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 $T$ 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 that

$RT^{n}=P_{H}SU^{n}\vert _{H}\;\forall n\geq 0,$

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, 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.