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 $n \times n$ 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 ${L_p}(C)$. The notion of local factorization of A relative to ${L_p}$ at a point ${t_0}$ in C is introduced. It is shown that A admits a factorization relative to ${L_p}(C)$ if and only if A admits a local factorization relative to ${L_p}$ at each point ${t_0}$ in C. Several problems connected with local factorizations relative to ${L_p}$ are raised.