On the factorization of matrix and operator Wiener-Hopf integral equations

Let (K) over cap be a Wiener-Hopf operator, (K) over cap f(x) = integral(infinity)(0) K(x-t) f(t) dt, x >= 0, and let (K) over cap* be the adjoint operator, (f (K) over cap*)(t) = integral(infinity)(0) f(x) K(x-t) dx, t >= 0, where K(x) belongs to the Banach space L-1(G, (-infinity, infinity)) of Bochner strongly integrable functions with values in a Banach algebra G. We consider the canonical factorization problem I - (K) over cap = (I - (V) over cap (-))(I - (V) over cap (+)), where I is the identity operator and (V) over cap (-) ( resp. (V) over cap (+)) is a left (resp. right) triangular convolution operator such that the operators I - (V) over cap (+/-) are invertible in the spaces L-p(G, (0, infinity)), 1 <= p <= infinity. We put forward a semi- inverse factorization method and prove that the canonical factorization exists if and only if the operators I - (K) over cap and I - (K) over cap* are invertible in L-1(G, (0, infinity)).