Invertibility issues for Toeplitz plus Hankel operators and their close relatives. (English) Zbl 07393027
Bastos, M. Amélia (ed.) et al., Operator theory, functional analysis and applications. Proceedings of the 30th international workshop on operator theory and its applications, IWOTA 2019, Lisbon, Portugal, July 22–26, 2019. Cham: Birkhäuser. Oper. Theory: Adv. Appl. 282, 113-156 (2021).
In the paper the authors describes various approaches to the invertibility of Toeplitz plus Hankel operators in Hardy and \(l_{p}\)-spaces, integral and difference Wiener-Hopf plus Hankel operators and generalized Toeplitz plus Hankel operators. First they make a brief presentation of the classical approach of I. Gohberg, N. Krupnik and G. Litvinchuk, as well as, Basor-Ehrhardt Approach. A second large presentation is dedicated to a recently newly method developed by the authors of the paper. Not all proofs of the results are given. The interested readers can consult the paper for more details.
For the entire collection see [Zbl 1471.47002].
For the entire collection see [Zbl 1471.47002].
Reviewer: Dumitrŭ Popa (Constanţa)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47B38 | Linear operators on function spaces (general) |
| 47B33 | Linear composition operators |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
References:
| [1] | Baik, J.; Rains, EM, Algebraic aspects of increasing subsequences, Duke Math. J., 109, 1-65 (2001) · Zbl 1007.05096 · doi:10.1215/S0012-7094-01-10911-3 |
| [2] | Basor, EL; Ehrhardt, T., On a class of Toeplitz + Hankel operators, New York J. Math., 5, 1-16 (1999) · Zbl 0927.47019 |
| [3] | Basor, EL; Ehrhardt, T., Factorization theory for a class of Toeplitz + Hankel operators, J. Oper. Theory, 51, 411-433 (2004) · Zbl 1085.47032 |
| [4] | Basor, EL; Ehrhardt, T., Fredholm and invertibility theory for a special class of Toeplitz + Hankel operators, J. Spectral Theory, 3, 171-214 (2013) · Zbl 1282.47035 · doi:10.4171/JST/42 |
| [5] | E.L. Basor, T. Ehrhardt, Asymptotic formulas for determinants of a special class of Toeplitz + Hankel matrices, in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, vol. 259. The Albrecht Böttcher Anniversary Volume. Operator Theory: Advances and Applications (Birkhäuser, Basel, 2017), pp. 125-154 · Zbl 1472.47021 |
| [6] | E. Basor, Y. Chen, T. Ehrhardt, Painlevé V and time-dependent Jacobi polynomials. J. Phys. A 43, 015204, 25 (2010) · Zbl 1202.33030 |
| [7] | A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd edn. Springer Monographs in Mathematics (Springer, Berlin, 2006) · Zbl 1098.47002 |
| [8] | A. Böttcher, Y.I. Karlovich, I.M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions (Birkhäuser, Basel, 2002) · Zbl 1011.47001 |
| [9] | Castro, LP; Nolasco, AP, A semi-Fredholm theory for Wiener-Hopf-Hankel operators with piecewise almost periodic Fourier symbols, J. Oper. Theory, 62, 3-31 (2009) · Zbl 1211.47012 |
| [10] | Castro, LP; Silva, AS, Wiener-Hopf and Wiener-Hopf-Hankel operators with piecewise-almost periodic symbols on weighted Lebesgue spaces, Mem. Diff. Equ. Math. Phys., 53, 39-62 (2011) · Zbl 1273.47052 |
| [11] | K. Clancey, I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators (Birkhäuser, Basel, 1981) · Zbl 0474.47023 |
| [12] | Coburn, LA; Douglas, RG, Translation operators on the half-line, Proc. Nat. Acad. Sci. USA, 62, 1010-1013 (1969) · Zbl 0181.40801 · doi:10.1073/pnas.62.4.1010 |
| [13] | Didenko, VD; Silbermann, B., Index calculation for Toeplitz plus Hankel operators with piecewise quasi-continuous generating functions, Bull. London Math. Soc., 45, 633-650 (2013) · Zbl 1280.47038 · doi:10.1112/blms/bds126 |
| [14] | Didenko, VD; Silbermann, B., The Coburn-Simonenko Theorem for some classes of Wiener-Hopf plus Hankel operators, Publ. de l’Institut Mathématique, 96, 110, 85-102 (2014) · Zbl 1349.47041 |
| [15] | Didenko, VD; Silbermann, B., Some results on the invertibility of Toeplitz plus Hankel operators, Ann. Acad. Sci. Fenn. Math., 39, 443-461 (2014) · Zbl 1304.47037 · doi:10.5186/aasfm.2014.3919 |
