41 research outputs found
RiemannâHilbert problems, Toeplitz operators and Q-classes
We generalize the notion of Q-classes C(Q1,Q2) , which was introduced in the context of WienerâHopf factorization, by considering very general 2 Ă 2 matrix functions Q1, Q2. This allows us to use a
mainly algebraic approach to obtain several equivalent representations for each class, to study the intersections of Q-classes and to explore their close connection with certain non-linear scalar equations. The results are applied to various factorization problems and to the study of Toeplitz operators with symbol in a Q-class. We conclude with a group theoretic interpretation of some of the main results.Fundação para a Ciência e a Tecnologia (FCT/Portugal), through Project PTDC/MAT/121837/2010 and Project Est- C/MAT/UI0013/2011. The first author was also supported by the Center for Mathematical
Analysis, Geometry, and Dynamical Systems and the second author was also supported by the Centre of Mathematics of the University of Minho through the FEDER Funds Programa Operacional Factores de Competitividade COMPET
Factorization in a torus and Riemann-Hilbert problems
A new concept of meromorphic -factorization, for H\"{o}lder
continuous functions defined on a contour that is the pullback of
(or the unit circle) in a Riemann surface of genus
1, is introduced and studied, and its relations with holomorphic
-factorization are discussed. It is applied to study and solve some
scalar Riemann-Hilbert problems in and vectorial Riemann-Hilbert
problems in , including Wiener-Hopf matrix factorization, as well
as to study some properties of a class of Toeplitz operators with
matrix symbols.Comment: accepted for publication in Journal of Mathematical Analysis and
Application
WienerâHopf factorization for a group of exponentials of nilpotent matrices
AbstractA complete study of the generalized factorization for a group of 2Ă2 matrix functions of the form G=I+ÎłN, where ÎłâC(RĚ), I denotes the 2Ă2 identity matrix and N represents a rational nilpotent matrix function, is presented. A closely related class involving the same matrix N is also studied. The canonical and non-canonical factorizations are considered and explicit formulas are obtained for the partial indices and the factors in such factorizations. It is shown in particular that only one of the columns in the factors needs to be determined, as a solution to a homogeneous linear RiemannâHilbert problem, the other column being expressed in terms of the first. Necessary and sufficient conditions for existence of a canonical factorization within the same class are established, as well as explicit formulas for the factors in this case