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.

Key words Singular integral operators with shift, operator factorization, factorization of matrix functions MSC (2000) Primary: 47G10; Secondary: 47A68 Dedicated to Professor Frank-Olme Speck on the occasion of his 60th birthdayThis paper deals with what we call modified singular integral operators. When dealing with (pure) singular integral operators on the unit circle with coefficients belonging to a decomposing algebra of continuous functions it is known that a factorization of the symbol induces a factorization of the original operator, which is a representation of the operator as a product of three singular integral operators where the outer operators in that representation are invertible.The main purpose of this paper is to obtain a similar operator factorization for the case of singular integral operators with a backward shift and to extract from there some consequences for their Fredholm characteristics. At the end of the paper it is shown that the operator factorization is also possible for other classes of singular integral operators, namely those including either a conjugation operator or a composition of a conjugation with a forward shift operator.

SynopsisA systematic approach is proposed for the generalised factorisation of certain non-rational n × n matrix functions. The first main result consists in a transformation of a meromorphic into a generalised factorisation by algebraic means. It closes a gap between the classical Wiener-Hopf procedure and the operator theoretic method of generalised factorisation. Secondly, as examples we consider certain matrix functions of Jones form or of N-part form, which are equivalent to each other, in a sense. The factorisation procedure is complete and explicit, based only on the factorisation of scalar functions, of rational matrix functions and upon linear algebra. Applications in elastodynamic diffraction theory are treated in detail and in a most effective way.

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In this paper, we generalize our recent results concerning scalar singular integral operators with a Carleman backward shift, allowing more general coefficients, bounded measurable functions on the unit circle. Our aim is to obtain an operator factorization for singular integral operators with a backward shift and bounded measurable coefficients, from which such Fredholm characteristics as the kernel and the cokernel can be described. The main tool is the factorization of matrix functions. In the course of the analysis performed, we obtain several useful representations, which allow us to characterize completely the set of invertible operators in that class, thus providing explicit examples of such operators