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The part of my path I walked together with Sergei Grudsky. (English) Zbl 1349.01022

Summary: This is an essay containing personal reminiscences and describing selected topics of joint mathematical work of the author and Sergei Grudsky. The topics have their focus on Toeplitz operators and large finite Toeplitz matrices.

MSC:

01A70 Biographies, obituaries, personalia, bibliographies
15-03 History of linear algebra
47-03 History of operator theory
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Biographic References:

Grudsky, Sergei
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References:

[1] Batalshchikov, A.A., Grudsky, S., Stukopin, V.A.: Asymptotics of eigenvalues of symmetric Toeplitz band matrices. Linear Algebra Appl. 469, 464-486 (2015) · Zbl 1318.47043 · doi:10.1016/j.laa.2014.11.034
[2] Bogoya, J.M., Böttcher, A., Grudsky, S.: Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices. Oper. Theory Adv. Appl. 220, 77-95 (2012) · Zbl 1390.15094
[3] Bogoya, J.M., Böttcher, A., Grudsky, S., Maksimenko, E.A.: Eigenvectors of Hessenberg Toeplitz matrices and a problem by Dai, Geary, and Kadanoff. Linear Algebra Appl. 436, 3480-3492 (2012) · Zbl 1260.15009 · doi:10.1016/j.laa.2011.12.012
[4] Bogoya, J.M., Böttcher, A., Grudsky, S., Maximenko, E.A.: Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. J. Math. Anal. Appl. 422, 1308-1334 (2015) · Zbl 1302.65086 · doi:10.1016/j.jmaa.2014.09.057
[5] Bogoya, J.M., Böttcher, A., Grudsky, S., Maximenko, E.A.: Maximum norm versions of the Szegő and Avram-Parter theorems for Toeplitz matrices. J. Approx. Theory 196, 79-100 (2015) · Zbl 1320.15028 · doi:10.1016/j.jat.2015.03.003
[6] Böttcher, A.: Wiener-Hopf determinants with rational symbols. Math. Nachr. 144, 39-64 (1989) · Zbl 0691.45004 · doi:10.1002/mana.19891440105
[7] Böttcher, A.: Pseudospectra and singular values of large convolution operators. J. Integral Equ. Appl. 6, 267-301 (1994) · Zbl 0819.45002 · doi:10.1216/jiea/1181075815
[8] Böttcher, A., Brunner, H., Iserles, A., Nørsett, S.P.: On the singular values and eigenvalues of the Fox-Li and related operators. N. Y. J. Math. 16, 539-561 (2010) · Zbl 1233.47022
[9] Böttcher, A., Embree, M., Trefethen, L.N.: Piecewise continuous Toeplitz matrices and operators: slow approach to infinity. SIAM J. Matrix Anal. Appl. 24, 484-489 (2002) · Zbl 1022.47018 · doi:10.1137/S0895479800376971
[10] Böttcher, A., Grudsky, S.: Toeplitz operators with discontinuous symbols: phenomena beyond piecewise discontinuity. Oper. Theory Adv. Appl. 90, 55-118 (1996) · Zbl 0887.47026
[11] Böttcher, A., Grudsky, S.: On the condition numbers of large semi-definite Toeplitz matrices. Linear Algebra Appl. 279, 285-301 (1998) · Zbl 0934.15005 · doi:10.1016/S0024-3795(98)00015-9
[12] Böttcher, A., Grudsky, S.: Toeplitz band matrices with exponentially growing condition numbers. Electron. J. Linear Algebra 5, 104-125 (1999) · Zbl 0940.15005
[13] Böttcher, A., Grudsky, S.: Condition numbers of large Toeplitz-like matrices. Contemp. Math. 280, 273-299 (2001) · Zbl 1001.47015 · doi:10.1090/conm/280/04635
[14] Böttcher, A., Grudsky, S.: Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis. Hindustan Book Agency, New Delhi (2000); reprinted by Birkhäuser Verlag, Basel (2000) · Zbl 0969.47022
[15] Böttcher, A., Grudsky, S.: Can spectral value sets of Toeplitz band matrices jump? Linear Algebra Appl. 351/352, 99-116 (2002) · Zbl 1005.47032
[16] Böttcher, A., Grudsky, S.: Asymptotic spectra of dense Toeplitz matrices are unstable. Numer. Algorithms 33, 105-112 (2003) · Zbl 1038.65030 · doi:10.1023/A:1025547501771
[17] Böttcher, A., Grudsky, S.: Toeplitz matrices with slowly growing pseudospectra. Factorization, Singular Operators and Related Problems (Funchal, 2002), pp. 43-54. Kluwer Acad. Publ, Dordrecht (2003) · Zbl 1037.47017
[18] Böttcher, A., Grudsky, S.: The norm of the product of a large matrix and a random vector. Electron. J. Probab. 8 paper no. 7 (2003) · Zbl 1065.15032
