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
:$R T^n = P_H S U^n \vert_H \; \forall n \geq 0,$

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''.

{{Functional analysis

Operator theory
Theorems in functional analysis