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Symbolic computation applied to the study of the kernel of special classes of paired singular integral operators. (English) Zbl 1491.68275

Summary: Operator 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. On the study of the operator’s kernel some progress has been achieved for some specific classes of singular integral operators whose properties allow the use of particular strategies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes of singular integral operators but are not designed to be implemented on a computer. The main goal of this paper is to show how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel of special classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle. It is described how some factorization algorithms can be used to compute the dimension of the kernel of special classes of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar], and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. For the essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The paper contains some interesting and nontrivial examples obtained with the use of a computer algebra system.

MSC:

68W30 Symbolic computation and algebraic computation
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47G10 Integral operators
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[1] Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society: Lecture Note Series, vol. 149. Cambridge University Press, Cambridge (1991) · Zbl 0762.35001
[2] Aktosun, T.; Klaus, M.; van der Mee, C., Explicit Wiener-Hopf factorization for certain non-rational matrix functions, Integral Equ. Oper. Theory, 15, 6, 879-900 (1992) · Zbl 0790.47012
[3] Ball, JA; Clancey, KF, An elementary description of partial indices of rational matrix functions, Integral Equ. Oper. Theory, 13, 3, 316-322 (1990) · Zbl 0719.47014
[4] Calderón, AP, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA, 74, 4, 1324-1327 (1997) · Zbl 0373.44003
[5] Câmara, MC; dos Santos, AF, Generalised factorization for a class of \(n\times n\) matrix functions—partial indices and explicit formulas, Integral Equ. Oper. Theory, 20, 2, 198-230 (1994) · Zbl 0815.47008
[6] Câmara, MC; dos Santos, AF; Carpentier, M., Explicit Wiewer-Hopf factorisation and non-linear Riemann-Hilbert problems, Proc. R. Soc. Edinb. Sect. (A), 132, 1, 45-74 (2002) · Zbl 1012.47005
[7] Clancey, K.; Gohberg, I., Factorization of matrix functions and singular integral operators, Oper. Theory Adv. Appl., 3, 1981 (1981) · Zbl 0474.47023
[8] Conceição, A.C.: Factorization of Some Classes of Matrix Functions and its Applications (in Portuguese). Ph.D. thesis, University of Algarve, Faro (2007)
[9] Conceição, A.C.: Computing the kernel of special classes of paired singular integral operators with Mathematica software. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) Proceedings of the 4th International Conference on Numerical and Symbolic Computation: Developments and Applications, Porto - Portugal (2019)
[10] Conceição, AC; Kravchenko, VG, About explicit factorization of some classes of non-rational matrix functions, Math. Nachr., 280, 9-10, 1022-1034 (2007) · Zbl 1131.47016
[11] Conceição, AC; Kravchenko, VG; Pereira, JC, Computing some classes of Cauchy type singular integrals with Mathematica software, Adv. Comput. Math., 39, 2, 273-288 (2013) · Zbl 1273.30029
[12] 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, 13 pp. Instituto Superior Técnico, Lisboa, Portugal, April 2-3 (2012)
[13] Conceição, AC; Kravchenko, VG; Pereira, JC; Ball, J.; Bolotnikov, V.; Rodman, L.; Helton, J.; Spitkovsky, I., Factorization algorithm for some special non-rational matrix functions, Topics in Operator Theory, Operator Theory: Advances and Applications, 87-109 (2010), Basel: Birkhäuser, Basel
[14] Conceição, AC; Kravchenko, VG; Teixeira, FS; Samko, S.; Lebre, A.; dos Santos, AF, Factorization of some classes of matrix functions and the resolvent of a Hankel operator, FSORP2003 Factorization, Singular Operators and Related Problems, Funchal, Portugal, 101-110 (2003), Dordrecht: Kluwer, Dordrecht
