BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Wiener-Hopf factorisation through an intermediate space and applic
ations to diffraction theory - Frank Speck (Universidade de Lisboa)
DTSTART:20190812T090000Z
DTEND:20190812T100000Z
UID:TALK128338@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:An operator factorisation conception is investigated for
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
p
rojectors and $A in mathcal{L}(X\,Y)$ is invertible. Namely we study a
particular factorisation of $A = A_- C A_+$ where $A_+ : X ightarrow Z$ a
nd $A_-
: Z ightarrow Y$ have certain invariance properties and the cr
oss factor $C :
Z ightarrow Z$ splits the "intermediate space" $Z$ int
o
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 gene
ralised
invertibility of the Wiener-Hopf operator and this kind of fact
orisation
(provided $P_1 sim
P_2$) which implies a formula for a
generalised inverse
of $W$. It embraces I.B. Simonenko'\;s generali
sed factorisation of matrix
measurable functions in $L^p$ spaces and va
rious 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 tw
o parts\, $Sigma$ and $Sigma'\;$\,
with different transmission condi
tions of first and second kind. These two
parts are half-spaces of $mat
hbb{R}^{n-1}$ (half-planes for $n = 3$). We
construct explicitly resolv
ent operators acting from the interface data into
the energy space $H^1
(Omega)$. The approach is based upon the present
factorisation concepti
on 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.
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR