Approximation by Analytic Matrix Functions: The Four Block Problem

The four block problem is a generalization of Nehariʹs problem for matrix functions. It plays an important role inH∞-optimal control theory. It is well known that Nehariʹs problem for a continuous scalar function has a unique solution. However, in the matrix case the situation is quite different. V. V. Peller and N. J. Young (1994,J. Funct. Anal.120, 300–343) studiedsuperoptimal solutionsof Nehariʹs problem. They minimize not only theL∞-norm of the corresponding matrix function but also the essential suprema of all further singular values. It was shown that forH∞+Cmatrix functions Nehariʹs problem has a unique superoptimal solution. In this paper we study superoptimal solutions of the four block problem and we find a natural condition under which such a superoptimal solution is unique. Our result is new even in the case of Nehariʹs problem. We study some related problems such as thematic factorizations, invariance of indices, and inequalities between the singular values of the four block operator and the superoptimal singular values.