Local and global factorizations of matrix-valued functions

Abstract

Let C be a simple closed Liapounov contour in the complex plane and A an invertible n × n n \times n matrix-valued function on C with bounded measurable entries. There is a well-known concept of factorization of the matrix function A relative to the Lebesgue space L p ( C ) {L_p}(C) . The notion of local factorization of A relative to L p {L_p} at a point t 0 {t_0} in C is introduced. It is shown that A admits a factorization relative to L p ( C ) {L_p}(C) if and only if A admits a local factorization relative to L p {L_p} at each point t 0 {t_0} in C. Several problems connected with local factorizations relative to L p {L_p} are raised.