Commutant lifting theorem
In operator theory, the commutant lifting theorem, due to Sz.-Nagy and Foias, is a powerful theorem used to prove several interpolation results.
Statement
The commutant lifting theorem states that if [math]\displaystyle{ T }[/math] is a contraction on a Hilbert space [math]\displaystyle{ H }[/math], [math]\displaystyle{ U }[/math] is its minimal unitary dilation acting on some Hilbert space [math]\displaystyle{ K }[/math] (which can be shown to exist by Sz.-Nagy's dilation theorem), and [math]\displaystyle{ R }[/math] is an operator on [math]\displaystyle{ H }[/math] commuting with [math]\displaystyle{ T }[/math], then there is an operator [math]\displaystyle{ S }[/math] on [math]\displaystyle{ K }[/math] commuting with [math]\displaystyle{ U }[/math] such that
- [math]\displaystyle{ R T^n = P_H S U^n \vert_H \; \forall n \geq 0, }[/math]
and
- [math]\displaystyle{ \Vert S \Vert = \Vert R \Vert. }[/math]
Here, [math]\displaystyle{ P_H }[/math] is the projection from [math]\displaystyle{ K }[/math] onto [math]\displaystyle{ H }[/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.
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