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On the Markov-Dyck shifts of vertex type
Dichotomy spectra of triangular equations
1. | Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt |
  Based on operator-theoretical tools, this paper provides various sufficient and verifiable criteria to obtain a corresponding diagonal significance for finite-dimensional difference equations in the following sense: Spectral and continuity properties of the diagonal elements extend to the whole triangular system.
References:
[1] |
Y. Abramovich and C. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics, 50. American Mathematical Society, Providence, RI, 2002.
doi: 10.1090/gsm/050. |
[2] |
P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer, Dordrecht, 2004. |
[3] |
P. Aiena, Semi-Fredholm Operators, Perturbation Theory and Localized SVEP, XX Escuela Venezolana de Matemáticas, Caracas, Venezuela, 2007. |
[4] |
P. Aiena, T. Miller and M. Neumann, On a localized single-valued extension property, Math. Proc. R. Ir. Acad., 104A (2004), 17-34.
doi: 10.3318/PRIA.2004.104.1.17. |
[5] |
B. Aulbach, N. Van Minh and P. Zabreiko, The concept of spectral dichotomy for linear difference equations, J. Math. Anal. Appl., 185 (1994), 275-287.
doi: 10.1006/jmaa.1994.1248. |
[6] |
B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, In: López-Fenner J., e.a. (ed.) Proceedings of the 5th Intern. Conference of Difference Eqns. and Application (Temuco, Chile, 2000), 45-55. Taylor & Francis, London, 2002. |
[7] |
M. Barraa and M. Boumazgour, A note on the spectrum of an upper triangular operator matrix, Proc. Am. Math. Soc., 131 (2006), 3083-3088.
doi: 10.1090/S0002-9939-03-06862-X. |
[8] |
L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics 148, AMS, Providence, RI, 2013. |
[9] |
F. Battelli and K. Palmer, Criteria for exponential dichotomy for triangular systems, Journal of Mathematical Analysis and Applications, 428 (2015), 525-543.
doi: 10.1016/j.jmaa.2015.03.029. |
[10] |
A. Ben-Artzi and I. Gohberg, Dichotomies of perturbed time varying systems and the power method, Indiana Univ. Math. J., 42 (1993), 699-720.
doi: 10.1512/iumj.1993.42.42031. |
[11] |
A. Bourhim and C. Chidume, The single-valued extension property for bilateral operator weighted shifts, Proc. Am. Math. Soc., 133 (2005), 485-491.
doi: 10.1090/S0002-9939-04-07535-5. |
[12] |
J. Cushing, S. LeVarge, N. Chitnis and S. Henson, Some discrete competition models and the competitive exclusion principle, J. Difference Equ. Appl., 10 (2004), 1139-1151.
doi: 10.1080/10236190410001652739. |
[13] |
L. Dieci and E. van Vleck, Lyapunov and other spectra: A survey, In Collected Lectures on the Preservation of Stability under Discretization, SIAM, (2002), 197-218. |
[14] |
L. Dieci, C. Elia and E. van Vleck, Exponential dichotomy on the real line: SVD and QR methods, J. Differ. Equations, 248 (2010), 287-308.
doi: 10.1016/j.jde.2009.07.004. |
[15] |
S. Djordjević and Y. Han, A note on Weyl's theorem for operator matrices, Proc. Am. Math. Soc., 131 (2003), 2543-2547.
doi: 10.1090/S0002-9939-02-06808-9. |
[16] |
B. Duggal, Upper triangular operators with SVEP: Spectral properties, Filomat, 21 (2007), 25-37.
doi: 10.2298/FIL0701025D. |
[17] |
H. Elbjaoui and E. Zerouali, Local spectral theory for $2\times 2$ operator matrices, Int. J. Math. Math. Sci., 42 (2003), 2667-2672.
doi: 10.1155/S0161171203012043. |
[18] |
J. Han, H. Lee and W. Lee, Invertible completitions of $2\times 2$ upper triangular operator matrices, Proc. Am. Math. Soc., 128 (2000), 119-123.
doi: 10.1090/S0002-9939-99-04965-5. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840. Springer, Berlin etc., 1981. |
[20] |
D. Hinrichsen and A. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness, Texts in Applied Mathematics, 48. Springer, Heidelberg etc., 2005.
doi: 10.1007/b137541. |
[21] |
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013. |
[22] |
D. Hong-Ke and P. Jin, Perturbations of spectrums of $2\times 2$ operator matrices, Proc. Am. Math. Soc., 121 (1994), 761-766.
doi: 10.2307/2160273. |
[23] |
T. Hüls, Numerical computation of dichotomy rates and projectors in discrete time, Discrete Contin. Dyn. Syst. (Series B), 12 (2009), 109-131.
doi: 10.3934/dcdsb.2009.12.109. |
[24] |
T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM Journal on Numerical Analysis, 48 (2010), 2043-2064.
