Mishuris G, Rogosin S
Department of Mathematics, Aberystwyth University, Penglais, SY23 3BZ Aberystwyth, UK.
Department of Economics, Belarusian State University, Nezavisimosti Avenue, 4, 220030 Minsk, Belarus.
Proc Math Phys Eng Sci. 2018 Jan;474(2209):20170279. doi: 10.1098/rspa.2017.0279. Epub 2018 Jan 17.
From the classic work of Gohberg & Krein (1958 , 3-72. (Russian).), 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.
从戈德堡和克莱因的经典著作（1958年，第3 - 72页，俄语）可知，非奇异矩阵函数的部分指标集可能会根据原矩阵的性质而改变。更确切地说，已证明如果最大部分指标与最小部分指标之差大于1，那么在原矩阵函数的任何邻域内，都存在另一个具有不同部分指标集的矩阵函数。因此，矩阵函数的因式分解本身就是一个极其困难的过程，即使在规范因式分解的情况下也是如此，而在部分指标集不稳定的情况下，因式分解仍然无法解决甚至存在疑问。反过来，这种情况已成为因式分解技术应用中不可避免的障碍。本文着手回答一个比有效分解具有不稳定部分指标集的矩阵函数这个问题野心稍小的问题，而是专注于确定这样的条件：在已知极限矩阵函数因式分解的情况下，这些条件允许构造另一个具有相同起源的矩阵函数族，该族保留不稳定的部分指标并且接近原矩阵函数集。
