Commutant lifting theorem

Let $ T _ {1} $ be a contraction on a Hilbert space $ {\mathcal H} _ {1} $, that is, $ \| {T _ {1} } \| \leq 1 $. Recall that $ U $ is an isometric dilation of $ T _ {1} $ if $ U $ is an isometry (cf. Isometric operator) on a Hilbert space $ {\mathcal K} \supseteq {\mathcal H} _ {1} $ and $ {\mathcal H} _ {1} $ is an invariant subspace for $ U ^ {*} $ satisfying $ U ^ {*} \mid {\mathcal H} _ {1} = T _ {1} ^ {*} $. 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 $ U $ on $ {\mathcal K} $ be an isometric dilation for a contraction $ T _ {1} $ on $ {\mathcal H} _ {1} $. Let $ A $ be an operator from the Hilbert space $ {\mathcal H} $ into $ {\mathcal H} _ {1} $ and $ T $ an isometry on $ {\mathcal H} $ satisfying $ T _ {1} A = AT $. Then there exists an operator $ B $ from $ {\mathcal H} $ into $ {\mathcal K} $ satisfying the following three conditions: $ UB = BT $, $ \| B \| = \| A \| $ and $ PB = A $, where $ P $ is the orthogonal projection onto $ {\mathcal H} _ {1} $.

The commutant lifting theorem was inspired by seminal work of D. Sarason [a3] on $ H ^ \infty $ interpolation. It can be used to solve many classical and modern $ H ^ \infty $ 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 $ H ^ \infty $ 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

$$ d _ \infty = \inf \left \{ {\left \| {f - h } \right \| _ \infty } : {h \in H ^ \infty } \right \} , $$

where $ f $ is a given function in $ L ^ \infty $. Here, $ L ^ \infty $ is the Banach space of all Lebesgue-measurable functions $ g $ on the unit circle whose norm $ \| g \| _ \infty = { \mathop{\rm ess} \sup } \{ {| {g ( e ^ {it } ) } | } : {0 \leq t < 2 \pi } \} $ is finite, and $ H ^ \infty $ is the subspace of $ L ^ \infty $ consisting of all functions $ g $ in $ L ^ \infty $ whose Fourier coefficients at $ e ^ {int } $ are zero for all $ n < 0 $. Likewise, $ L ^ {2} $ is the Hilbert space of all Lebesgue-measurable, square-integrable functions on the unit circle, and $ H ^ {2} $ is the subspace of $ L ^ {2} $ consisting of all functions in $ L ^ {2} $ whose Fourier coefficients at $ e ^ {int } $ vanish for all $ n < 0 $. Now, let $ A $ be the Hankel operator from $ {\mathcal H} = H ^ {2} $ into $ {\mathcal H} _ {1} = L ^ {2} \ominus H ^ {2} $ defined by $ Ax = Pfx $ for $ x $ in $ H ^ {2} $. Let $ T $ be the isometry on $ H ^ {2} $ and $ U $ the unitary operator on $ {\mathcal K} = L ^ {2} $ defined by $ Tx = e ^ {it } x $ and $ Uy = e ^ {it } y $, respectively. Let $ T _ {1} $ be the contraction on $ {\mathcal H} _ {1} $ defined by $ T _ {1} h _ {1} = PUh _ {1} $ for $ h _ {1} $ in $ {\mathcal H} _ {1} $. Since $ T _ {1} ^ {*} = U ^ {*} \mid {\mathcal H} _ {1} $, it follows that $ U $ is an isometric lifting of $ T _ {1} $. By applying the commutant lifting theorem, there exists an operator $ B $ from $ H ^ {2} $ into $ L ^ {2} $ satisfying $ UB = BT $, $ \| B \| = \| A \| $ and $ PB = A $. Therefore, the error $ d _ \infty = \| A \| $, and there exists an $ h \in H ^ \infty $ such that $ d _ \infty = \| {f - h } \| _ \infty $.

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 $H^\infty$" 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)