Misplaced Pages

Contraction (operator theory): Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editNext edit →Content deleted Content addedVisualWikitext
Revision as of 22:55, 30 January 2012 editMathsci (talk | contribs)Autopatrolled, Extended confirmed users, Pending changes reviewers66,107 editsm Undid revision 474115482 by 94.196.111.116 (talk) rv sock of banned user Echigo mole← Previous edit Revision as of 22:57, 30 January 2012 edit undo94.196.111.116 (talk) Undid revision 474115767 by Mathsci (talk) - rm obviously false assertionNext edit →
Line 5: Line 5:
Consider the special case where ''T'' is a contraction acting on a ] <math>\mathcal{H}</math>. We define some basic objects associated with ''T''. Consider the special case where ''T'' is a contraction acting on a ] <math>\mathcal{H}</math>. We define some basic objects associated with ''T''.


The '''defect operators''' of ''T'' are the operators ''D<sub>T</sub>''&nbsp;=&nbsp;(1&nbsp;&minus;&nbsp;''T*T'')<sup>&frac12;</sup> and ''D<sub>T*</sub>''&nbsp;=&nbsp;(1&nbsp;&minus;&nbsp;''TT*'')<sup>&frac12;</sup>. The square root is the ] given by the ]. The '''defect spaces''' <math>\mathcal{D}_T</math> and <math>\mathcal{D}_{T*}</math> are the ranges Ran(''D<sub>T</sub>'') and Ran(''D<sub>T*</sub>'') respectively. The positive operator ''D<sub>T</sub>'' induces an inner product on <math>\mathcal{H}</math>. The space <math>\mathcal{D}_T</math> can be identified naturally with <math>\mathcal{H}</math>, with the induced inner product.{{cn|date=December 2011}} The same can be said for <math>\mathcal{D}_{T*}</math>. The '''defect operators''' of ''T'' are the operators ''D<sub>T</sub>''&nbsp;=&nbsp;(1&nbsp;&minus;&nbsp;''T*T'')<sup>&frac12;</sup> and ''D<sub>T*</sub>''&nbsp;=&nbsp;(1&nbsp;&minus;&nbsp;''TT*'')<sup>&frac12;</sup>. The square root is the ] given by the ]. The '''defect spaces''' <math>\mathcal{D}_T</math> and <math>\mathcal{D}_{T*}</math> are the ranges Ran(''D<sub>T</sub>'') and Ran(''D<sub>T*</sub>'') respectively.


The '''defect indices''' of ''T'' are the pair The '''defect indices''' of ''T'' are the pair
Line 21: Line 21:


Contractions on Hilbert spaces can be viewed as the operator analogs of cos&thinsp;&theta; and are called '''operator angles''' in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices. Contractions on Hilbert spaces can be viewed as the operator analogs of cos&thinsp;&theta; and are called '''operator angles''' in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.

==Dilation theorem== ==Dilation theorem==
], proved in 1953, states that for any contraction ''T'' on a Hilbert space ''H'', there is a ] ''U'' on a larger Hilbert space ''K'' ⊇ ''H'' such that if ''P'' is the orthogonal projection of ''K'' onto ''H'' then ''T''<sup>''n''</sup> =''P'' ''U''<sup>''n''</sup> ''P'' for all ''n'' > 0. The operator ''U'' is called a ] of ''T'' and is uniquely determined if ''U'' is mininal, i.e. ''K'' is the smallest closed subspace invariant under ''U'' and ''U''* containing ''H''. ], proved in 1953, states that for any contraction ''T'' on a Hilbert space ''H'', there is a ] ''U'' on a larger Hilbert space ''K'' ⊇ ''H'' such that if ''P'' is the orthogonal projection of ''K'' onto ''H'' then ''T''<sup>''n''</sup> =''P'' ''U''<sup>''n''</sup> ''P'' for all ''n'' > 0. The operator ''U'' is called a ] of ''T'' and is uniquely determined if ''U'' is mininal, i.e. ''K'' is the smallest closed subspace invariant under ''U'' and ''U''* containing ''H''.

