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Wiener-Hopf factorisation through an intermediate space and applications to diffraction theory

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WHTW01 - Factorisation of matrix functions: New techniques and applications

An operator factorisation conception is investigated for
a general Wiener-Hopf operator $W = P_2 A |$ where $X,Y$ are Banach

$P_1 in mathcal{L}(X), P_2 in mathcal{L}(Y)$ are
projectors and $A in mathcal{L}(X,Y)$ is invertible. Namely we study a
particular factorisation of $A = A
- C A $ where $A : X ightarrow Z$ 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, $W sim P C |$.

The main result shows equivalence between the generalised
invertibility of the Wiener-Hopf operator and this kind of factorisation
(provided $P_1 sim

P_2$) which implies a formula for a generalised inverse
of $W$. It embraces I.B. Simonenko's generalised factorisation of matrix
measurable functions in $Lp$ spaces and various other factorisation

As applications we consider interface problems in weak
formulation for the n-dimensional Helmholtz equation in $Omega =
+ cup mathbb{R}n_-$ (due to $x_n > 0$ or $x_n < 0$,
respectively), where the interface $Gamma = partial Omega$ is identified
with $mathbb{R}
{n-1}$ and divided into two parts, $Sigma$ and $Sigma'$,
with different transmission conditions of first and second kind. These two
parts are half-spaces of $mathbb{R}$ (half-planes for $n = 3$). We
construct explicitly resolvent operators acting from the interface data into
the energy space $H
1(Omega)$. The approach is based upon the present
factorisation conception and avoids an interpretation of the factors as
unbounded operators. In a natural way, we meet anisotropic Sobolev spaces which
reflect the edge asymptotic of diffracted waves.

This talk is part of the Isaac Newton Institute Seminar Series series.

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