doi: 10.1137/090754509. |
[25] |
T. Hüls and C. Pötzsche, Qualitative analysis of a nonautonomous Beverton-Holt Ricker model, SIAM Journal of Applied Dynamical Systems, 13 (2014), 1442-1488.
doi: 10.1137/140955434. |
[26] |
C. Jiang and Z. Wang, Structure of Hilbert Space Operators, World Scientific, New Jersey, 2006 |
[27] |
R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33.
doi: 10.1137/0518001. |
[28] |
K. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. AMS, Providence, RI, 2011.
doi: 10.1090/surv/176. |
[29] |
K. Laursen and M. Neumann, An Introduction to Local Spectral Theory, Oxford Science Publications, Oxford, 2000. |
[30] |
W. Lee, Weyl spectra of operator matrices, Proc. Am. Math. Soc., 129 (2000), 131-138.
doi: 10.1090/S0002-9939-00-05846-9. |
[31] |
J. Li, The single valued extension property for operator weighted shifts, Northeast. Math. J., 10 (1994), 99-103. |
[32] |
J. Li, Y. Ji and S. Sun, The essential spectrum and Banach reducibility of operator weighted shifts, Acta Mathematica Sinica, 17 (2001), 413-424.
doi: 10.1007/s101149900033. |
[33] |
T. Miller, V. Miller and M. Neumann, Local spectral properties of weighted shifts, J. Operator Theory, 51 (2004), 71-88. |
[34] |
K. Palmer, A diagonal dominance criterion for exponential dichotomy, Bull. Austral. Math. Soc., 17 (1977), 363-374.
doi: 10.1017/S0004972700010649. |
[35] |
G. Papaschinopoulos, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109.
doi: 10.1080/00036819108839996. |
[36] |
C. Pötzsche, A note on the dichotomy spectrum, J. Difference Equ. Appl., 15 (2009), 1021-1025, (see also the corrigendum in J. Difference Equ. Appl., 18 (2009), 1257-1261 (2012)).
doi: 10.1080/10236190802320147. |
[37] |
C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Oper. Theory, 73 (2012), 107-151.
doi: 10.1007/s00020-012-1959-7. |
[38] |
C. Pötzsche, Continuity of the dichotomy spectrum on the half line, Submitted, 2014. |
[39] |
C. Pötzsche and E. Russ, Continuity and invariance of the dichotomy spectrum, Submitted, 2014. |
[40] |
W. Ridge, Approximate point spectrum of a weighted shift, Trans. Am. Math. Soc., 147 (1970), 349-356.
doi: 10.1090/S0002-9947-1970-0254635-5. |
[41] |
R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differ. Equations, 27 (1978), 320-358.
doi: 10.1016/0022-0396(78)90057-8. |
[42] |
S. Sánchez-Perales and S. Djordjević, Continuity of spectrum and approximate point spectrum on operator matrices, J. Math. Anal. Appl., 378 (2011), 289-294.
doi: 10.1016/j.jmaa.2011.01.062. |
[43] |
S. Siegmund, Normal forms for nonautonomous differential equations, J. Differ. Equations, 178 (2002), 541-573.
doi: 10.1006/jdeq.2000.4008. |
[44] |
S. Siegmund, Normal forms for nonautonomous difference equations, Comput. Math. Appl., 45 (2003), 1059-1073.
doi: 10.1016/S0898-1221(03)00085-3. |
[45] |
E. Zerouali and H. Zguitti, Perturbation of spectra of operator matrices and local spectral theory, J. Math. Anal. Appl., 324 (2006), 992-1005.
doi: 10.1016/j.jmaa.2005.12.065. |
[46] |
Y. Zhang, H. Zhong and L. Lin, Browder spectra and essential spectra of operator matrices, Acta Mathematica Sinica, 24 (2008), 947-955.
doi: 10.1007/s10114-007-6339-x. |
show all references
References:
[1] |
Y. Abramovich and C. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics, 50. American Mathematical Society, Providence, RI, 2002.
doi: 10.1090/gsm/050. |
[2] |
P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer, Dordrecht, 2004. |
[3] |
P. Aiena, Semi-Fredholm Operators, Perturbation Theory and Localized SVEP, XX Escuela Venezolana de Matemáticas, Caracas, Venezuela, 2007. |
[4] |
P. Aiena, T. Miller and M. Neumann, On a localized single-valued extension property, Math. Proc. R. Ir. Acad., 104A (2004), 17-34.
doi: 10.3318/PRIA.2004.104.1.17. |
[5] |
B. Aulbach, N. Van Minh and P. Zabreiko, The concept of spectral dichotomy for linear difference equations, J. Math. Anal. Appl., 185 (1994), 275-287.
