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Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

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Abstract

Spectral theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to show how the symbolic computation capabilities of Mathematica allow us to explore the spectra of several classes of singular integral operators. For the one-dimensional case, nontrivial rational examples, computed with the automated process called [ASpecPaired-Scalar], are presented. For the matrix case, nontrivial essentially bounded and rational examples, computed with the analytical algorithms [AFact], [SInt], and [ASpecPaired-Matrix], are presented. In both cases, it is possible to check, for each considered paired singular integral operator, if a complex number (chosen arbitrarily) belongs to its spectrum.

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References

  1. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society: Lecture Note Series 149. Cambridge University Press, Cambridge (1991)

  2. Aktosun, T., Klaus, M., van der Mee, C.: Explicit Wiener–Hopf factorization for certain non-rational matrix functions. Integral Eq. Oper. Theory 15(6), 879–900 (1992)

    Article  MATH  Google Scholar 

  3. Asch, M., Lebeau, G.: The Spectrum of the damped wave operator for a bounded domain in \(R^2\). Exp. Math. 12(2), 227–241 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ball, J.A., Clancey, K.F.: An elementary description of partial indices of rational matrix functions. Integral Eq. Oper. Theory. 13(3), 316–322 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bastos, M.A., Bravo, A., Karlovich, Y.I., Spitkovsky, I.M.: On the factorization of some block triangular almost periodic matrix functions. Oper. Theory Adv. Appl. 242, 25–52 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bastos, M.A., Karlovich, Yu.I., Spitkovsky, I.M., Tishin, P.M.: On a new algorithm for almost periodic factorization. Oper. Theory Adv. Appl. 103, 53–74 (Birkhäuser) (1998)

  7. Böttcher, A., Karlovich, Yu.I.: Cauchy’s singular integral operator and its beautiful spectrum. Oper. Theory Adv. Appl. 129, 109–142 (Birkhäuser, Basel) (2001)

  8. Câmara, M.C., dos Santos, A.F.: Generalised factorization for a class of \(n\times n\) matrix functions–partial indices and explicit formulas. Integral Eq. Oper. Theory. 20(2), 198–230 (Birkhäuser) (1994)

  9. Câmara, M.C., dos Santos, A.F., Carpentier, M.: Explicit Wiewer–Hopf factorisation and non-linear Riemann–Hilbert problems. Proc. R. Soc. Edinb. Sect. (A) 132(1), 45–74 (2002)

    Article  MATH  Google Scholar 

  10. Clancey, K., Gohberg, I.: Factorization of matrix functions and singular integral operators. In: Ball, J.A., Dym, H., Kaashoek, M., Langer, H., Tretter, C. (eds) Operator Theory: Advances and Applications. vol. 3. Birkhäuser, Basel (1981)

  11. Conceição, A.C.: Factorization of some classes of matrix functions and its applications (in portuguese). Ph.D thesis, University of Algarve, Faro (2007)

  12. Conceição, A.C., Kravchenko, V.G.: About explicit factorization of some classes of non-rational matrix functions. Math. Nachr. 280(9–10), 1022–1034 (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) (2007)

  13. Conceição, A.C., Kravchenko, V.G.: Factorization algorithm for some special matrix functions. Oper. Theory Adv. Appl. 181, 173–185 (Birkhäuser) (2008)

  14. Conceição, A.C., Kravchenko, V.G., Pereira, J.C.: Factorization algorithm for Some special non-rational matrix functions. In: Ball, J., Bolotnikov, V., Rodman, L., Helton, J., Spitkovsky, I. (eds.) Topics in Operator Theory, Operator Theory: Advances and Applications, vol. 202, pp. 87–109. Birkhäuser, Basel (2010)

    Chapter  Google Scholar 

  15. Conceição, A.C., Kravchenko, V.G., Pereira, J.C.: Rational functions factorization algorithm: a symbolic computation for the scalar and matrix cases. In: Proceedings of the 1st National Conference on Symbolic Computation in Education and Research (CD-ROM), P02, pp. 13. Instituto Superior Tcnico, Lisboa, Portugal, April 2–3 (2012)

  16. Conceição, A.C., Kravchenko, V.G., Pereira, J.C.: Computing some classes of Cauchy type singular integrals with Mathematica software. Adv. Comput. Math. 39(2), 273–288 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Conceição, A.C., Kravchenko, V.G., Teixeira, F.S.: Factorization of some classes of matrix functions and the resolvent of a Hankel operator. FSORP2003 Factorization, Singular Operators and Related Problems. In: Samko, S., Lebre, A., dos Santos, A.F. (eds.) Funchal, Portugal, pp. 101–110. Kluwer Academic Publishers (2003)

  18. Conceição, A.C., Kravchenko, V.G., Teixeira, F.S.: Factorization of matrix funtions and the resolvents of certain operators Oper. Theory Adv. Appl. 142, 91–100 (2003)

