ON THE FREDHOLM INDICES OF ASSOCIATED SYSTEMS OF WIENER-HOPF EQUATIONS

Every matrix function $\mathrm{A}\in {\mathrm{L}}_{\mathrm{n}\times \mathrm{n}}^{\mathrm{\infty }}\left(\mathbf{R}\right)$ generates a Wiener-Hopf integral operator on ${\mathrm{L}}_{\mathrm{n}}^{2}\left({\mathbf{R}}_{+}\right)$, the direct sum of n copies of L2(R+). The associated Wiener-Hopf integral operator is the operator W(Ã) where Ã(x) := A(−x). We discuss the connection between the Fredholm indices Ind W(A) and Ind W(Ã). Our main result says that if A has at most a finite number d of discontinuities on R ⋃ {∞} and both W(A) and W(Ã) are Fredholm, then |Ind W(A) + Ind W(Ã)| ≤ d(n – 1); conversely, given integers κ and ν satisfying |κ+ν| ≤ d(n – 1), there exist $\mathrm{A}\in {\mathrm{L}}_{\mathrm{n}\times \mathrm{n}}^{\mathrm{\infty }}\left(\mathbf{R}\right)$ with at most d discontinuities such that W(A) is Fredholm of index κ and W(Ã) is Fredholm of index ν.