Substitution matrix

2014-02-24T01:37:45Z

115.248.50.22: /* BLOSUM */

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Subbayya Sivasankaranarayana Pillai

2013-12-05T10:04:51Z

115.248.50.22:

In [[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]] for ''T'', then the compression of ''T'' to ''K'' is the [[restriction|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 [[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 [[identity function]] <math>I: W -> H</math>, <math>V^* = I^*=P_K : H -> 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]]

{{mathanalysis-stub}}

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Substitution matrix

Subbayya Sivasankaranarayana Pillai