Commutant lifting theorem

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

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

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 ${\displaystyle M_{\phi }^{\ast }}$ satisfies

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

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

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

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 ${\displaystyle M_{\phi }^{\ast }}$ of a multiplication operator on the entirety of ${\displaystyle H^2(\mathbb{D})}$ for some ${\displaystyle \phi }$, where ${\displaystyle \phi }$ 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 ${\displaystyle S}$ is an invariant subspace of ${\displaystyle M_z^{\ast }}$, that ${\displaystyle T}$ commutes with the restriction of ${\displaystyle M_z^{\ast }}$ to ${\displaystyle S}$, and that ${\displaystyle M_z^{\ast }}$ 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 ${\displaystyle M_z^{\ast }}$. Then in particular ${\displaystyle \tilde{T}}$ commutes with ${\displaystyle M_p}$ for any polynomial ${\displaystyle p}$. By setting ${\displaystyle \phi =\tilde{T}\mathbf{1}}$, 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 ${\displaystyle {\tilde{T}}^{\ast }=M_{\phi }}$, so that ${\displaystyle \tilde{T}=M_{\phi }^{\ast }}$. This function must then interpolate the points, as

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

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

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

showing that ${\displaystyle \Vert T\Vert \le 1}$ by showing that ${\displaystyle I-T^{\ast }T}$ 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 ${\displaystyle \phi }$ is the desired interpolating function.