Nonselfadjoint operators in diffraction and scattering

On the basis property for the root vectors of some nonselfadjoint operators

“…While this statement can be found in the literature [ 1,4,6] we give its proof for the convenience of the reader after the proof of the theorem.…”

“…Let us assume that L = Q;' exists (without loss of generality, see [6]). Then l,(L) -UZ"~, c = const, so that p = 0.5, where p is defined in ( 1).…”

“…Then l,(L) -UZ"~, c = const, so that p = 0.5, where p is defined in ( 1). Since in the theorem the unperturbed operator is unbounded we denote A = (Q,+Q,)-'=(Z+LQ,>-lL=L+T, TS -(I t LQ,)-'LQ,L, we assumed that (Q, + Q,))' exists again without loss of generality; where k > 0 and k* is not an eigenvalue of the Laplace operator for the interior Dirichlet problem in the domain D with the boundary r it is easy to prove that (QO t Q,)-' exists [6]. Since ord LQ,L = -co we can take the number a in (2) negative and large, so that p(1 -a) > 2.…”

Back Matter

“…For the selfadjoint operator B, this method goes back to Picard (1881) (see [74]). If B is non self-adjoint, then the problem is very difficult in general: (1) B might be quasinilpotent (e.g., Volterra operator) and then it has no eigenvalues, (2) the closure of the linear span of the eigenvectors (or root vectors) of B may not coincide with the whole space H, (3) if it does, the root system of B, i.e., the set of all linearly independent root vectors of B, may not form a basis of H. There is a considerable interest in applications to EEM (see [17,208,212,216,230,238,240,470]). At this time, there is an extensive bibliography of the engineering work on EEM and SEM, hundreds of references can be found in the special issue of the journal "Electromagnetics" 1, N4, (1981).…”

Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering

“…For more general domains than the circle, little is known about spectral decompositions, however understanding the spectral properties and normality of these boundary integral operators has both theoretical and practical implications. On the theoretical side, in Ramm (1973Ramm ( , 1980 Ramm asked when the eigensystem of the single layer potential operator in 3-d forms a complete basis in L 2 (Γ ) and gave normality as a sufficient condition. On the practical side, whether an operator is normal or not, and, if it is not normal, the degree of nonnormality, affect the convergence of iterative solvers such as GMRES.…”