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 " 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)