en>ChrisGualtieri: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

2013-12-19T05:43:10Z

Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

New page

In [[operator theory]], the '''commutant lifting theorem''', due to [[Béla Szőkefalvi-Nagy|Sz.-Nagy]] and [[Ciprian Foias|Foias]], is a powerful theorem used to prove several interpolation results.

==Statement==
The commutant lifting theorem states that if ''T'' is a [[contraction (operator theory) | contraction]] on a [[Hilbert space]] ''H'', ''U'' is its minimal unitary [[Dilation_(operator_theory)|dilation]] acting on some Hilbert space ''K'' (which can be shown to exist by [[Sz.-Nagy's dilation theorem]]), and ''R'' is an operator on ''H'' commuting with ''T'', then there is an operator ''S'' on ''K'' commuting with ''U'' such that
:<math>R T^n = P_H S U^n \vert_H \; \forall n \geq 0,</math>

and

:<math>\Vert S \Vert = \Vert R \Vert.</math>

In other words, an operator from the [[commutant]] of ''T'' can be "lifted" to an operator in the commutant of the unitary dilation of ''T''.

==Applications==
The commutant lifting theorem can be used to prove the left [[Nevanlinna-Pick interpolation]] theorem, the [[Sarason interpolation theorem]], and the two-sided Nudelman theorem, among others.

==References==
*Vern Paulsen, ''Completely Bounded Maps and Operator Algebras'' 2002, ISBN 0-521-81669-6
*B Sz.-Nagy and C. Foias, "The "Lifting theorem" for intertwining operators and some new applications", ''Indiana Univ. Math. J'' 20 (1971): 901-904
*Foiaş, Ciprian, ed. ''Metric Constrained Interpolation, Commutant Lifting, and Systems. Vol. 100. Springer, 1998''.
[[Category:Operator theory]]
[[Category:Theorems in functional analysis]]

Spherical design - Revision history

en>ChrisGualtieri: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB