Abstract
The present chapter is devoted to the study of the behaviour of the factorization factors under small perturbations of the matrix function G. Here we encounter another qualitative difference between the matrix case (n > 1) and the scalar one (n = 1). It appears that the partial indices of a matrix function and the factorization factors G± are unstable, in general. This fact gives rise to principle difficulties in the solution of the two most important problems in the factorization theory of a matrix function G and the solvability theory of the vector-valued Riemann boundary value problem with this matrix: 1) the problems of calculating the partial indices (of calculating the defect numbers of the Riemann problem) and 2) the problems of constructing an approximate factorization of a matrix function (of finding an approximate solution of the Riemann boundary value problem).
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