In operator theory, the commutant lifting theorem, due to Sz.-Nagy and Foias, is a powerful theorem used to prove several interpolation results.
The commutant lifting theorem states that if
is its minimal unitary dilation acting on some Hilbert space
(which can be shown to exist by Sz.-Nagy's dilation theorem), 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. 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.