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. {{math theorem|name=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: •
There exists a holomorphic function \varphi:\mathbb D\to\mathbb D with \varphi(z_j)=w_j for j=1,\dots,n. •
The Pick matrix \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 H^2(\mathbb D) of the disc \mathbb D and use that this is a
reproducing kernel Hilbert space with multipliers the space H^\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\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 \varphi\in H^\infty(\mathbb D), the
operator M_\varphi on H^2(\mathbb D) given by multiplication by \varphi, meaning that :M_\varphi f(z)=\varphi(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_\varphi^* satisfies :M_\varphi^*k(\,\cdot\,,z)=\overline{\varphi(z)}k(\,\cdot\,,z) An important special case of this is when \varphi(z)=z, in which case we write M_z for its multiplication operator. Consider next the finite-dimensional
subspace :S=\operatorname{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)=\overline w_jk(\,\cdot\,,z_j) The idea is now to extend the operator T to the adjoint M_\varphi^* of a multiplication operator on the entirety of H^2(\mathbb D) for some \varphi, where \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 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 \varphi=\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_\varphi, so that \tilde T=M_\varphi^*. This function must then interpolate the points, as :\overline{\varphi(z_j)}k(\,\cdot\,,z_j)=M_\varphi^*k(\,\cdot\,,z_j)=\tilde Tk(\,\cdot\,,z_j)=Tk(\,\cdot\,,z_j)=\overline w_j k(\,\cdot\,,z_j), from which we get \varphi(z_j)=w_j. That \varphi(\mathbb D)\subseteq\mathbb D is then a consequence of computing :\sup_{z\in\mathbb D}|\varphi(z)|=\lVert M_\varphi\rVert=\lVert\tilde T\rVert=\lVert T\rVert, showing that \lVert T\rVert\leq1 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 \varphi is the desired interpolating function. ==References==