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Riemann–Hilbert Problems, Toeplitz Operators and \({\mathfrak{Q}}\)-Classes

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Abstract

We generalize the notion of \({\mathfrak{Q}}\)-classes \({C_{{Q_1} {Q_2}}}\), which was introduced in the context of Wiener–Hopf factorization, by considering very general 2 × 2 matrix functions Q 1, Q 2. This allows us to use a mainly algebraic approach to obtain several equivalent representations for each class, to study the intersections of \({\mathfrak{Q}}\)-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 \({\mathfrak{Q}}\)-class. We conclude with a group theoretic interpretation of some of the main results.

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Authors and Affiliations

  1. Departamento de Matemática, Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, 1049-001, Lisbon, Portugal

    M. C. Câmara

  2. Departamento de Matemática e Aplicações, Universidade do Minho, Campus de Azurém, 4800, Guimarães, Portugal

    M. T. Malheiro

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  1. M. C. Câmara
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  2. M. T. Malheiro
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Correspondence to M. C. Câmara.

Additional information

This work was partially supported by 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 COMPETE.

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Câmara, M.C., Malheiro, M.T. Riemann–Hilbert Problems, Toeplitz Operators and \({\mathfrak{Q}}\)-Classes. Integr. Equ. Oper. Theory 80, 239–264 (2014). https://doi.org/10.1007/s00020-014-2140-2

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  • DOI: https://doi.org/10.1007/s00020-014-2140-2

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