On a vectorial riemann boundary value problem with infinite defect numbers, and related factorization of matrix-valued functions

The author presents a method for the factorization of a matrix-valued function, specified on a closed contour, in which functions meromorphic inside and outside of (with, generally speaking, an infinite number of zeros and poles), and products of such functions, are allowed as the diagonal elements of the central factor. The concept of partial indices of such a factorization is introduced, and their invariance is established. Conditions for solvability are obtained, as well as the general form of the solution, and criteria for the closure of the image of a vectorial Riemann boundary value problem whose matrix coefficient is factored in the sense indicated.