Commutant lifting 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  is a  contraction on a Hilbert space ,  is its minimal unitary dilation acting on some Hilbert space  (which can be shown to exist by Sz.-Nagy's dilation theorem), and  is an operator on  commuting with , then there is an operator  on  commuting with  such that

and

Here,  is the projection from  onto . 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. References
Vern Paulsen, Completely Bounded Maps and Operator Algebras 2002, 
B Sz.-Nagy and C. Foias, "The "Lifting theorem" for intertwining operators and some new applications", Indiana Univ. Math. J 20 (1971): 901-904
Foiaş, Ciprian, ed. Metric Constrained Interpolation, Commutant Lifting, and Systems. Vol. 100. Springer, 1998. Operator theory
Theorems in functional analysis