commutant lifting theorem

commutant lifting theorem

1. (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 ${\displaystyle RT^{n}=P_{H}SU^{n}\vert _{H}\;\forall n\geq 0,}$ and ${\displaystyle \Vert S\Vert =\Vert R\Vert }$. 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.