On the spectrum of operator families on discrete groups over minimal dynamical systems

Abstract

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in \(\ell ^p ({\mathbb {Z}})\). Here, we generalize this to a large class of bounded linear operator families on Banach-space valued \(\ell ^p\)-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

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References

  1. 1.

    Beckus, S., Lenz, D., Lindner, M., Seifert, C.: Note on spectra of non-selfadjoint operators over dynamical systems. Proc. Edinb. Math. Soc (to appear). arXiv-Preprint: arXiv:1412.5926

  2. 2.

    Béguin, C., Valette, A., Zuk, a: On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper’s operator. J. Geom. Phys. 21(4), 337–356 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Bellissard, J., Iochum, B., Scoppola, E., Testard, D.: Spectral properties of one-dimensional quasi-crystals. Comm. Math. Phys. 125, 527–543 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Chandler-Wilde, S.N., Lindner, M.: Sufficiency of Favard’s condition for a class of band-dominated operators on the axis. J. Funct. Anal. 254, 1146–1159 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Chandler-Wilde, S.N., Lindner, M.: Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices, vol. 210. American Mathematical Society, Providence, Rhode Island (2011)

  6. 6.

    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics. Spinger, Berlin (1987)

    Google Scholar 

  7. 7.

    Davies, E.B.: Spectral theory of pseudo-ergodic operators. Commun. Math. Phys. 216, 687–704 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Johnson, R.: Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients. J. Differ. Equ. 61, 54–78 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)

    Google Scholar 

  10. 10.

    Lange, B.V., Rabinovich, V.S.: On the Noether property of multidimensional discrete convolutions. Mater. Zametki 37(3), 407–421 (1985). (Russian, English translation: Math. Notes 37 (1985), 228–237)

  11. 11.

    Lenz, D.: Random operators and crossed products. Math. Phys. Anal. Geom. 2, 197–220 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Lenz, D., Seifert, C., Stollmann, P.: Zero measure Cantor spectra for continuum one-dimensional quasicrystals. J. Differ. Equ. 256(6), 1905–1926 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Lindner, M.: Infinite Matrices and Their Finite Sections: An Introduction to the Limit Operator Method. Birkhäuser, Basel (2006)

    Google Scholar 

  14. 14.

    Lindner, M., Seidel, M.: An affirmative answer to a core issue on limit operators. J. Funct. Anal. 267, 901–917 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Muhamadiev, E.M.: On normal solvability and Noether property of elliptic operators in spaces of functions on \({\mathbb{R}}^n\), Part I. Zapiski nauchnih sem. LOMI 110, 120–140 (1981). (Russian)

    Google Scholar 

  16. 16.

    Prössdorf, S., Silbermann, B.: Numerical Analysis for Integral and Related Operator Equations. Akademie-Verlag/Birkhäuser Verlag, Berlin/Basel (1991)

  17. 17.

    Rabinovich, V.S., Roch, S., Roe, J.: Fredholm indices of band-dominated operators. Integr. Equ. Oper. Theory 49(2), 221–238 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Rabinovich, V.S., Roch, S., Silbermann, B.: Fredholm theory and finite section method for band-dominated operators. Integr. Equ. Oper. Theory 30, 452–495 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and Their Applications in Operator Theory. Birkhäuser, Basel (2004)

  20. 20.

    Roch, S., Silbermann, B.: Non-strongly converging approximation methods. Demonstr. Math. 22(3), 651–676 (1989)

    MATH  Google Scholar 

  21. 21.

    Roe, J.: Band-dominated Fredholm operators on discrete groups. Integr. Equ. Oper. Theory 51, 411–416 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Seidel, M., Silbermann, B.: Banach algebras of operator sequences. Oper. Matr. 6(3), 385–432 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Seidel, M.: Fredholm theory for band-dominated and related operators: a survey. Linear Algebra Appl. 445, 373–394 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Seifert, C.: Constancy of spectra of equivariant (non-selfadjoint) operators over minimal dynamical systems. Proc. Appl. Math. Mech. 15, 697–698 (2015)

    Article  Google Scholar 

  25. 25.

    Simonenko, B.: On multidimensional discrete convolutions. Mat. Issled. 3(1), 108–127 (1968). (in Russian)

    MathSciNet  Google Scholar 

  26. 26.

    Špakula, J., Willett, R.: A metric approach to limit operators. Trans. AMS (to appear). arXiv:1408.0678

  27. 27.

    Stollmann, P.: Caught by Disorder, Bound States in Random Media. Progress in Mathematical Physics, vol. 20. Birkhäuser, Boston (2001)

    Google Scholar 

  28. 28.

    Trefethen, L.N., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)

    Google Scholar 

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Correspondence to Christian Seifert.

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Beckus, S., Lenz, D., Lindner, M. et al. On the spectrum of operator families on discrete groups over minimal dynamical systems. Math. Z. 287, 993–1007 (2017). https://doi.org/10.1007/s00209-017-1856-5

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Keywords

  • Minimal dynamical system
  • Pseudo-ergodicity
  • Spectrum
  • \({\mathcal {P}}\)-Theory

Mathematics Subject Classification

  • 47A10
  • 47A35
  • 47B37