Amarino B. Lebre

Factorization of Singular Integral Operators with a Carleman Shift via Factorization of Matrix Functions

This paper is devoted to singular integral operators with a linear fractional Carleman shift of arbitrary order preserving the orientation on the unit circle. The main goal is to obtain a special factorization of the operator with the help of a factorization of a related matrix function in a suitable algebra. This factorization allows us to characterize the kernel and the range of the operator under consideration, similarly to the case of singular integral operators without shift.

Factorization of Singular Integral Operators with a Carleman Backward Shift: The Vector Case

On the vector Lebesgue space on the unit circle \({L}^{n}_{p} ({p} \in (1, \infty), {n} \in \mathbb{N}\) we consider singular integral operators with a Carleman backward shift of linear fractional type, of the form \({T}_{A,B} = {AP}_{+} + {BP}_{-}\) with A = aI + bU, B = cI + dU, where \({a, b, c, d} \in {L}^{n \times n}, {P}_{\pm} = \frac{1}{2}({I} {\pm} {S})\) are the Cauchy projectors in \({L}^{n}_{p}\) defined componentwise, and U is an involutory shift operator associated with the given Carleman backward shift also defined componentwise. By generalization to the vector case (n > 1) of the previously obtained results for the scalar case (n = 1), it is shown that whenever a certain 2n × 2n matrix function, associated with the original singular integral operator, admits a bounded factorization in \({L}^{2n}_{p}\) the Fredholm characteristics of the paired operator TA,B can be obtained in terms of that factorization, in particular the dimensions of the kernel and of the cokernel.

Semi-Fredholmness of Weighted Singular Integral Operators with Shifts and Slowly Oscillating Data

Let α, β be orientation-preserving homeomorphisms of [0,∞] onto itself, which have only two fixed points at 0 and ∞, and whose restrictions to ℝ+ = (0,∞) are diffeomorphisms, and let Uα, Uβ be the corresponding isometric shift operators on the space Lp(ℝ+) given by \({U}_{\mu}{f} = ({\mu}^\prime)^{1/p}(f\circ\mu)\) for \({\mu} \in \{\alpha, \beta\}\). We prove sufficient conditions for the right and left Fredholmness on Lp(ℝ+) of singular integral operators of the form \({A}_{+}{P}^{+}_{\gamma} \ {+} \ {A}_{-}{P}^{-}_{\gamma}\) , where \({P}^{\pm}_{\gamma} = ({I} \ {\pm} \ {S}_{\gamma})/2, \ {S}_{\gamma}\) is a weighted Cauchy singular integral operator, \({A}_{+} = \sum\nolimits_{k\in\mathbb{Z}} {a}_{k}{U}^{k}_{\alpha}\) and \({A}_{-} = \sum\nolimits_{k\in\mathbb{Z}} {b}_{k}{U}^{k}_{\beta}\) are operators in the Wiener algebras of functional operators with shifts. We assume that the coefficients ak, bk for \({k} \in \mathbb{Z}\) and the derivatives of the shifts \({\alpha}^\prime, {\beta}^\prime\) are bounded continuous functions on ℝ+ which may have slowly oscillating discontinuities at 0 and ∞.

On some Generalized Riemann Boundary Value Problems with Shift on the Real Line

On the real line we consider singular integral operators with a linear Carleman shift and complex conjugation, acting in \( \tilde{L}_2(\mathbb{R})\), the real space of all Lebesgue measurable complex value functions on ℝ with p = 2 power. We show that the original singular integral operator with shift and conjugation is, after extension, equivalent to a singular integral operator without shift and with a 4 × 4 matrix coefficients. By exploiting the properties of the factorization of the symbol of this last operator, it is possible to describe the solution of a generalized Riemann boundary value problem with a Carleman shift.

Singular Integral Operators with Linear Fractional Shifts on the Unit Circle

In this paper we propose a classification to the linear-fractional shifts and consider a class of paired singular integral operators with a shift of that class. We show how the study of the this type of operators can be reduced to the study of paired operators with, what we call, a canonical shift. Some of the results obtained are used to construct explicit solutions for a class of singular integral equations with a non-Carleman shift.