| [16] | Didenko, VD; Silbermann, B., Structure of kernels and cokernels of Toeplitz plus Hankel operators, Integr. Equ. Oper. Theory, 80, 1-31 (2014) · Zbl 1314.47041 · doi:10.1007/s00020-014-2170-9 |
| [17] | Didenko, VD; Silbermann, B., Generalized inverses and solution of equations with Toeplitz plus Hankel operators, Bol. Soc. Mat. Mex., 22, 645-667 (2016) · Zbl 1394.47034 · doi:10.1007/s40590-016-0101-2 |
| [18] | Didenko, VD; Silbermann, B., Generalized Toeplitz plus Hankel operators: kernel structure and defect numbers, Compl. Anal. Oper. Theory, 10, 1351-1381 (2016) · Zbl 1394.47033 · doi:10.1007/s11785-015-0524-1 |
| [19] | Didenko, VD; Silbermann, B., Invertibility and inverses of Toeplitz plus Hankel operators, J. Oper. Theory, 72, 293-307 (2017) · Zbl 1399.47091 |
| [20] | V.D. Didenko, B. Silbermann, Kernels of Wiener-Hopf plus Hankel operators with matching generating functions, in Recent Trends in Operator Theory and Partial Differential Equations, vol. 258. The Roland Duduchava Anniversary Volume. Operator Theory: Advances and Applications (Birkhäuser, Basel, 2017), pp. 111-127 · Zbl 1381.47021 |
| [21] | V.D. Didenko, B. Silbermann, Kernels of a class of Toeplitz plus Hankel operators with piecewise continuous generating functions, in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, ed. by J. Dick, F.Y. Kuo, H. Woźniakowski (eds). (Springer, Cham, 2018), pp. 317-337 · Zbl 1405.47009 |
| [22] | Didenko, VD; Silbermann, B., The invertibility of Toeplitz plus Hankel operators with subordinated operators of even index, Linear Algebra Appl., 578, 425-445 (2019) · Zbl 1476.47024 · doi:10.1016/j.laa.2019.05.028 |
| [23] | V.D. Didenko, B. Silbermann, Invertibility issues for a class of Wiener-Hopf plus Hankel operators. J. Spectral Theory 11 (2021) · Zbl 07393027 |
| [24] | Duduchava, RV, Wiener-Hopf integral operators with discontinuous symbols, Dokl. Akad. Nauk SSSR, 211, 277-280 (1973) · Zbl 0294.47026 |
| [25] | Duduchava, RV, Integral operators of convolution type with discontinuous coefficients, Math. Nachr., 79, 75-98 (1977) · Zbl 0362.45004 · doi:10.1002/mana.19770790108 |
| [26] | Duduchava, RV, Integral Equations with Fixed Singularities (1979), Leipzig: BSB B. G. Teubner Verlagsgesellschaft, Leipzig · Zbl 0439.45002 |
| [27] | R. Edwards, Fourier Series. A Modern Introduction, vol. 1. Graduate Texts in Mathematics, vol. 85 (Springer, Berlin, 1982) · Zbl 0599.42001 |
| [28] | Ehrhardt, T., Factorization theory for Toeplitz+Hankel operators and singular integral operators with flip (2004), Habilitation Thesis: Technische Universität Chemnitz, Habilitation Thesis · Zbl 1063.47020 |
| [29] | Ehrhardt, T., Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip, J. Funct. Anal., 208, 64-106 (2004) · Zbl 1063.47020 · doi:10.1016/S0022-1236(03)00113-7 |
| [30] | Forrester, PJ; Frankel, NE, Applications and generalizations of Fisher-Hartwig asymptotics, J. Math. Phys., 45, 2003-2028 (2004) · Zbl 1071.82014 · doi:10.1063/1.1699484 |
| [31] | Gohberg, IC; Feldman, IA, Convolution Equations and Projection Methods for Their Solution (1974), Providence: American Mathematical Society, Providence · Zbl 0278.45008 |
| [32] | I. Gohberg, N. Krupnik, One-Dimensional Linear Singular Integral Equations. I, vol. 53. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1992) · Zbl 0743.45004 |
| [33] | I. Gohberg, N. Krupnik, One-Dimensional Linear Singular Integral Equations. II, vol. 54. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1992) · Zbl 0781.47038 |
| [34] | Grudsky, S.; Rybkin, A., On positive type initial profiles for the KdV equation, Proc. Am. Math. Soc., 142, 2079-2086 (2014) · Zbl 1294.35122 · doi:10.1090/S0002-9939-2014-11943-5 |
| [35] | Grudsky, S.; Rybkin, A., Soliton theory and Hankel operators, SIAM J. Math. Anal., 47, 2283-2323 (2015) · Zbl 1332.35325 · doi:10.1137/151004926 |
| [36] | Grudsky, SM; Rybkin, AV, On the trace-class property of Hankel operators arising in the theory of the Korteweg-de Vries equation, Math. Notes, 104, 377-394 (2018) · Zbl 07035816 · doi:10.1134/S0001434618090067 |
