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SUMMARY:Wiener-Hopf factorisation through an intermediate space and applic
 ations to diffraction theory - Frank Speck (Universidade de Lisboa)
DTSTART:20190812T090000Z
DTEND:20190812T100000Z
UID:TALK128341@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:An operator factorisation conception is investigated for a gen
 eral Wiener-Hopf operator $W = P_2 A |_{P_1 X}$ where $X\,Y$ are Banach sp
 aces\,  $P_1 \\in \\mathcal{L}(X)\, P_2 \\in \\mathcal{L}(Y)$ are projecto
 rs and $A \\in \\mathcal{L}(X\,Y)$ is invertible. Namely we study a partic
 ular factorisation of $A = A_- C A_+$ where $A_+ : X \\rightarrow Z$ and $
 A_- : Z \\rightarrow Y$ have certain invariance properties and the cross f
 actor $C : Z \\rightarrow Z$ splits the "intermediate space" $Z$ into comp
 lemented subspaces closely related to the kernel and cokernel of $W$\, suc
 h that $W$ is equivalent to a "simpler" operator\, $W \\sim P C|_{P Z}$.  
 &nbsp\;  <br><br>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 inve
 rse of $W$. It embraces I.B. Simonenko&#39\;s generalised factorisation of
  matrix measurable functions in $L^p$ spaces and various other factorisati
 on approaches.  &nbsp\;  <br><br>As applications we consider interface pro
 blems in weak formulation for the n-dimensional Helmholtz equation in $\\O
 mega = \\mathbb{R}^n_+ \\cup \\mathbb{R}^n_-$ (due to $x_n &gt\; 0$ or $x_
 n &lt\; 0$\, respectively)\, where the interface $\\Gamma = \\partial \\Om
 ega$ is identified with $\\mathbb{R}^{n-1}$ and divided into two parts\, $
 \\Sigma$ and $\\Sigma&#39\;$\, with different transmission conditions of f
 irst 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 a
 pproach is based upon the present factorisation conception and avoids an i
 nterpretation of the factors as unbounded operators. In a natural way\, we
  meet anisotropic Sobolev spaces which reflect the edge asymptotic of diff
 racted waves.
LOCATION:Seminar Room 1\, Newton Institute
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