restricted]] 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|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 [[Hermitian adjoint|adjoint]] of ''V''. If ''T'' is a [[Self-adjoint operator|self-adjoint operator]], then the compression <math>T_W</math> is also self-adjoint.
When ''V'' is replaced by the [[Inclusion map|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)|Dilation]]

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

[[Category:Functional analysis]]



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