| [37] | Junghanns, P.; Kaiser, R., A note on Kalandiya’s method for a crack problem, Appl. Numer. Math., 149, 52-64 (2020) · Zbl 1471.74065 · doi:10.1016/j.apnum.2019.05.002 |
| [38] | Karapetiants, NK; Samko, SG, On Fredholm properties of a class of Hankel operators, Math. Nachr., 217, 75-103 (2000) · Zbl 0966.45010 · doi:10.1002/1522-2616(200009)217:1<75::AID-MANA75>3.0.CO;2-J |
| [39] | Karapetiants, NK; Samko, SG, Equations with Involutive Operators (2001), Boston: Birkhäuser Boston Inc., Boston · Zbl 0990.47011 · doi:10.1007/978-1-4612-0183-0 |
| [40] | Kravchenko, VG; Lebre, AB; Rodríguez, JS, Factorization of singular integral operators with a Carleman shift via factorization of matrix functions: the anticommutative case, Math. Nachr., 280, 1157-1175 (2007) · Zbl 1130.47028 · doi:10.1002/mana.200510543 |
| [41] | Kravchenko, VG; Lebre, AB; Rodríguez, JS, Factorization of singular integral operators with a Carleman backward shift: the case of bounded measurable coefficients, J. Anal. Math., 107, 1-37 (2009) · Zbl 1190.47054 · doi:10.1007/s11854-009-0001-8 |
| [42] | N.Y. Krupnik, Banach Algebras with Symbol and Singular Integral Operators, vol. 26. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1987) · Zbl 0641.47031 |
| [43] | Lebre, AB; Meister, E.; Teixeira, FS, Some results on the invertibility of Wiener-Hopf-Hankel operators, Z. Anal. Anwend., 11, 57-76 (1992) · Zbl 0787.47022 · doi:10.4171/ZAA/626 |
| [44] | G.S. Litvinchuk, I.M. Spitkovskii, Factorization of Measurable Matrix Functions, vol. 25. Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1987) · Zbl 0651.47010 |
| [45] | E. Meister, F. Penzel, F.-O. Speck, F.S. Teixeira, Two-media scattering problems in a half-space, in Partial Differential Equations with Real Analysis. Dedicated to Robert Pertsch Gilbert on the occasion of his 60th birthday (Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1992), pp. 122-146 · Zbl 0791.35133 |
| [46] | Meister, E.; Speck, F-O; Teixeira, FS, Wiener-Hopf-Hankel operators for some wedge diffraction problems with mixed boundary conditions, J. Integral Equ. Appl., 4, 229-255 (1992) · Zbl 0756.45005 · doi:10.1216/jiea/1181075683 |
| [47] | V.V. Peller, Hankel Operators and Their Applications. Springer Monographs in Mathematics (Springer, New York, 2003) · Zbl 1030.47002 |
| [48] | Power, SC, C*-algebras generated by Hankel operators and Toeplitz operators, J. Funct. Anal., 31, 52-68 (1979) · Zbl 0401.46036 · doi:10.1016/0022-1236(79)90097-1 |
| [49] | S. Roch, B. Silbermann, Algebras of convolution operators and their image in the Calkin algebra, vol. 90. Report MATH (Akademie der Wissenschaften der DDR Karl-Weierstrass-Institut für Mathematik, Berlin, 1990) · Zbl 0717.47019 |
| [50] | Roch, S.; Silbermann, B., A handy formula for the Fredholm index of Toeplitz plus Hankel operators, Indag. Math., 23, 663-689 (2012) · Zbl 1272.47041 · doi:10.1016/j.indag.2012.06.008 |
| [51] | S. Roch, P.A. Santos, B. Silbermann, Non-Commutative Gelfand Theories. A Tool-Kit for Operator Theorists and Numerical Analysts. Universitext (Springer, London, 2011) · Zbl 1209.47002 |
| [52] | B. Silbermann, The C^∗-algebra generated by Toeplitz and Hankel operators with piecewise quasicontinuous symbols. Integr. Equ. Oper. Theory 10, 730-738 (1987) · Zbl 0647.47037 |
| [53] | Simonenko, IB, Some general questions in the theory of Riemann boundary problem, Math. USSR Izvestiya, 2, 1091-1099 (1968) · Zbl 0186.13601 · doi:10.1070/IM1968v002n05ABEH000706 |
| [54] | I.J. Šneı̆berg, Spectral properties of linear operators in interpolation families of Banach spaces. Mat. Issled. 9, 2(32), 214-229 (1974) (in Russian) · Zbl 0314.46033 |
| [55] | I.M. Spitkovskiı̆, The problem of the factorization of measurable matrix-valued functions. Dokl. Akad. Nauk SSSR 227, 576-579 (1976) (in Russian) · Zbl 0345.47022 |
| [56] | Teixeira, FS, Diffraction by a rectangular wedge: Wiener-Hopf-Hankel formulation, Integr. Equ. Oper. Theory, 14, 436-454 (1991) · Zbl 0734.35023 · doi:10.1007/BF01218506 |
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