Commutant lifting theorem

Let be a contraction on a Hilbert space , that is, . Recall that is an isometric dilation of if is an isometry (cf. Isometric operator) on a Hilbert space and is an invariant subspace for satisfying . The Sz.-Nagy–Shäffer construction shows that all contractions admit an isometric dilation [a1], [a5]. This sets the stage for the following result, known as the Sz.-Nagy–Foias commutant lifting theorem [a1], [a4], [a5].

Let on be an isometric dilation for a contraction on . Let be an operator from the Hilbert space into and an isometry on satisfying . Then there exists an operator from into satisfying the following three conditions: , and , where is the orthogonal projection onto .

The commutant lifting theorem was inspired by seminal work of D. Sarason [a3] on interpolation. It can be used to solve many classical and modern interpolation problems, including the Carathéodory, Nevanlinna–Pick, Hermite–Féjer, Nudelman, Nehari, and Löwner interpolation problems in both their classical and tangential form (see [a1] and also Carathéodory interpolation; Nevanlinna–Pick interpolation). The commutant lifting theorem can also be used to solve problems in control theory and inverse scattering [a1], [a2].

There is a one-to-one correspondence between the set of all solutions in the commutant lifting theorem and a certain choice sequence of contractions. This choice sequence is a generalization of the Schur numbers used to solve the Carathéodory interpolation problem or the reflection coefficients appearing in inverse scattering problems for layered media in geophysics. There is also a one-to-one correspondence between the sets of all solutions for the commutant lifting theorem and a certain contractive analytic function in the open unit disc. This characterization of all solutions has several different network interpretations [a1].

As an illustration of the commutant lifting theorem, consider the Nehari interpolation problem

where is a given function in . Here, is the Banach space of all Lebesgue-measurable functions on the unit circle whose norm is finite, and is the subspace of consisting of all functions in whose Fourier coefficients at are zero for all . Likewise, is the Hilbert space of all Lebesgue-measurable, square-integrable functions on the unit circle, and is the subspace of consisting of all functions in whose Fourier coefficients at vanish for all . Now, let be the Hankel operator from into defined by for in . Let be the isometry on and the unitary operator on defined by and , respectively. Let be the contraction on defined by for in . Since , it follows that is an isometric lifting of . By applying the commutant lifting theorem, there exists an operator from into satisfying , and . Therefore, the error , and there exists an such that .

References

[a1]	C. Foias, A.E. Frazho, "The commutant lifting approach to interpolation problems" , Operator Theory: Advances and Applications , 44 , Birkhäuser (1990)
[a2]	C. Foias, H. Özbay, A. Tannenbaum, "Robust control of infinite-dimensional systems" , Springer (1996)
[a3]	D. Sarason, "Generalized interpolation in " Trans. Amer. Math. Soc. , 127 (1967) pp. 179–203
[a4]	B. Sz.-Nagy, C. Foias, "Dilatation des commutants d'opérateurs" C.R. Acad. Sci. Paris , A266 (1968) pp. 493–495
[a5]	B. Sz.-Nagy, C. Foias, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970)