Approximation by analytic matrix functions. The four block problem

1995/12/22 by Vladimir Peller, Sergei Treil, Peller, Vladimir +1
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #math.CA

paper · pdf · doi:10.48550/arxiv.math/9512223

arxiv created 1995/12/22 · openalex publication_date 1995/12/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the problem of finding a superoptimal solution to the four block problem. Given a bounded block matrix function (Φ₁₁ · Φ₁₂
Φ₂₁ · Φ₂₂) on the unit circle the four block problem is to minimize the L^∞ norm of (Φ₁₁-F · Φ₁₂
Φ₂₁ · Φ₂₂) 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.

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