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

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The commutant lifting theorem states that if   is a contraction on a Hilbert space  ,   is its minimal unitary dilation acting on some Hilbert space   (which can be shown to exist by Sz.-Nagy's dilation theorem), and   is an operator on   commuting with  , then there is an operator   on   commuting with   such that

 

and

 

Here,   is the projection from   onto  . 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

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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

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  • 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.