Local and global factorizations of matrix-valued functions

Clancey, K. F.; Gohberg, I.

doi:10.1090/S0002-9947-1977-0454742-4

Local and global factorizations of matrix-valued functions
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by K. F. Clancey and I. Gohberg PDF

Trans. Amer. Math. Soc. 232 (1977), 155-167 Request permission

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.

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