Commutant lifting theorem

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.

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.

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.

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.

• 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