commutant lifting theorem

== English == === Proper noun === the commutant lifting theorem (mathematics) 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 | H ∀ n ≥ 0 , {\displaystyle RT^{n}=P_{H}SU^{n}\vert _{H}\;\forall n\geq 0,} and ‖ S ‖ = ‖ R ‖ {\displaystyle \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.