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

• Frank Speck (Universidade de Lisboa)
• Monday 12 August 2019, 10:00-11:00
• Seminar Room 1, Newton Institute.

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
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 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
approaches.

As applications we consider 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 \$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.