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

Add to your list(s) Download to your calendar using vCal

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

If you have a question about this talk, please contact INI IT.

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 $L<sup>p$ spaces and various other factorisation
approaches.



As applications we consider interface problems in weak
formulation for the n-dimensional Helmholtz equation in $Omega =
mathbb{R}</sup>n+ cup mathbb{R}^{n_-$ (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.

This talk is included in these lists:

• All CMS events
• Featured lists
• INI info aggregator
• Isaac Newton Institute Seminar Series
• School of Physical Sciences
• Seminar Room 1, Newton Institute
• bld31

Note that ex-directory lists are not shown.