Commutant lifting theorem

In operator theory, the commutant lifting theorem, due to Sz.-Nagy and Foias, 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

$R T^n = P_H S U^n \vert_H \; \forall n \geq 0,$

and

$\Vert S \Vert = \Vert R \Vert.$

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.

References

• Vern Paulsen, Completely Bounded Maps and Operator Algebras 2002, ISBN 0-521-81669-6

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