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Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation

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Abstract

On the Hilbert space \(\widetilde{L}_{2}(\mathbb {T})\) the singular integral operator with non-Carleman shift and conjugation \(K=P_{+}+(aI+AC)P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(A=\sum \nolimits _{j=0}^{m}a_{j}U^{j}\), \(a,a_{j}\), \(j=\overline{1,m}\), are continuous functions on the unit circle \(\mathbb {T}\), U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.

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Authors and Affiliations

  1. Center for Functional Analysis, Linear Structures and Applications (CEAFEL), Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, 8005-139, Faro, Portugal

    Ana C. Conceição, Rui C. Marreiros & José C. Pereira

  2. Center for Environmental and Sustainability Research (CENSE), Departamento de Engenharia Electrónica e Informática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, 8005-139, Faro, Portugal

    José C. Pereira

Authors
  1. Ana C. Conceição
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  2. Rui C. Marreiros
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  3. José C. Pereira
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Correspondence to Rui C. Marreiros.

Additional information

This research was supported by Fundação para a Ciência e Tecnologia (Portugal) through Centro de Análise Funcional, Estruturas Lineares e Aplicações of Instituto Superior Técnico.

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Conceição, A.C., Marreiros, R.C. & Pereira, J.C. Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation. Math.Comput.Sci. 10, 365–386 (2016). https://doi.org/10.1007/s11786-016-0271-3

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  • DOI: https://doi.org/10.1007/s11786-016-0271-3

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