Revision as of 22:57, 30 January 2012

In operator theory, a bounded operator T: XY between normed vector spaces X and Y is said to be a contraction if its operator norm ||T|| ≤ 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators.

Contractions on a Hilbert space

Consider the special case where T is a contraction acting on a Hilbert space H {\displaystyle {\mathcal {H}}} . We define some basic objects associated with T.

The defect operators of T are the operators DT = (1 − T*T) and DT* = (1 − TT*). The square root is the positive semidefinite one given by the spectral theorem. The defect spaces D T {\displaystyle {\mathcal {D}}_{T}} and D T {\displaystyle {\mathcal {D}}_{T*}} are the ranges Ran(DT) and Ran(DT*) respectively.

The defect indices of T are the pair

( dim D T , dim D T ) . {\displaystyle ({\mbox{dim}}{\mathcal {D}}_{T},{\mbox{dim}}{\mathcal {D}}_{T^{*}}).}

The defect operators and the defect indices are a measure of the non-unitarity of T.

A contraction T on a Hilbert space can be canonically decomposed into an orthogonal direct sum

T = Γ U {\displaystyle T=\Gamma \oplus U}

where U is a unitary operator and Γ is completely non-unitary in the sense that it has no reducing subspaces on which its restriction is unitary. If U = 0, T is said to be a completely non-unitary contraction. A special case of this decomposition is the Wold decomposition for an isometry, where Γ is a proper isometry.


Contractions on Hilbert spaces can be viewed as the operator analogs of cos θ and are called operator angles in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.

Dilation theorem

Sz.-Nagy's dilation theorem, proved in 1953, states that for any contraction T on a Hilbert space H, there is a unitary operator U on a larger Hilbert space KH such that if P is the orthogonal projection of K onto H then T =P U P for all n > 0. The operator U is called a dilation of T and is uniquely determined if U is mininal, i.e. K is the smallest closed subspace invariant under U and U* containing H.

In fact define

H = H H H , {\displaystyle \displaystyle {{\mathcal {H}}=H\oplus H\oplus H\oplus \cdots ,}}

the orthogonal direct sum of countably many copies of H.

Let V be the isometry on H {\displaystyle {\mathcal {H}}} defined by

V ( ξ 1 , ξ 2 , ξ 3 , ) = ( T ξ 1 , I T T ξ 1 , ξ 2 , ξ 3 , ) . {\displaystyle \displaystyle {V(\xi _{1},\xi _{2},\xi _{3},\dots )=(T\xi _{1},{\sqrt {I-T^{*}T}}\xi _{1},\xi _{2},\xi _{3},\dots ).}}

Let

K = H H . {\displaystyle \displaystyle {{\mathcal {K}}={\mathcal {H}}\oplus {\mathcal {H}}.}}

Define a unitary W on K {\displaystyle {\mathcal {K}}} by

W ( x , y ) = ( V x + ( I V V ) y , V y ) . {\displaystyle \displaystyle {W(x,y)=(Vx+(I-VV^{*})y,-V^{*}y).}}

W is then a unitary dilation of T with H considered as the first component of H K {\displaystyle {\mathcal {H}}\subset {\mathcal {K}}} .

The minimal dilation U is obtained by taking the restriction of W to the closed subspace generated generated by powers of W applied to H.

Notes

  1. Sz.-Nagy et al. 2010, p. 10-14

References

  • Sz.-Nagy, B.; Foias, C.; Bercovici, H.; Kérchy, L. (2010), Harmonic analysis of operators on Hilbert space, Universitext (Second ed.), Springer, ISBN 978-1-4419-6093-1
  • Riesz, F.; Sz.-Nagy, B. (1995), Functional analysis. Reprint of the 1955 original, Dover Books on Advanced Mathematics, Dover, pp. 466–472, ISBN 0-486-66289-6
Category: