TY - JOUR

T1 - Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

AU - Mishuris, Gennady

AU - Rogosin, Sergei

PY - 2018/1/31

Y1 - 2018/1/31

N2 - From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk. XIII, 3–72. (Russian).), it is well known that the set of partial indices of a nonsingular
matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to
answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions

AB - From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk. XIII, 3–72. (Russian).), it is well known that the set of partial indices of a nonsingular
matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to
answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions

KW - asymptotic methods

KW - unstable partial indices

KW - factorization of matrix-functions

U2 - 10.1098/rspa.2017.0279

DO - 10.1098/rspa.2017.0279

M3 - Article

C2 - 29434502

SN - 0308-2105

VL - 474

JO - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

IS - 2209

M1 - 0170279

ER -