Commutant lifting theorem

Short description: Operator 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 $T$ is a contraction on a Hilbert space $H$, $U$ is its minimal unitary dilation acting on some Hilbert space $K$ (which can be shown to exist by Sz.-Nagy's dilation theorem), and $R$ is an operator on $H$ commuting with $T$, then there is an operator $S$ on $K$ commuting with $U$ such that

$R{T}^n=P_{H}S{U}^n{|}_H\mathrm{\forall }n\ge 0,$

and

$\Vert S\Vert =\Vert R\Vert .$

Here, $P_H$ is the projection from $K$ onto $H$. 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 $z_1,\dots ,z_n\in \mathbb{𝔻}$ and $w_1,\dots ,w_n\in \mathbb{𝔻}$. The following are equivalent:

1. There exists a holomorphic function $\phi :\mathbb{𝔻}\to \mathbb{𝔻}$ with $\phi (z_j)=w_j$ for $j=1,\dots ,n$.
2. The Pick matrix ${\left(\frac{1-w_i{\bar{w}}_j}{1-z_i{\bar{z}}_j}\right)}_{i,j=1}^n$ is positive semi-definite.

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

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

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. Suppose that the Pick matrix is positive semi-definite. Consider, for $\phi \in H^{\mathrm{\infty }}(\mathbb{𝔻})$, the operator $M_{\phi }$ on $H^2(\mathbb{𝔻})$ given by multiplication by $\phi$, meaning that

$M_{\phi }f(z)=\phi (z)f(z)$

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

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

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

$S=\mathrm{span}\{k(\cdot ,z_1),\dots ,k(\cdot ,z_n)\}$

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

$Tk(\cdot ,z_j)={\bar{w}}_{j}k(\cdot ,z_j)$

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

$\overline{\phi (z_j)}k(\cdot ,z_j)=M_{\phi }^{*}k(\cdot ,z_j)=\tilde{T}k(\cdot ,z_j)=Tk(\cdot ,z_j)={\bar{w}}_{j}k(\cdot ,z_j),$

from which we get $\phi (z_j)=w_j$. That $\phi (\mathbb{𝔻})\subseteq \mathbb{𝔻}$ is then a consequence of computing

$\underset{z\in \mathbb{𝔻}}{\sup }|\phi (z)|=\Vert M_{\phi }\Vert =\Vert \tilde{T}\Vert =\Vert T\Vert ,$

showing that $\Vert T\Vert \le 1$ by showing that $I-T^{*}T$ 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 $\phi$ 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