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Note on spectra of non-selfadjoint operators over dynamical systems. (English) Zbl 07091610
Summary: We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

47A10 Spectrum, resolvent
47A35 Ergodic theory of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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[1] Avron, J. and Simon, B., Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys.82(1) (1981/82), 101-120. · Zbl 0484.35069
[2] Avron, J. and Simon, B., Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J.50(1) (1983), 369-391. · Zbl 0544.35030
[3] Bellissard, J., Iochum, B., Scoppola, E. and Testard, D., Spectral properties of one-dimensional quasi-crystals, Comm. Math. Phys.125 (1989), 527-543. · Zbl 0825.58010
[4] Böttcher, A., Embree, M. and Sokolov, V. I., The spectra of large Toeplitz band matrices with a randomly perturbed entry, Math. Comp.72 (2003), 1329-1348. · Zbl 1022.47019
[5] Böttcher, A., Grudsky, S. and Spitkovsky, I., On the essential spectrum of Toeplitz operators with semi-almost periodic symbols, In Singular integral operators, factorization and applications, Operatory Theory: Advances and Applications, Volume 142, pp. 59-77 (Birkhäuser, Basel, 2003).
[6] Carmona, R. and Lacroix, J., Spectral theory of random Schrödinger operators (Birkhäuser, Boston, MA, 1990). · Zbl 0717.60074
[7] Chandler-Wilde, S. N., Chonchaiya, R. and Lindner, M., On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, Oper. Matrices7 (2013), 739-775. · Zbl 1303.47054
[8] Chandler-Wilde, S. N. and Lindner, M., Sufficiency of Favard’s condition for a class of band-dominated operators on the axis. J. Funct. Anal.254 (2008), 1146-1159. · Zbl 1136.47009
[9] Chandler-Wilde, S. N. and Lindner, M., Limit operators, collective compactness, and the spectral theory of infinite matrices, Memoirs of the American Mathematical Society, Volume 210 (American Mathematical Society, Providence, RI, 2011). · Zbl 1219.47001
[10] Corduneanu, C., Almost periodic functions, (1989), Chelsea Publishers: Chelsea Publishers, New York · Zbl 0672.42008
[11] Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B., Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics (Springer, Berlin, 1987). · Zbl 0619.47005
[12] Dahmen, H. A., Nelson, D. R. and Shnerb, N. M., Population dynamics and non-hermitian localization, In Statistical mechanics of biocomplexity, Lecture Notes in Physics, Volume 527, pp. 124-151 (Springer, 1999). · Zbl 0958.92025
[13] Damanik, D., Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in mathematical quasicrystals, CRM Monograph Series, Volume 13, pp. 277-305 (American Mathematical Society, Providence, RI, 2000). · Zbl 0989.81025
[14] Damanik, D., The Spectrum of the Almost Mathieu Operator, (arxiv.org/abs/0908.1093; 2009). · Zbl 1169.82009
[15] Damanik, D., Embree, M. and Gorodetski, A., Spectral properties of Schrödinger operators arising in the study of quasicrystals’, In Mathematics of aperiodic order (ed. Kellendonk, J., Lenz, D. and Savinien, J.) Progress in Mathematics, Volume 309, pp. 307-370 (Birkhäuser, 2015). · Zbl 1378.81031
[16] Davies, E. B., Spectral theory of pseudo-ergodic operators, Commun. Math. Phys., 216, 687-704, (2001) · Zbl 1044.47031
[17] Favard, J., Sur les equations differentielles lineaires a coefficients presque-periodiques, Acta Math., 51, 31-81, (1927) · JFM 53.0409.02
[18] Feinberg, J. and Zee, A., Spectral curves of non-Hermitean Hamiltonians, Nucl. Phys. B552 (1999), 599-623. · Zbl 0944.82017
[19] Goldsheid, I. Ya. and Khoruzhenko, B. A., Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model, Comm. Math. Phys.238 (2003), 505-524. · Zbl 1032.82004
[20] Hatano, N. and Nelson, D. R., Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett.77 (1996), 570-573.
