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Regular approximate factorization of a class of matrix-function with an unstable set of partial indices. (English) Zbl 1402.15010

Summary: From the classic work of I. C. Gochberg and M. G. Krejn [Usp. Mat. Nauk 13, No. 2(80), 3–72 (1958; Zbl 0098.07402)], it is well known that the set of partial indices of a non-singular 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.

MSC:

15A23 Factorization of matrices

Citations:

Zbl 0098.07402
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References:

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