Commutant lifting theorem explained

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

T is a contraction on a Hilbert space H , U is its minimal unitary 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 thatR T ^{n} = P _{H} S U ^{n} \vert _{H} \forall n \geq 0 ,

and

\Vert S \Vert = \Vert R \Vert.

Here,

P _{H} is the projection from K onto H . 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,
• 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.