[21] Jitomirskaya, S. Ya., Almost everything about the almost Mathieu operator. II, in XIth International Congress of Mathematical Physics (Paris, 1994), pp. 373-382 (International Press, Cambridge, MA, 1995). · Zbl 1052.82539
[22] Johnson, R., Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations, 61, 54-78, (1986) · Zbl 0608.34056
[23] Kato, T., Perturbation theory for linear operators, (1980), Springer
[24] Kurbatov, V. G., Functional differential operators and equations, (1999), Kluwer Academic · Zbl 0926.34053
[25] Lange, B. V. and Rabinovich, V. S., On the Noether property of multidimensional discrete convolutions, Mat. Zametki37(3) (1985), 407-421 (in Russian); English translation: Math. Notes 37 (1985), 228-237.
[26] Last, Y., Almost everything about the Almost Mathieu Operator. I., in XIth International Congress of Mathematical Physics (Paris, 1994), pp. 366-372 (International Press, Cambridge, MA, 1995). · Zbl 1052.82541
[27] Last, Y. and Simon, B., The essential spectrum of Schrödinger, Jacobi and CMV operators, J. Anal. Math.98 (2006), 183-220. · Zbl 1145.34052
[28] Lenz, D., Singular continuous spectrum for certain quasicrystal Schrödinger operators, In Complex analysis and dynamical systems, Contemporary Mathematics, Volume 364, pp. 169-180 (American Mathematical Society, Providence, RI, 2004).
[29] Lenz, D., Random operators and crossed products, Math. Phys. Anal. Geom., 2, 197-220, (1999) · Zbl 0965.47024
[30] Lindner, M., Limit operators and applications on the space of essentially bounded functions. Dissertation Thesis, TU Chemnitz, 2003, http://nbn-resolving.de/urn:nbn:de:swb: ch1-200301569.
[31] Lindner, M., Infinite matrices and their finite sections: an introduction to the limit operator method, (2006), Birkhäuser · Zbl 1107.47001
[32] 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
[33] Lindner, M. and Roch, S., Finite sections of random Jacobi operators, SIAM J. Numer. Anal.50 (2012), 287-306. · Zbl 1241.65015
[34] Lindner, M. and Seidel, M., An affirmative answer to a core issue on limit operators, J. Funct. Anal.267 (2014), 901-917. · Zbl 1292.47020
[35] Martinez, C., Spectral estimates for the one-dimensional non-self-adjoint Anderson model, J. Operator Theory, 56, 59-88, (2006) · Zbl 1134.47304
[36] Nelson, D. R. and Shnerb, N. M., Non-Hermitian localization and population biology, Phys. Rev. E58, (1998), 1383-1403.
[37] Pastur, L. A. and Figotin, A., Spectra of random and almost-periodic operators (Springer, Berlin, 1992). · Zbl 0752.47002
[38] Rabinovich, V. S., Roch, S. and Silbermann, B., Fredholm theory and finite section method for band-dominated operators, Integral Equations Operator Theory30 (1998), 452-495. · Zbl 0909.47023
[39] Rabinovich, V. S., Roch, S. and Silbermann, B., Limit operators and their applications in operator theory (Birkhäuser, 2004). · Zbl 1077.47002
[40] Reed, M. and Simon, B., Methods of modern mathematical physics I: functional analysis (Academic Press, 1980). · Zbl 0459.46001
[41] Seidel, M., Fredholm theory for band-dominated and related operators: A survey, Linear Algebra Appl., 445, 373-394, (2014) · Zbl 1301.47016
[42] Seifert, C., Constancy of spectra of equivariant (non-selfadjoint) operators over minimal dynamical systems, Proc. Appl. Math. Mech., 15, 697-698, (2015)
[43] Shubin, M. A., Almost periodic functions and partial differential operators, Russian Math. Surv., 33, 1-52, (1978) · Zbl 0408.47039
[44] Stollmann, P., Caught by disorder, bound states in random media, Progress in Mathematical Physics, Volume 20 (Birkhäuser, Boston, 2001). · Zbl 0983.82016
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