2022/03/15 by G. J. Groenewald, Groenewald, G. J., M. A. Kaashoek +3
Mathematics · #47A35 #47B35 #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2203.07821
openalex publication_date 2022/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
As in the paper [G. Groenewald, M.A. Kaashoek, A.C.M. Ran, Wiener-Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators. Indag. Math. 28 (2017) 694--710] our aim is to obtain explicitly the Wiener-Hopf indices of a rational m× m matrix function R(z) that has no poles and no zeros on the unit circle \mathbbT but, in contrast with that paper, the function R(z) is not required to be unitary on the unit circle. On the other hand, using a Douglas-Shapiro-Shields type of factorization, we show that R(z) factors as R(z)=Ξ(z)Ψ(z), where Ξ(z) and Ψ(z) are rational m× m matrix functions, Ξ(z) is unitary on the unit circle and Ψ(z) is an invertible outer function. Furthermore, the fact that Ξ(z) is unitary on the unit circle allows us to factor as Ξ(z) =V(z)W^*(z) where V(z) and W(z) are rational bi-inner m× m matrix functions. The latter allows us to solve the Wiener-Hopf indices problem. To derive explicit formulas for the functions V(z) and W(z) requires additional realization properties of the function Ξ(z) which are given in the last two sections.