Commutant Lifting Theorem Definition

A theorem in operator theory , stating that, if T is a contraction on a Hilbert space H, and U is its minimal unitary dilation acting on some Hilbert space K, 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. 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.