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 ${\displaystyle T}$ is a contraction on a Hilbert space ${\displaystyle H}$, ${\displaystyle U}$ is its minimal unitary dilation acting on some Hilbert space ${\displaystyle K}$ (which can be shown to exist by Sz.-Nagy's dilation theorem), and ${\displaystyle R}$ is an operator on ${\displaystyle H}$ commuting with ${\displaystyle T}$, then there is an operator ${\displaystyle S}$ on ${\displaystyle K}$ commuting with ${\displaystyle U}$ such that

${\displaystyle RT^{n}=P_{H}SU^{n}\vert _{H}\;\forall n\geq 0,}$

and

${\displaystyle \Vert S\Vert =\Vert R\Vert .}$

Here, ${\displaystyle P_{H}}$ is the projection from ${\displaystyle K}$ onto ${\displaystyle H}$. 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, ISBN 0-521-81669-6
• 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.