[15] Conceição, AC; Kravchenko, VG; Teixeira, FS; Büttcher, A.; Kaashoek, MA; Lebre, AB; dos Santos, AF; Speck, F-O, Factorization of matrix funtions and the resolvents of certain operators Singular Integral Operators, Factorization and Applications—Operator Theory: Advances and Applications, 91-100 (2003), Basel: Birkhäuser, Basel · Zbl 1057.47025
[16] Conceição, AC; Marreiros, RC, On the kernel of a singular integral operator with non-Carleman shift and conjugation, Oper. Matrices, 9, 2, 433-456 (2015) · Zbl 1316.47041
[17] Conceição, AC; Marreiros, RC; Pereira, JC, Symbolic computation applied to the study of the kernel of a singular integral operator with non-Carleman shift and conjugation, Math. Comput. Sci., 10, 3, 365-386 (2016) · Zbl 1369.47059
[18] Conceição, A.C., Pereira, J.C.: Using wolfram mathematica in spectral theory. In: Loja, A., Barbosa, J.I., Rodrigues, J.A. (eds.) Proceedings of the 3rd International Conference on Numerical and Symbolic Computation: Developments and Applications, Guimarães, Portugal, pp. 295-304 (2017)
[19] Conceição, AC; Pereira, JC, Exploring the spectra of some classes of singular integral operators with symbolic computation, Math. Comput. Sci., 10, 2, 291-309 (2016) · Zbl 1380.47038
[20] Conceição, AC; Pereira, JC, Exploring the spectra of some classes of paired singular integral operators: the scalar and matrix cases, Lib. Math. (new series), 34, 2, 35 (2014) · Zbl 1342.47006
[21] Davis, G., Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. Ecole Norm. S., 17, 1, 157-189 (1984) · Zbl 0537.42016
[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) · Zbl 1008.47016
[23] Faddeev, LD; Tkhatayan, LA, Hamiltonian Methods in the Theory of Solitons (1987), Berlin: Springer, Berlin · Zbl 1111.37001
[24] Feldman, I.; Gohberg, I.; Krupnik, N., An explicit factorization algorithm, Integral Equ. Oper. Theory, 49, 2, 149-164 (2004) · Zbl 1074.15014
[25] Feldman, I.; Marcus, A., On some properties of factorization indices, Integral Equ. Oper Theory, 30, 3, 326-337 (1998) · Zbl 0905.47012
[26] Garnett, JB, Bounded Analytic Functions. Graduate Texts in Mathematics (2007), Berlin: Springer, Berlin
[27] Gohberg, I.; Kaashoek, MA; Spitkovsky, IM; Gohberg, I.; Manojloviv, N.; dos Santos, AF, An overview of matrix factorization theory and operator applications, Factorization and Integrable Systems, Operator Theory: Advances and Applications, 1-102 (2003), Basel: Birkhäuser, Basel · Zbl 1049.47001
[28] Gohberg, I.; Krupnik, N., One-dimensional linear singular integral equations, Oper. Theory Adv. Appl., 54, 1992 (1992) · Zbl 0781.47038
[29] Gohberg, I.; Krupnik, N., One-dimensional linear singular integral equations, Oper. Theory Adv. Appl., 53, 1992 (1992) · Zbl 0781.47038
[30] Janashia, G.; Lagvilava, E., On factorization and partial indices of unitary matrix-functions of one class, Georgian Math. J., 4, 5, 439-442 (1997) · Zbl 0893.47011
[31] Kravchenko, VG; Litvinchuk, GS, Introduction to the Theory of Singular Integral Operators with Shift. Mathematics and its Applications (1994), Dordrecht: Kluwer, Dordrecht · Zbl 0811.47049
[32] Kravchenko, VG; Migdal’skii, AI, A regularization algorithm for some boundary-value problems of linear conjugation, Dokl. Math., 52, 319-321 (1995)
[33] Litvinchuk, GS, Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and its Applications (2000), Dordrecht: Kluwer, Dordrecht · Zbl 0980.45001
[34] Litvinchuk, GS; Spitkovskii, IM, Factorization of measurable matrix functions, Oper. Theory Adv. Appl., 25, 1987 (1987)
[35] Mikhlin, SG; Prössdorf, S., Singular Integral Operators (1986), Berlin: Springer, Berlin
[36] Nikol’skii, NK, Treatise on the Shift Operator. Spectral Function Theory. Grundlehren der mathematischen Wissenschaften (1986), Berlin: Springer, Berlin
[37] Plemelj, J., Riemannshe Funktionenscharen mit gegebener Monodromiegruppe, Monat. Math. Phys., 19, 211-245 (1908) · JFM 39.0461.01
[38] Prössdorf, S., Some Classes of Singular Equations (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0416.45003
[39] Rudin, W., Real and Complex Analysis (1987), New York: McGraw-Hill, New York · Zbl 0925.00005
[40] Voronin, AF, A method for determining the partial indices of symmetric matrix functions, Sib. Math. J., 52, 41-53 (2011) · Zbl 1223.15013
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