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.