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A C*-algebra of Singular Integral Operators with Shifts Similar to Affine Mappings

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Operator Theory, Operator Algebras and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 242))

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Abstract

Representations on Hilbert spaces for the nonlocal C*-algebra \(\mathfrak{B}\) of singular integral operators with piecewise slowly oscillating coefficients, which is extended by the unitary shift operatorsU g associated with the solvable discrete group G of diffeomorphisms \(g\;:\;\mathbb{T}\rightarrow\mathbb{T}\) that are similar to affine mappings on the real line, are constructed. Such shifts may change or preserve the orientation on t and have both common fixed points for all \(g\;\in\;G\) and distinct fixed points for different shifts. Using the theory developed for C*- algebras of singular integral operators with shifts preserving the orientation of a contour, a Fredholm symbol calculus for the C*-algebra \(\mathfrak{B}\) is constructed and a Fredholm criterion for the operators \(B\;\in\;\mathfrak{B}\) is established.

Mathematics Subject Classification (2010). Primary 47A53; Secondary 47A67, 47B33, 47G10, 47L15, 47L30.

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Author information

Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2825, Monte de Caparica, Portugal

    Cláudio A. Fernandes

  3. Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209, Cuernavaca, Morelos, México

    Yuri I. Karlovich

Authors
  1. M. Amélia Bastos
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  2. Cláudio A. Fernandes
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  3. Yuri I. Karlovich
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Corresponding author

Correspondence to M. Amélia Bastos .

Editor information

Editors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal

    Amarino Lebre

  3. Departamento de Matemática, Universidade do Algarve, Faro, Portugal

    Stefan Samko

  4. Department of Mathematics, College of William & Mary, Williamsburg, Virginia, USA

    Ilya M. Spitkovsky

Additional information

Dedicated to Professor António Ferreira dos Santos

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Bastos, M.A., Fernandes, C.A., Karlovich, Y.I. (2014). A C*-algebra of Singular Integral Operators with Shifts Similar to Affine Mappings. In: Bastos, M., Lebre, A., Samko, S., Spitkovsky, I. (eds) Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, vol 242. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0816-3_3

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