A. B. Lebre

The diffraction problem for a half plane with different face impedances revisited

Abstract The problem of the diffraction of an electromagnetic wave by a half plane with different face impedances is dealt with, following a rigorous approach based on the [L2+( R ]2 theory of systems of Wiener-Hopf equations with piecewise continuous presymbols. The corresponding operator is defined in spaces of physically admissible solutions, the Sobolev spaces Hα+( R )×Hα−1+( R ) for α> 1 2 , and its Fredholm characteristics are determined. For 1 2 it is shown that the operators are invertible and their inverses are calculated. In the final section the inverse of a related operator presented by Meister and Speck is also obtained.

Generalized factorization for a class of non-rational 2×2 matrix functions

In this paper the generalized factorization for a class of 2×2 piecewise continuous matrix functions on ℝ is studied. Using a space transformation the problem is reduced to the generalized factorization of a scalar piecewise continuous function on a contour in the complex plane. Both canonical and non-canonical generalized factorization of the original matrix function are studied.

Factorization in the Wiener algebra of a class of 2×2 matrix functions

In this paper the problem of the factorization in the Wiener algebra of a class of 2×2 matrix functions of Daniele type is considered. For these matrix functions necessary and sufficient conditions for the existence of a canonical factorization are obtained and, provided these conditions are fulfilled, the factors of such factorization can be derived in an explicit form.

Generalised factorisation for a class of Jones form matrix functions

A 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.

Spectrum Problems for Singular Integral Operators with Carleman Shift

In this paper we are concerned with the complete spectral analysis for the operator = in the space Lp() ( denoting the unit circle), where is the characteristic function of some arc of , is the singular integral operator with Cauchy kernel and is a Carleman shift operator which satisfies the relations 2 = I and = ±, where the sign + or — is taken in dependence on whether is a shift operator on Lp() preserving or changing the orientation of . This includes the identification of the Fredholm and essential parts of the spectrum of the operator , the determination of the defect numbers of — λI, for λ in the Fredholm part of the spectrum, as well as a formula for the resolvent operator.

Invertibility of Functional Operators with Slowly Oscillating Non-Carleman Shifts

We prove criteria for the invertibility of the binomial functional operator

Operator algebras, operator theory and applications

A = al - bW\alpha

A normalization problem for a class of singular integral operators with carleman shift and unbounded coefficients

in the Lebesgue spaces L p (0,1), 1 < p < ∞, where a and b are continuous functions on (0, 1), I is the identity operator, W,, is the shift operator, W α f=f ○ α, generated by a non-Carleman shift α : [0,1] → [0,1] which has only two fixed points 0 and 1. We suppose that log a’ is bounded and continuous on (0, 1) and that a, b, a’ slowly oscillate at 0 and 1. The main difficulty connected with slow oscillation is overcome by using the method of limit operators.