Approximation by analytic matrix functions. The four block problem

Abstract

We study the problem of finding a superoptimal solution to the four block problem. Given a bounded block matrix function $\left(\begin{array}{cc}\Phi_{11} &\Phi_{12}\\\Phi_{21}&\Phi_{22}\end{array}\right)$ on the unit circle the four block problem is to minimize the $L^\infty$ norm of $\left(\begin{array}{cc} \Phi_{11}-F&\Phi_{12}\\\Phi_{21}&\Phi_{22}\end{array}\right)$ over $F\in H^\infty$. Such a minimizing $F$ (an optimal solution) is almost never unique. We consider the problem to find a superoptimal solution which minimizes not only the supremum of the matrix norms but also the suprema of all further singular values. We give a natural condition under which the superoptimal solution is unique.