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.

The Nevanlinna-Pick interpolation theorem

A classical application of the commutant lifting theorem is in solving the Nevanlinna-Pick interpolation problem. The points for which the interpolation problem has a solution can be characterized precisely in terms of the positive semi-definiteness of a certain matrix constructed from the points.

Theorem (Nevanlinna-Pick interpolation)—Let ${\displaystyle z_{1},\dots ,z_{n}\in \mathbb {D} }$ and ${\displaystyle w_{1},\dots ,w_{n}\in \mathbb {D} }$. The following are equivalent:

1. There exists a holomorphic function ${\displaystyle \varphi$ :\mathbb {D} \to \mathbb {D} } with ${\displaystyle \varphi (z_{j})=w_{j}}$ for ${\displaystyle j=1,\dots ,n}$.
2. The Pick matrix ${\displaystyle \left({\frac {1-w_{i}{\overline {w}}_{j}}{1-z_{i}{\overline {z}}_{j}}}\right)_{i,j=1}^{n}}$ is positive semi-definite.

The main idea behind the proof is to consider the Hardy space ${\displaystyle H^{2}(\mathbb {D} )}$ of the disc ${\displaystyle \mathbb {D} }$ and use that this is a reproducing kernel Hilbert space with multipliers the space ${\displaystyle H^{\infty }(\mathbb {D} )}$ of bounded holomorphic functions on ${\displaystyle \mathbb {D} }$. The reproducing kernel of ${\displaystyle H^{2}(\mathbb {D} )}$ is the function

${\displaystyle k(z,w)={\frac {1}{1-z{\overline {w}}}}}$

commonly referred to as the Szegő kernel. The tricky part of the proof is showing that the condition of positive semi-definiteness implies the existence of said interpolating function. Following J. Agler and J. McCarthy the idea of the proof is as follows. ^[1] Suppose that the Pick matrix is positive semi-definite. Consider, for ${\displaystyle \varphi \in H^{\infty }(\mathbb {D} )}$, the operator ${\displaystyle M_{\varphi }}$ on ${\displaystyle H^{2}(\mathbb {D} )}$ given by multiplication by ${\displaystyle \varphi }$, meaning that

${\displaystyle M_{\varphi }f(z)=\varphi (z)f(z)}$

for ${\displaystyle f\in H^{2}(\mathbb {D} )}$. This is a bounded operator on ${\displaystyle H^{2}(\mathbb {D} )}$, and one can show that its adjoint ${\displaystyle M_{\varphi }^{*}}$ satisfies

${\displaystyle M_{\varphi }^{*}k(\,\cdot \,,z)={\overline {\varphi (z)}}k(\,\cdot \,,z)}$

An important special case of this is when ${\displaystyle \varphi (z)=z}$, in which case we write ${\displaystyle M_{z}}$ for its multiplication operator. Consider next the finite-dimensional subspace

${\displaystyle S=\operatorname {span} \{k(\,\cdot \,,z_{1}),\dots ,k(\,\cdot \,,z_{n})\}}$

of ${\displaystyle H^{2}(\mathbb {D} )}$. Define an operator ${\displaystyle T}$ on ${\displaystyle S}$ by letting

${\displaystyle Tk(\,\cdot \,,z_{j})={\overline {w}}_{j}k(\,\cdot \,,z_{j})}$

The idea is now to extend the operator ${\displaystyle T}$ to the adjoint ${\displaystyle M_{\varphi }^{*}}$ of a multiplication operator on the entirety of ${\displaystyle H^{2}(\mathbb {D} )}$ for some ${\displaystyle \varphi }$, where ${\displaystyle \varphi }$ will then be the solution to the interpolation problem. This is where the commutant lifting theorem comes into play. In particular, one can verify that ${\displaystyle S}$ is an invariant subspace of ${\displaystyle M_{z}^{*}}$, that ${\displaystyle T}$ commutes with the restriction of ${\displaystyle M_{z}^{*}}$ to ${\displaystyle S}$, and that ${\displaystyle M_{z}^{*}}$ is co-isometric (meanining that its adjoint is isometric). Applying the commutant lifting theorem we can then find an operator ${\displaystyle {\tilde {T}}}$ on ${\displaystyle H^{2}(\mathbb {D} )}$ which agrees with ${\displaystyle T}$ on ${\displaystyle S}$, which has the same norm as ${\displaystyle T}$, and which commutes with ${\displaystyle M_{z}^{*}}$. Then in particular ${\displaystyle {\tilde {T}}}$ commutes with ${\displaystyle M_{p}}$ for any polynomial ${\displaystyle p}$. By setting ${\displaystyle \varphi ={\tilde {T}}\mathbf {1} }$, where ${\displaystyle \mathbf {1} }$ is the constant function equal to ${\displaystyle 1}$, and using that the polynomials are dense in ${\displaystyle H^{2}(\mathbb {D} )}$, one can then show that ${\displaystyle {\tilde {T}}^{*}=M_{\varphi }}$, so that ${\displaystyle {\tilde {T}}=M_{\varphi }^{*}}$. This function must then interpolate the points, as

${\displaystyle {\overline {\varphi (z_{j})}}k(\,\cdot \,,z_{j})=M_{\varphi }^{*}k(\,\cdot \,,z_{j})={\tilde {T}}k(\,\cdot \,,z_{j})=Tk(\,\cdot \,,z_{j})={\overline {w}}_{j}k(\,\cdot \,,z_{j}),}$

from which we get ${\displaystyle \varphi (z_{j})=w_{j}}$. That ${\displaystyle \varphi (\mathbb {D} )\subseteq \mathbb {D} }$ is then a consequence of computing

${\displaystyle \sup _{z\in \mathbb {D} }|\varphi (z)|=\lVert M_{\varphi }\rVert =\lVert {\tilde {T}}\rVert =\lVert T\rVert ,}$

showing that ${\displaystyle \lVert T\rVert \leq 1}$ by showing that ${\displaystyle I-T^{*}T}$ is positive (which is where the positive semi-definiteness of the Pick matrix comes in), and then finally appealing to the open mapping theorem. As such ${\displaystyle \varphi }$ is the desired interpolating function.

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.
• J. Agler and J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, 44, Amer. Math. Soc., Providence, RI, 2002

1. J. Agler and J. E. McCarthy (2002). Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics. Vol. 44. American Mathematical Society, Providence, RI. pp. xx+308. doi:10.1090/gsm/044. ISBN   0-8218-2898-3.