Meromorphic Factorization, Partial Index Estimates and Elastodynamic Diffraction Problems¹)

Abstract

This work is motivated by some elliptic boundary and transmission problems in mathematical physics, in particular by elastodynamic wave propagation. The analytical solution of the boundary pseudodifferential equations requires a generalized factorization of the lifted FOURIER symbol which is a non-rational matrix-function. In the factorization procedure poles and increasing terms appear, and cause enormous practical and theoretical problems due to the possible occurance of partial indices different from zero. The paper presents an approach which avoids those difficulties by use of a factorization of the symbol matrix into meromorphic factors. An operator theoretic interpretation yields resolvents up to finite dimensional operators, whose ranks are closely related to partial indices, order of the algebraic increase at infinity, and the multiplicities of the poles in the factors.

Minimal Normalization of Wiener–Hopf Operators in Spaces of Bessel Potentials

“…The following results are mainly collected from [7] and [21], but see also [2,19,24] and other references in particular cases. Consider the WHOs (or PDOs) defined in (1.2).…”

Meromorphic Factorization Revisited and Application to Some Groups of Matrix Functions

“…R). Meromorphic factorization appears in a natural way for some classes of matrix functions, for instance in elastodynamic diffraction problems [7,18]. This kind of factorization of matrix symbols does not give, as least directly, the same amount of information on Toeplitz operators as the Wiener-Hopf factorization defined above, namely as regards invertibility.…”

“…An example is meromorphic factorization [7,16] in which the factors and their inverses are allowed to have poles in the corresponding domain in the complex plane (C − or C + ) and algebraic growth at ∞. For a matrix function G ∈ G(C μ ( .…”

Factorization of a class of matrices generated by Sommerfeld diffraction problems with oblique derivatives

“…where g*"r > [g>!g\, g>#g\]23[H\ (1 > )] are combinations of the given data defined on the half-line, i.e., HP (1 > ) are restrictions to 1 > of spaces HP"HP (1) of Bessel potentials of order r31. Further A"F\ .…”

“…( #i t\) , (1.2) 31, t( )"( !k), which will be abbreviated by t for shortness, Im k'0 and "b> #b\ , "b> #b\ , "b> !b\ , "b> !b\ are linear combinations of the complex coefficients of the normal (b $ ) and tangential derivatives (b $ ). In (1.1) the unknown data > are distributions on 1 M > : (1.3) so that the zero extensions of the jumps of the Dirichlet and Neumann data on 1 belong to H (1) and H\ (1), respectively. The restriction r > to 1 > and the zero extension operator l to the whole line are defined on the space of distributions S(1) and S > :S(1 > )"S(1 > ), respectively [5].…”