Exact conditions for preservation of the partial indices of a perturbed triangular 2 × 2 matrix function

Victor M Adukov; Gennady Mishuris; Sergei V Rogosin

doi:10.1098/rspa.2020.0099

Exact conditions for preservation of the partial indices of a perturbed triangular 2 × 2 matrix function

Proc Math Phys Eng Sci. 2020 May;476(2237):20200099. doi: 10.1098/rspa.2020.0099. Epub 2020 May 27.

Authors

Victor M Adukov¹, Gennady Mishuris², Sergei V Rogosin³

Affiliations

¹ Institute of Natural Sciences and Mathematics, South Ural State University, 454080 Chelyabinsk, Russia.
² Institute of Mathematics, Physics and Computer Science, Aberystwyth University, Aberystwyth SY23 3BZ, UK.
³ Department of Economics, Belarusian State University, 220030 Minsk, Belarus.

Abstract

The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.

Keywords: Toeplitz matrix; Wiener algebra; essential polynomials; factorization of matrix functions; triangular matrices.