Invertibility of matrix Wiener-Hopf plus Hankel operators with APW Fourier symbols

Operators of Wiener-Hopf plus Hankel type have been receiving an increasing attention in the last years (see [1, 2, 4, 6, 10, 12–16]). Some of the interest in their study arises directly from concrete applications where these kind of operators appear. This is the case in problems of wave diffraction by some particular rectangular geometries which originate specific boundary-transmission value problems that may be equivalently translated by systems of integral equations that lead to such kind of operators (see, e.g., [5, 7, 8]). A great part of the study in this kind of operators is concentrated in the description of their Fredholm and invertibility properties. In particular, for some classes of the so-called Fourier symbols of the operators, their invertibility properties are already known (cf. the above references). Despite those advances, for some other classes of Fourier symbols, a complete description of the Fredholm and invertibility properties is still missing. In this way, some of the ongoing researches try to achieve the best possible factorization procedures of the involved Fourier symbols in such a way that a representation of the (generalized) inverses of the Wiener-Hopf plus Hankel operators will be possible to obtain when in the presence of a convenient factorization. Within this spirit, the main aim of the present work is to provide an invertibility criterion for the Wiener-Hopf plus Hankel operators of the form