APPROXIMATION BY ANALYTIC MATRIX FUNCTIONS. THE FOUR BLOCK PROBLEM

VLADIMIR PELLER SERGEI TREIL

Abstract. We study the problem of finding a superoptimal solution to the four

block problem. Given a bounded block matrix function ( \begin{matrix}Φ_{11} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} ) on the unit

circle the four block problem is to minimize the L^{∞} norm of ( \begin{matrix}Φ_{11} − F & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix} )

over F ∈ H^{∞}. 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.