commutant lifting theorem

Proper Noun

• 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 $R\; T^n\; =\; P\_H\; S\; U^n\; \backslash vert\_H\; \backslash ;\; \backslash forall\; n\; \backslash geq\; 0,$ and $\backslash Vert\; S\; \backslash Vert\; =\; \backslash Vert\; R\; \backslash Vert$. 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.