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

Let $\widehat{K}$ be a Wiener–Hopf operator, $\widehat{K}f(x)=\int_0^{\infty}K(x-t)f(t)\,dt$, $x\geqslant 0$, and let $\widehat{K}^*$ be the adjoint operator, $(f\widehat{K}^*)(t)=\int_0^{\infty}f(x)K(x-t)\,dx$, $t\geqslant 0$, where $K(x)$ belongs to the Banach space $L_1 (G,(-\infty,\infty))$ of Bochner strongly integrable functions with values in a Banach algebra $G$. We consider the canonical factorization problem $I-\widehat{K}=(I-\widehat{V}_-)(I-\widehat{V}_+)$, where $I$ is the identity operator and $\widehat{V}_-$ (resp. $\widehat{V}_+ $) is a left (resp. right) triangular convolution operator such that the operators $I-\widehat{V}_{\pm}$ are invertible in the spaces $L_{p} (G,(0,\infty))$, $1\leqslant p\leqslant \infty$. We put forward a semi-inverse factorization method and prove that the canonical factorization exists if and only if the operators $I-\widehat{K}$ and $I-\widehat{K}^*$ are invertible in $L_1 (G,(0,\infty))$.