[19] Böttcher, A., Grudsky, S.: Asymptotically good pseudomodes for Toeplitz matrices and Wiener-Hopf operators. Oper. Theory Adv. Appl. 147, 175-188 (2004) · Zbl 1073.47036
[20] Böttcher, A., Grudsky, S.: Structured condition numbers of large Toeplitz matrices are rarely better than usual condition numbers. Numer. Linear Algebra Appl. 12, 95-102 (2005) · Zbl 1164.15309 · doi:10.1002/nla.401
[21] Böttcher, A., Grudsky, S.: Spectral Properties of Banded Toeplitz Matrices. SIAM, Philadelphia (2005) · Zbl 1089.47001 · doi:10.1137/1.9780898717853
[22] Böttcher, A., Grudsky, S., Huybrechs, D., Iserles, A.: First-order trace formulae for the iterates of the Fox-Li operator. Oper. Theory Adv. Appl. 218, 207-224 (2012) · Zbl 1270.47024
[23] Böttcher, A., Grudsky, S., Iserles, A.: Spectral theory of large Wiener-Hopf operators with complex-symmetric kernels and rational symbols. Math. Proc. Camb. Philos. Soc. 151, 161-191 (2011) · Zbl 1227.47015 · doi:10.1017/S0305004111000259
[24] Böttcher, A.; Grudsky, S.; Iserles, A.; Pardalos, PM (ed.); Rassias, TM (ed.), The Fox-Li operator as a test and a spur for Wiener-Hopf theory, 37-48 (2012), Heidelberg · Zbl 1317.47027 · doi:10.1007/978-3-642-28821-0_3
[25] Böttcher, A., Grudsky, S., Kozak, A., Silbermann, B.: Convergence speed estimates for the norms of the inverses of large truncated Toeplitz matrices. Calcolo 36, 103-122 (1999) · Zbl 0964.15027 · doi:10.1007/s100920050025
[26] Böttcher, A., Grudsky, S., Kozak, A., Silbermann, B.: Norms of large Toeplitz band matrices. SIAM J. Matrix Anal. Appl. 21, 547-561 (1999) · Zbl 0945.15020 · doi:10.1137/S0895479898343012
[27] Böttcher, A., Grudsky, S., Maksimenko, E.A.: Pushing the envelope of the test functions in the Szegő and Avram-Parter theorems. Linear Algebra Appl. 429, 346-366 (2008) · Zbl 1153.15012 · doi:10.1016/j.laa.2008.02.031
[28] Böttcher, A., Grudsky, S., Maksimenko, E.A.: The Szegő and Avram-Parter theorems for general test functions. C. R. Math. Acad. Sci. Paris 346, 749-752 (2008) · Zbl 1145.15007 · doi:10.1016/j.crma.2008.06.002
[29] Böttcher, A., Grudsky, S., Maksimenko, E.A.: On the asymptotics of all the eigenvalues of Hermitian Toeplitz band matrices. Dokl. Akad. Nauk 428, 153-156 (2009) (Russian); Engl. translation in Dokl. Math. 80, 662-664 (2009) · Zbl 1181.15010
[30] Böttcher, A., Grudsky, S., Maksimenko, E.A.: Inside the eigenvalues of certain Hermitian Toeplitz band matrices. J. Comput. Appl. Math. 233, 2245-2264 (2010) · Zbl 1195.15009 · doi:10.1016/j.cam.2009.10.010
[31] Böttcher, A., Grudsky, S., Schwartz, M.: Some problems concerning the test functions in the Szegő and Avram-Parter theorems. Oper. Theory Adv. Appl. 187, 81-93 (2009) · Zbl 1168.47023
[32] Böttcher, A., Grudsky, S., Silbermann, B.: Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices. N. Y. J. Math. 3, 1-31 (1997) · Zbl 0887.47025
[33] Böttcher, A., Grudsky, S., Spitkovsky, I.: The spectrum is discontinuous on the manifold of Toeplitz operators. Arch. Math. (Basel) 75, 46-52 (2000) · Zbl 0966.47015 · doi:10.1007/s000130050472
[34] Böttcher, A., Grudsky, S., Spitkovsky, I.: Matrix functions with arbitrarily prescribed left and right partial indices. Integr. Equ. Oper. Theory 36, 71-91 (2000) · Zbl 0960.47012 · doi:10.1007/BF01236287
[35] Böttcher, A., Grudsky, S., Spitkovsky, I.: On the Fredholm indices of associated systems of Wiener-Hopf equations. J. Integral Equ. Appl. 12, 1-29 (2000) · Zbl 0991.47012 · doi:10.1216/jiea/1020282131
[36] Böttcher, A., Grudsky, S., Spitkovsky, I.: Toeplitz operators with frequency modulated semi-almost periodic symbols. J. Fourier Anal. Appl. 7, 523-535 (2001) · Zbl 0996.47035 · doi:10.1007/BF02511224
[37] Böttcher, A., Grudsky, S., Unterberger, J.: Asymptotic pseudomodes of Toeplitz matrices. Oper. Matrices 2, 525-541 (2008) · Zbl 1172.47021 · doi:10.7153/oam-02-33
[38] Böttcher, A., Hurák, Z., Šebek, M.: Minimum distance to the range of a banded lower triangular Toeplitz operator in \[\ell^1\] ℓ1 and application in \[\ell^1\] ℓ1-optimal control. SIAM J. Control Optim. 45, 107-122 (2006) · Zbl 1116.47025 · doi:10.1137/S0363012903437940
[39] Böttcher, A., Karlovich, Yu.I., Spitkovsky, I.M.: Convolution Operators and Factorization of Almost Periodic Matrix Functions. Birkhäuser Verlag, Basel (2002) · Zbl 1011.47001
[40] Böttcher, A., Potts, D.: Probability against condition number and sampling of multivariate trigonometric random polynomials. Electron. Trans. Numer. Anal. 26, 178-189 (2007) · Zbl 1171.65389
[41] Böttcher, A., Potts, D., Wenzel, D.: A probability argument in favor of ignoring small singular values. Oper. Matrices 1, 31-43 (2007) · Zbl 1120.65055 · doi:10.7153/oam-01-02
[42] Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators, 2nd edn 2006. Springer-Verlag, Berlin (1990) · Zbl 0732.47029
[43] Böttcher, A., Wenzel, D.: How big can the commutator of two matrices be and how big is it typically? Linear Algebra Appl. 403, 216-228 (2005) · Zbl 1077.15020 · doi:10.1016/j.laa.2005.02.012
[44] Carrada-Herrera, R., Grudsky, S., Palomino-Jiménez, C., Porter, R.M.: Asymptotics of European double-barrier option with compound Poisson component. Commun. Math. Anal. 14, 40-66 (2013) · Zbl 1267.91066
[45] Dai, H., Geary, Z., Kadanoff, L.P.: Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. J. Stat. Mech.-Theory Exp. 2009(5), 1-25, Art ID P05012 (2009) · Zbl 1456.15005
[46] Deift, P., Its, A., Krasovsky, I.: Eigenvalues of Toeplitz matrices in the bulk of the spectrum. Bull. Inst. Math. Acad. Sin. (N.S.) 7, 437-461 (2012) · Zbl 1292.15029
[47] Dybin, V.B., Grudsky, S.: Introduction to the Theory of Toeplitz Operators with Infinite Index. Birkhäuser Verlag, Basel (2002) · Zbl 1030.47001 · doi:10.1007/978-3-0348-8213-2
[48] Gohberg, I., Feldman, I.A.: Convolution Equations and Projection Methods for Their Solution. Amer. Math. Soc. Transl. Math. Monographs, vol. 41. Providence (1974) · Zbl 1320.15028
[49] Grudsky, S.: Singular integral equations and the Riemann boundary value problem with infinite index in the space \[L_p(\Gamma ,\varrho )\] Lp(Γ,ϱ). Izv. Akad. Nauk SSSR 49, 55-80 (1985) (Russian) · Zbl 1302.65086
[50] Grudsky, S.: Singular integral operators with infinite index and Blaschke products. Math. Nachr. 129, 313-331 (1986) (Russian) · Zbl 0612.30039
[51] Grudsky, S., Rybkin, A.: Soliton theory and Hankel operators. SIAM J. Math. Anal. 47, 2283-2323 (2015) · Zbl 1332.35325 · doi:10.1137/151004926
[52] Grudsky, S., Shargorodsky, E.: Applications of Blaschke products to the spectral theory of Toeplitz operators. Fields Inst. Commun. 65, 1-30 (2013) · Zbl 1277.47042 · doi:10.1007/978-1-4614-5341-3_1
[53] Landau, H.J.: The notion of approximate eigenvalues applied to an integral equation of laser theory. Q. Appl. Math. 35, 165-172 (1977/78) · Zbl 0366.45003
[54] Landau, H.J., Widom, H.: Eigenvalue distribution of time and frequency limiting. J. Math. Anal. Appl. 77, 469-481 (1980) · Zbl 0471.47029 · doi:10.1016/0022-247X(80)90241-3
[55] Reichel, L., Trefethen, L.N.: Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. Linear Algebra Appl. 162/164, 153-185 (1992) · Zbl 0748.15010
[56] Serra Capizzano, S.: Test functions, growth conditions and Toeplitz matrices. Rend. Circ. Mat. Palermo, Ser. II 68(Suppl.), 791-795 (2002) · Zbl 1025.15014
[57] Shargorodsky, E.: On the level sets of the resolvent norm of a linear operator. Bull. Lond. Math. Soc. 40, 493-504 (2008) · Zbl 1147.47007 · doi:10.1112/blms/bdn038
[58] Treil, S.: Invertibility of Toelitz operators does not imply applicability of the finite section method. Dokl. Akad. Nauk SSSR 292, 563-567 (1987) (Russian) · Zbl 0887.47026
[59] Trench, W.F.: An elementary view of Weyl’s theory of equal distribution. Am. Math. Mon. 119, 852-861 (2012) · Zbl 1275.11105 · doi:10.4169/amer.math.monthly.119.10.852
[60] Vainshtein, L.A.: Open resonance for lasers. Sov. Phys. JETP 40, 709-719 (1963)
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