a general Wiener-Hopf operator $W = P_2 A |_{P_1 X}$ where $X\,Y$ are Banach

spaces\,

$P_1 in mathcal{L}(X)\, P_2 in mathcal{L}(Y) $ are

projectors and $A in mathcal{L}(X\,Y)$ is invertible. Namely we study a

particular facto risation of $A = A_- C A_+$ where $A_+ : X ightar row Z$ and $A_-

: Z ightarrow Y$ have certain invariance properties and the cross factor $C :

Z ightarrow Z$ splits the "intermediate space" $ Z$ into

complemented subspaces closely related to the kernel and cokernel of $W$\, such

that $ W$ is equivalent to a "simpler" operator\, $W sim P C|_{P Z}$.

The main result shows equivalence between the generalised

invertibili ty of the Wiener-Hopf operator and this kind of fa ctorisation

(provided $P_1 sim

P_2$) whi ch implies a formula for a generalised inverse

of $W$. It embraces I.B. Simonenko'\;s generali sed factorisation of matrix

measurable function s in $L^p$ spaces and various other factorisation< br>approaches.

As applications we c onsider interface problems in weak

formulation for the n-dimensional Helmholtz equation in $Omega =

mathbb{R}^n_+ cup mathbb{R}^n_-$ (due to $x_ n > 0$ or $x_n < 0$\,

respectively)\, where the interface $Gamma = partial Omega$ is identified

with different transmission conditions of first and second kind. These two

parts are half-spaces of $mathbb{R}^{ n-1}$ (half-planes for $n = 3$). We

construct e xplicitly resolvent operators acting from the inte rface data into

the energy space $H^1(Omega)$. The approach is based upon the present

factoris ation conception and avoids an interpretation of t he factors as

unbounded operators. In a natural way\, we meet anisotropic Sobolev spaces which

reflect the edge asymptotic of diffracted waves. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR