Factorization of Block Triangular Matrix Functions in Wiener Algebras on Ordered Abelian Groups

The notion of Wiener-Hopf type factorization is introduced in the abstract framework of Wiener algebras of matrix-valued functions on connected compact abelian groups. Factorizations of 2 x 2 block triangular matrix functions with elementary functions on the main diagonal are studied in detail. A conjectl,lre is formulated concerning characterization of dual groups with the property that every invertible matrix function in a Wiener algebra admits a factorization. Applications of factorization are given to systems of difference equations and orthogonal families of functions. 1. Wiener Algebras Let G be a (multiplicative) connected compact abelian group and let r be its (additive) character group. Recall that r consists of all continuous homomorphisms of G into the group of unimodular complex numbers. Since G is compact, r is discrete. I~ applications, often r is an additive subgroup of JR, the group of real numbers, or of JRk , and G is the Bohr compactification of r. The group G can be also thought of as the character group of r, an observation that will be often used. The group G has a unique invariant measure v satisfying v(G) = 1, while r is equipped with the discrete topology and the (translation invariant) counting measure. It is well-known [31] that, because G is connected, r can be made into a linearly ordered group. So let ~ be a linear order such that (r, ~) is an ordered group, i. e., if x, y, z E r and x ~ y, then x + z ~ y + z. Throughout the paper it will be assumed that r is ordered with a fixed linear order ~. The notation -<, !,:, >-, Received by the editors April 14, 2003; revised August 30, 2003. Submitted by J. A. Ball. Mathematics Subject Classification (2000). Primary 47 A68. Secondary 43A17, 42C05.