## Exploring the spectra of some classes of singular integral operators with symbolic computation.(English)Zbl 1380.47038

The paper shows the possibilities of the Mathematica computer algebra system to explore the spectra of several classes of singular integral operators. For a singular integral operator with rational coefficients, a pseudo code is given which permits to check if a given complex number is in the spectrum of the given operator. The matrix case is also discussed. Singular integral operators with essentially bounded coefficients of some special type are investigated. The paper contains numerous examples.

### MSC:

 47G10 Integral operators 47A10 Spectrum, resolvent 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 68W30 Symbolic computation and algebraic computation 45P05 Integral operators 45E05 Integral equations with kernels of Cauchy type

### Software:

SInt; ASpecPaired-Matrix; Mathematica; ASpecPaired-Scalar
Full Text:

### 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) · Zbl 0762.35001 [2] Aktosun, T; Klaus, M; Mee, C, Explicit Wiener-Hopf factorization for certain non-rational matrix functions, Integral Eq. Oper. Theory, 15, 879-900, (1992) · Zbl 0790.47012 [3] Asch, M; Lebeau, G, The spectrum of the damped wave operator for a bounded domain in $$R^2$$, Exp. Math., 12, 227-241, (2003) · Zbl 1061.35064 [4] Ball, JA; Clancey, KF, An elementary description of partial indices of rational matrix functions, Integral Eq. Oper. Theory., 13, 316-322, (1990) · Zbl 0719.47014 [5] Bastos, MA; Bravo, A; Karlovich, YI; Spitkovsky, IM, On the factorization of some block triangular almost periodic matrix functions, Oper. Theory Adv. Appl., 242, 25-52, (2014) · Zbl 1337.47028 [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) · Zbl 0903.47013 [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) · Zbl 1316.47041 [8] Câmara, M.C., dos Santos, A.F.: Generalised factorization for a class of $$n× n$$ matrix functions-partial indices and explicit formulas. Integral Eq. Oper. Theory. 20(2), 198-230 (Birkhäuser) (1994) · Zbl 0815.47008 [9] Câmara, MC; Santos, AF; Carpentier, M, Explicit wiewer-Hopf factorisation and non-linear Riemann-Hilbert problems, Proc. R. Soc. Edinb. Sect. (A), 132, 45-74, (2002) · Zbl 1012.47005 [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) · Zbl 0474.47023 [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) · Zbl 1207.47029 [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, AC; Kravchenko, VG; Pereira, JC; Ball, J (ed.); Bolotnikov, V (ed.); Rodman, L (ed.); Helton, J (ed.); Spitkovsky, I (ed.), Factorization algorithm for some special non-rational matrix functions, No. 202, 87-109, (2010), Basel · Zbl 1208.47017 [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) · Zbl 1273.30029 [16] Conceição, AC; Kravchenko, VG; Pereira, JC, Computing some classes of Cauchy type singular integrals with Mathematica software, Adv. Comput. Math., 39, 273-288, (2013) · Zbl 1273.30029 [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) · Zbl 0893.47011 [18] Conceição, AC; Kravchenko, VG; Teixeira, FS, Factorization of matrix funtions and the resolvents of certain operators oper, Theory Adv. Appl., 142, 91-100, (2003) · Zbl 1057.47025 [19] Conceição, AC; Marreiros, RC, On the kernel of a singular integral operator with non-Carleman shift and conjugation, Oper. Matrices, 9, 433-456, (2015) · Zbl 1316.47041 [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, 35, (2014) · Zbl 1342.47006 [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) · Zbl 1205.15016 [22] Ehrhardt, T; Speck, F-O, Transformation techniques towards the factorization of non-rational $$2× 2$$ matrix functions, Linear Algebra Appl., 353, 53-90, (2002) · Zbl 1008.47016 [23] Faddeev, L.D., Tkhatayan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987) · Zbl 1111.37001 [24] Feldman, I; Gohberg, I; Krupnik, N, An explicit factorization algorithm, Integral Eq. Oper. Theory., 49, 149-164, (2004) · Zbl 1074.15014 [25] Feldman, I; Marcus, A, On some properties of factorization indices, Integral Eq. Oper. Theory., 30, 326-337, (1998) · Zbl 0905.47012 [26] Gohberg, I; Kaashoek, MA; Spitkovsky, IM; Gohberg, I (ed.); Manojloviv, N (ed.); Santos, AF (ed.), An overview of matrix factorization theory and operator applications, No. 141, 1-102, (2003), Basel · Zbl 1049.47001 [27] Gohberg, I; Krupnik, N, The spectrum of singular integral operators in $$L_p$$ spaces, Oper. Theory Adv. Appl., 206, 111-125, (2010) · Zbl 1207.47029 [28] Gohberg, I., Krupnik, N.: One-dimensional linear singular integral equations. In: Operator Theory: Advances and Applications. vol. 53. Birkhäuser, Basel (1992) · Zbl 0781.47038 [29] Gohberg, I; Lerer, L; Rodman, L, Factorization indices for matrix polynomials, Bull. Am. Math. Soc., 84, 275-277, (1978) · Zbl 0381.30020 [30] Isgur, A; Spitkovsky, IM, On the spectra of some Toeplitz and Wiener-Hopf operators with almost periodic matrix symbols, Oper. Matrices, 2, 371-383, (2008) · Zbl 1168.47024 [31] Janashia, G; Lagvilava, E, On factorization and partial indices of unitary matrix-functions of one class, Georgian Math. J., 4, 439-442, (1997) · Zbl 0893.47011 [32] Kravchenko, VG; Lebre, AB; Litvinchuk, GS, Spectrum problems for singular integral operators with Carleman shift, Math. Nachr., 226, 129-151, (2001) · Zbl 0994.47047 [33] Kravchenko, VG; Lebre, AB; Rodriguez, JS, Factorization of singular integral operators with a Carleman shift and spectral problems, J. Integral Eq. Appl., 13, 339-383, (2001) · Zbl 1009.47034 [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) · Zbl 0811.47049 [35] Kravchenko, VG; Migdal’skii, AI, A regularization algorithm for some boundary-value problems of linear conjugation, Dokl. Math., 52, 319-321, (1995) · Zbl 0885.65152 [36] Kravchenko, VG; Nikolaichuk, AM, On partial indices of the Riemann problem for two pair of functions, Sov. Math. Dokl., 15, 438-442, (1974) · Zbl 0308.30010 [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) · Zbl 0980.45001 [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) · Zbl 0612.47024 [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) · Zbl 1168.47024 [41] Plemelj, J, Riemannshe funktionenscharen mit gegebener monodromiegruppe, Monat. Math. Phys., 19, 211-245, (1908) · JFM 39.0461.01 [42] Voronin, AF, A method for determining the partial indices of symmetric matrix functions, Sib. Math. J., 52, 41-53, (2011) · Zbl 1223.15013 [43] Voronin, AF, Partial indices of unitary and Hermitian matrix functions, Sib. Math. J., 51, 805-809, (2010) · Zbl 1205.15016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.