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

Presented by: 
Frank Speck
Monday 12th August 2019 - 10:00 to 11:00
INI Seminar Room 1

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 projectors and $A \in \mathcal{L}(X,Y)$ is invertible. Namely we study a particular factorisation of $A = A_- C A_+$ where $A_+ : X \rightarrow Z$ and $A_- : Z \rightarrow Y$ have certain invariance properties and the cross factor $C : Z \rightarrow Z$ 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 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 $L^p$ spaces and various other factorisation approaches.  

As applications we consider interface problems in weak formulation for the n-dimensional Helmholtz equation in $\Omega = \mathbb{R}^n_+ \cup \mathbb{R}^n_-$ (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 $\mathbb{R}^{n-1}$ (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.

Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons