Coburn’s lemma and the finite section method for random Jacobi operators. (English) Zbl 1328.65128
Summary: We study the spectra and pseudospectra of semi-infinite and bi-infinite tridiagonal random matrices and their finite principal submatrices, in the case where each of the three diagonals varies over a separate compact set, say \(U, V, W \subset \mathbb{C}\). Such matrices are sometimes termed stochastic Toeplitz matrices \(A_+\) in the semi-infinite case and stochastic Laurent matrices \(A\) in the bi-infinite case. Their spectra, \(\Sigma = \operatorname{spec} A\) and \(\Sigma_+ = \operatorname{spec} A_+\), are independent of \(A\) and \(A_+\) as long as \(A\) and \(A_+\) are pseudoergodic (in the sense of E. B. Davies [Commun. Math. Phys. 216, No. 3, 687–704 (2001; Zbl 1044.47031)]), which holds almost surely in the random case. This was shown in [loc. cit.] for \(A\); that the same holds for \(A_+\) is one main result of this paper. Although the computation of \(\Sigma\) and \(\Sigma_+\) in terms of \(U\), \(V\) and \(W\) is intrinsically difficult, we give upper and lower spectral bounds, and we explicitly compute a set \(G\) that fills the gap between \(\Sigma\) and \(\Sigma_+\) in the sense that \(\Sigma \cup G = \Sigma_+\). We also show that the invertibility of one (and hence all) operators \(A_+\) implies the invertibility – and uniform boundedness of the inverses – of all finite tridiagonal square matrices with diagonals varying over \(U\), \(V\) and \(W\). This implies that the so-called finite section method for the approximate solution of a system \(A_+ x = b\) is applicable as soon as \(A_+\) is invertible, and that the finite section method for estimating the spectrum of \(A_+\) does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. Indeed, one of our main tools is a new stochastic version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part of the paper we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over \(U\), \(V\) and \(W\). This, in particular, allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices.
MSC:
| 65J10 | Numerical solutions to equations with linear operators |
| 47A10 | Spectrum, resolvent |
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 47B80 | Random linear operators |
Keywords:
finite section method; random operator; Coburn lemma; Jacobi operator; spectral pollution; pseudoergodicCitations:
Zbl 1044.47031Software:
EigtoolReferences:
| [1] | Anderson, P. W., Absence of diffusion in certain random lattices, Phys. Rev., 109, 1492-1505 (1958) |
| [2] | Anderson, P. W., Localized magnetic states in metals, Phys. Rev., 124, 41-53 (1961) |
| [3] | Böttcher, A., Pseudospectra and singular values of large convolution operators, J. Integral Equations Appl., 6, 267-301 (1994) · Zbl 0819.45002 |
| [4] | Böttcher, A.; Embree, M.; Lindner, M., Spectral approximation of banded Laurent matrices with localized random perturbations, Integral Equations Operator Theory, 42, 142-165 (2002) · Zbl 0995.47021 |
| [5] | Böttcher, A.; Embree, M.; Sokolov, V. I., Infinite Toeplitz and Laurent matrices with localized impurities, Linear Algebra Appl., 343/344, 101-118 (2002) · Zbl 0995.15013 |
| [6] | Böttcher, A.; Embree, M.; Sokolov, V. I., On large Toeplitz band matrices with an uncertain block, Linear Algebra Appl., 366, 87-97 (2003) · Zbl 1021.15008 |
| [7] | Böttcher, A.; Embree, M.; Sokolov, V. I., The spectra of large Toeplitz band matrices with a randomly perturbed entry, Math. Comp., 72, 1329-1348 (2003) · Zbl 1022.47019 |
| [8] | Böttcher, A.; Grudsky, S. M., Spectral Properties of Banded Toeplitz Matrices (2005), SIAM: SIAM Philadelphia · Zbl 1089.47001 |
| [9] | Böttcher, A.; Grudsky, S. M.; Silbermann, B., Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices, New York J. Math., 3, 1-31 (1997) · Zbl 0887.47025 |
| [10] | Böttcher, A.; Silbermann, B., Introduction to Large Truncated Toeplitz Matrices (1999), Springer: Springer Berlin, Heidelberg · Zbl 0916.15012 |
| [11] | Chandler-Wilde, S. N.; Chonchaiya, R.; Lindner, M., Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator, Oper. Matrices, 5, 633-648 (2011) · Zbl 1301.47050 |
| [12] | Chandler-Wilde, S. N.; Chonchaiya, R.; Lindner, M., On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, Oper. Matrices, 7, 739-775 (2013) · Zbl 1303.47054 |
| [13] | Chandler-Wilde, S. N.; Davies, E. B., Spectrum of a Feinberg-Zee random hopping matrix, J. Spectr. Theory, 2, 147-179 (2012) · Zbl 1262.15007 |
| [14] | Chandler-Wilde, S. N.; Hagger, R., On symmetries of the Feinberg-Zee random hopping matrix, submitted for publication · Zbl 1367.47048 |
| [15] | Chandler-Wilde, S. N.; Lindner, M., Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices, Mem. Amer. Math. Soc., vol. 210 (989) (2011) · Zbl 1219.47001 |
| [16] | Cicuta, G. M.; Contedini, M.; Molinari, L., Non-Hermitian tridiagonal random matrices and returns to the origin of a random walk, J. Stat. Phys., 98, 685-699 (2000) · Zbl 0991.82015 |
| [17] | Cicuta, G. M.; Contedini, M.; Molinari, L., Enumeration of simple random walks and tridiagonal matrices, J. Phys. A: Math. Gen., 35, 1125-1146 (2002) · Zbl 0998.05003 |
| [18] | Coburn, L. A., Weyl’s theorem for non-normal operators, Michigan Math. J., 13, 285-288 (1966) · Zbl 0173.42904 |
| [19] | Davies, E. B., Spectral properties of non-self-adjoint matrices and operators, Proc. Roy. Soc. A., 457, 191-206 (2001) · Zbl 1014.47002 |
| [20] | Davies, E. B., Spectral theory of pseudo-ergodic operators, Comm. Math. Phys., 216, 687-704 (2001) · Zbl 1044.47031 |
| [21] | Feinberg, J.; Zee, A., Non-Hermitean localization and de-localization, Phys. Rev. E, 59, 6433-6443 (1999) |
| [22] | Feinberg, J.; Zee, A., Spectral curves of non-Hermitean Hamiltonians, Nuclear Phys. B, 552, 599-623 (1999) · Zbl 0944.82017 |
| [23] | Gohberg, I., On the number of solutions to homogeneous singular equations with continuous coefficients, Dokl. Akad. Nauk SSSR, 112, 327-330 (1958), (in Russian) · Zbl 0105.08401 |
| [24] | Gohberg, I.; Feldman, I. A., Convolution Equations and Projection Methods for Their Solution, Transl. Math. Monogr., vol. 41 (1974), Amer. Math. Soc. · Zbl 0278.45008 |
| [25] | Goldsheid, I.; Khoruzhenko, B., Eigenvalue curves of asymmetric tridiagonal random matrices, Electron. J. Probab., 5, 1-28 (2000) · Zbl 0983.82006 |
| [26] | Hagen, R.; Roch, S.; Silbermann, B., \(C^\ast \)-Algebras and Numerical Analysis (2001), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York, Basel · Zbl 0964.65055 |
| [27] | Hagger, R., On the spectrum and numerical range of tridiagonal random operators, J. Spectr. Theory (2015), in press, preprint at |
| [28] | Hagger, R., The eigenvalues of tridiagonal sign matrices are dense in the spectra of periodic tridiagonal sign operators, J. Funct. Anal., 269, 1563-1570 (2015) · Zbl 1329.15062 |
| [29] | Hagger, R., Symmetries of the Feinberg-Zee random hopping matrix, Random Matrices Theory Appl. (2015), in press, preprint at · Zbl 1408.47010 |
| [30] | Hagger, R.; Lindner, M.; Seidel, M., Essential pseudospectra and essential norms of band-dominated operators, submitted for publication, preprint at · Zbl 1336.47036 |
| [31] | Hatano, N.; Nelson, D. R., Vortex pinning and non-Hermitian quantum mechanics, Phys. Rev. B, 56, 8651-8673 (1997) |
| [32] | Hausdorff, F., Set Theory (1962), Chelsea · Zbl 0060.12401 |
| [33] | Heinig, G.; Hellinger, F., The finite section method for Moore-Penrose inversion of Toeplitz operators, Integral Equations Operator Theory, 19, 419-446 (1994) · Zbl 0817.47036 |
| [34] | Kurbatov, V. G., Functional Differential Operators and Equations (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0926.34053 |
| [35] | Lindner, M., Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method, Front. Math. (2006), Birkhäuser · Zbl 1107.47001 |
| [36] | Lindner, M., Fredholmness and index of operators in the Wiener algebra are independent of the underlying space, Oper. Matrices, 2, 297-306 (2008) · Zbl 1153.47008 |
| [37] | Lindner, M., A note on the spectrum of bi-infinite bi-diagonal random matrices, J. Anal. Appl., 7, 269-278 (2009) · Zbl 1181.47042 |
| [38] | Lindner, M., The finite section method and stable subsequences, Appl. Numer. Math., 60, 501-512 (2010) · Zbl 1197.47029 |
| [39] | Lindner, M.; Roch, S., Finite sections of random Jacobi operators, SIAM J. Numer. Anal., 50, 287-306 (2012) · Zbl 1241.65015 |
| [40] | Lindner, M.; Seidel, M., An affirmative answer to a core issue on limit operators, J. Funct. Anal., 267, 901-917 (2014) · Zbl 1292.47020 |
| [41] | Lindner, M.; Strang, G., The main diagonal of a permutation matrix, Linear Algebra Appl., 439, 524-537 (2013) · Zbl 1283.15092 |
| [42] | Marletta, M.; Naboko, S., The finite section method for dissipative operators, Mathematika, 60, 415-443 (2014) · Zbl 1297.47031 |
| [43] | Martínez Adame, C., On the spectral behaviour of a non-self-adjoint operator with complex potential, Math. Phys. Anal. Geom., 10, 81-95 (2007) · Zbl 1181.47026 |
| [44] | Nelson, D. R.; Shnerb, N. M., Non-Hermitian localization and population biology, Phys. Rev. E, 58, 1383-1403 (1998) |
| [45] | Prössdorf, S.; Silbermann, B., Numerical Analysis for Integral and Related Operator Equations (1991), Akademie-Verlag: Akademie-Verlag Berlin: Birkhäuser Verlag: Akademie-Verlag: Akademie-Verlag Berlin: Birkhäuser Verlag Basel, Boston, Berlin · Zbl 0763.65102 |
| [46] | Rabinovich, V. S.; Roch, S.; Roe, J., Fredholm indices of band-dominated operators, Integral Equations Operator Theory, 49, 2, 221-238 (2004) · Zbl 1068.47016 |
| [47] | Rabinovich, V. S.; Roch, S.; Silbermann, B., Fredholm theory and finite section method for band-dominated operators, Integral Equations Operator Theory, 30, 452-495 (1998) · Zbl 0909.47023 |
| [48] | Rabinovich, V. S.; Roch, S.; Silbermann, B., Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections, Integral Equations Operator Theory, 40, 3, 342-381 (2001) · Zbl 1032.47051 |
| [49] | Rabinovich, V. S.; Roch, S.; Silbermann, B., Limit Operators and Their Applications in Operator Theory (2004), Birkhäuser · Zbl 1077.47002 |
| [50] | Rabinovich, V. S.; Roch, S.; Silbermann, B., On finite sections of band-dominated operators, Oper. Theory Adv. Appl., 181, 385-391 (2008) · Zbl 1167.47015 |
| [51] | Reichel, L.; Trefethen, L. N., Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Linear Algebra Appl., 162-164, 153-185 (1992) · Zbl 0748.15010 |
| [52] | Roch, S., Band-dominated operators on \(\ell^p\)-spaces: Fredholm indices and finite sections, Acta Sci. Math., 70, 3-4, 783-797 (2004) · Zbl 1087.47013 |
| [53] | Roch, S.; Silbermann, B., Non-strongly converging approximation methods, Demonstratio Math., 22, 3, 651-676 (1989) · Zbl 0696.65055 |
| [54] | Seidel, M., On some Banach algebra techniques in operator theory (2011), TU Chemnitz, PhD Thesis |
| [55] | Seidel, M.; Silbermann, B., Finite sections of band-dominated operators - norms, condition numbers and pseudospectra, Oper. Theory Adv. Appl., 228, 275-390 (2013) · Zbl 1280.47022 |
| [56] | 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 |
| [57] | Trefethen, L. N.; Contedini, M.; Embree, M., Spectra, pseudospectra, and localization for random bidiagonal matrices, Comm. Pure Appl. Math., 54, 595-623 (2001) · Zbl 1025.15034 |
| [58] | Trefethen, L. N.; Embree, M., Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators (2005), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1085.15009 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
