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.