{"document_url": "https://en.wikipedia.org/wiki/Commutant_lifting_theorem"}
Operator 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
and
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.