    MATH  Google Scholar 

  19. Conceição, A.C., Marreiros, R.C.: On the kernel of a singular integral operator with non-Carleman shift and conjugation. Oper. Matrices 9(2), 433–456 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Conceição, A.C., Pereira, J.C.: Exploring the spectra of some classes of paired singular integral operators: the scalar and matrix cases. Lib. Math. (new series) 34(2), 35 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Conceição, A.C., Pereira, J.C.: An overview of symbolic computation on operator theory: SYMCOMP2013. In. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) Proceedings of the 1st international conference on algebraic and symbolic computation (ECCOMAS Thematic Conference), pp. 39–71. Lisboa, Portugal (2013)

  22. Ehrhardt, T., Speck, F.-O.: Transformation techniques towards the factorization of non-rational \(2\times 2\) matrix functions. Linear Algebra Appl. 353(1–3), 53–90 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Faddeev, L.D., Tkhatayan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)

    Book  Google Scholar 

  24. Feldman, I., Gohberg, I., Krupnik, N.: An explicit factorization algorithm. Integral Eq. Oper. Theory. 49(2), 149–164 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Feldman, I., Marcus, A.: On some properties of factorization indices. Integral Eq. Oper. Theory. 30(3), 326–337 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gohberg, I., Kaashoek, M.A., Spitkovsky, I.M.: An overview of matrix factorization theory and operator applications. In: Gohberg, I., Manojloviv, N., dos Santos, A.F. (eds.) Factorization and Integrable Systems, Operator Theory: Advances and Applications, vol. 141, pp. 1–102. Birkhäuser, Basel (2003)

    Chapter  Google Scholar 

  27. Gohberg, I., Krupnik, N.: The spectrum of singular integral operators in \(L_p\) spaces. Oper. Theory Adv. Appl. 206, 111–125 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Gohberg, I., Krupnik, N.: One-dimensional linear singular integral equations. In: Operator Theory: Advances and Applications. vol. 53. Birkhäuser, Basel (1992)

  29. Gohberg, I., Lerer, L., Rodman, L.: Factorization indices for matrix polynomials. Bull. Am. Math. Soc. 84(2), 275–277 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Isgur, A., Spitkovsky, I.M.: On the spectra of some Toeplitz and Wiener–Hopf operators with almost periodic matrix symbols. Oper. Matrices 2(3), 371–383 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Janashia, G., Lagvilava, E.: On factorization and partial indices of unitary matrix-functions of one class. Georgian Math. J. 4(5), 439–442 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kravchenko, V.G., Lebre, A.B., Litvinchuk, G.S.: Spectrum problems for singular integral operators with Carleman shift. Math. Nachr. 226, 129–151 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kravchenko, V.G., Lebre, A.B., Rodriguez, J.S.: Factorization of singular integral operators with a Carleman shift and spectral problems. J. Integral Eq. Appl. 13(4), 339–383 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kravchenko, V.G., Litvinchuk, G.S.: Introduction to the Theory of Singular Integral Operators with Shift. In: Hazewinkel, M. (ed) Mathematics and its Applications. vol. 289. Kluwer Academic Publishers, Dordrecht (1994)

  35. Kravchenko, V.G., Migdal’skii, A.I.: A regularization algorithm for some boundary-value problems of linear conjugation. Dokl. Math. 52, 319–321 (1995)

    Google Scholar 

  36. Kravchenko, V.G., Nikolaichuk, A.M.: On partial indices of the Riemann problem for two pair of functions. Sov. Math. Dokl. 15, 438–442 (1974)

    Google Scholar 

  37. Litvinchuk, G.S.: Solvability theory of boundary value problems and singular integral equations with shift. In: Hazewinkel, M. (ed) Mathematics and its Applications, vol. 523. Kluwer Academic Publishers, Dordrecht (2000)

  38. Litvinchuk, G.S., Spitkovskii, I.M.: Factorization of measurable matrix functions. In: Heinig, G. (ed) Operator Theory: Advances and Applications, vol. 25. Birkhäuser, Basel (1987)

  39. Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  40. Pereira, J.C., Conceição, A.C.: Exploring the spectra of singular integral operators with rational coefficients. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) SYMCOMP2013 proceedings of the 1st international conference on algebraic and symbolic computation (ECCOMAS thematic conference), pp. 175–194. Lisboa, Portugal (2013)

  41. Plemelj, J.: Riemannshe Funktionenscharen mit gegebener Monodromiegruppe. Monat. Math. Phys. 19, 211–245 (1908)

    Article  MathSciNet  MATH  Google Scholar 

  42. Voronin, A.F.: A method for determining the partial indices of symmetric matrix functions. Sib. Math. J. 52, 41–53 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. Voronin, A.F.: Partial indices of unitary and hermitian matrix functions. Sib. Math. J. 51, 805–809 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Ana C. Conceição

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

    José C. Pereira

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  1. Ana C. Conceição
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  2. José C. Pereira
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Correspondence to Ana C. Conceição.

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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., Pereira, J.C. Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation. Math.Comput.Sci. 10, 291–309 (2016). https://doi.org/10.1007/s11786-016-0264-2

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  • DOI: https://doi.org/10.1007/s11786-016-0264-2

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