Left versus right canonical wiener-hopf factorization for rational matrix functions: an alternative version

Abstract

It is an established fact that both canonical and non-canonical Wiener-hopf factorizations of matrix functions play an important role in various aspects of mathematical analysis and its applications. Indeed, for instance, the Fredholm properties of a block Toeplitz operator 7, with symbol W from the rn x rn matrix Wiener algebra Wnaxm over the unit circle T, may be read off from a (right) Wiener- hopf factorization W(^)-t\/-())r())14l+()) , )€1r, (0.1) where lVa and W- are in W*'*, the funct\on Wa has an analytic extension to the open unit disc D such that det Wa@) f 0 for z e D, th" function W- has an analytic extension to 0 U {-}\ D, such that det W-(z) I 0 for z € Q u {-}\ D, and D()) : diag ()K,)7, , (0.2) with rc1 t. . .t K^ integers. In particular, 7 is invertible if and only if the factorization is canonocal i.e., the indices Ktt . . . ) Krn are all equal to zero, and in this case the inverse of 7 may be constructed