doi: 10.1006/jmaa.1994.1248. |
[6] |
B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, In: López-Fenner J., e.a. (ed.) Proceedings of the 5th Intern. Conference of Difference Eqns. and Application (Temuco, Chile, 2000), 45-55. Taylor & Francis, London, 2002. |
[7] |
M. Barraa and M. Boumazgour, A note on the spectrum of an upper triangular operator matrix, Proc. Am. Math. Soc., 131 (2006), 3083-3088.
doi: 10.1090/S0002-9939-03-06862-X. |
[8] |
L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics 148, AMS, Providence, RI, 2013. |
[9] |
F. Battelli and K. Palmer, Criteria for exponential dichotomy for triangular systems, Journal of Mathematical Analysis and Applications, 428 (2015), 525-543.
doi: 10.1016/j.jmaa.2015.03.029. |
[10] |
A. Ben-Artzi and I. Gohberg, Dichotomies of perturbed time varying systems and the power method, Indiana Univ. Math. J., 42 (1993), 699-720.
doi: 10.1512/iumj.1993.42.42031. |
[11] |
A. Bourhim and C. Chidume, The single-valued extension property for bilateral operator weighted shifts, Proc. Am. Math. Soc., 133 (2005), 485-491.
doi: 10.1090/S0002-9939-04-07535-5. |
[12] |
J. Cushing, S. LeVarge, N. Chitnis and S. Henson, Some discrete competition models and the competitive exclusion principle, J. Difference Equ. Appl., 10 (2004), 1139-1151.
doi: 10.1080/10236190410001652739. |
[13] |
L. Dieci and E. van Vleck, Lyapunov and other spectra: A survey, In Collected Lectures on the Preservation of Stability under Discretization, SIAM, (2002), 197-218. |
[14] |
L. Dieci, C. Elia and E. van Vleck, Exponential dichotomy on the real line: SVD and QR methods, J. Differ. Equations, 248 (2010), 287-308.
doi: 10.1016/j.jde.2009.07.004. |
[15] |
S. Djordjević and Y. Han, A note on Weyl's theorem for operator matrices, Proc. Am. Math. Soc., 131 (2003), 2543-2547.
doi: 10.1090/S0002-9939-02-06808-9. |
[16] |
B. Duggal, Upper triangular operators with SVEP: Spectral properties, Filomat, 21 (2007), 25-37.
doi: 10.2298/FIL0701025D. |
[17] |
H. Elbjaoui and E. Zerouali, Local spectral theory for $2\times 2$ operator matrices, Int. J. Math. Math. Sci., 42 (2003), 2667-2672.
doi: 10.1155/S0161171203012043. |
[18] |
J. Han, H. Lee and W. Lee, Invertible completitions of $2\times 2$ upper triangular operator matrices, Proc. Am. Math. Soc., 128 (2000), 119-123.
doi: 10.1090/S0002-9939-99-04965-5. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840. Springer, Berlin etc., 1981. |
[20] |
D. Hinrichsen and A. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness, Texts in Applied Mathematics, 48. Springer, Heidelberg etc., 2005.
doi: 10.1007/b137541. |
[21] |
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013. |
[22] |
D. Hong-Ke and P. Jin, Perturbations of spectrums of $2\times 2$ operator matrices, Proc. Am. Math. Soc., 121 (1994), 761-766.
doi: 10.2307/2160273. |
[23] |
T. Hüls, Numerical computation of dichotomy rates and projectors in discrete time, Discrete Contin. Dyn. Syst. (Series B), 12 (2009), 109-131.
doi: 10.3934/dcdsb.2009.12.109. |
[24] |
T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM Journal on Numerical Analysis, 48 (2010), 2043-2064.
doi: 10.1137/090754509. |
[25] |
T. Hüls and C. Pötzsche, Qualitative analysis of a nonautonomous Beverton-Holt Ricker model, SIAM Journal of Applied Dynamical Systems, 13 (2014), 1442-1488.
doi: 10.1137/140955434. |
[26] |
C. Jiang and Z. Wang, Structure of Hilbert Space Operators, World Scientific, New Jersey, 2006 |
[27] |
R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33.
doi: 10.1137/0518001. |
[28] |
K. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. AMS, Providence, RI, 2011.
doi: 10.1090/surv/176. |
[29] |
K. Laursen and M. Neumann, An Introduction to Local Spectral Theory, Oxford Science Publications, Oxford, 2000. |
[30] |
W. Lee, Weyl spectra of operator matrices, Proc. Am. Math. Soc., 129 (2000), 131-138.
doi: 10.1090/S0002-9939-00-05846-9. |
[31] |
J. Li, The single valued extension property for operator weighted shifts, Northeast. Math. J., 10 (1994), 99-103. |
[32] |
J. Li, Y. Ji and S. Sun, The essential spectrum and Banach reducibility of operator weighted shifts, Acta Mathematica Sinica, 17 (2001), 413-424.
doi: 10.1007/s101149900033. |
[33] |
T. Miller, V. Miller and M. Neumann, Local spectral properties of weighted shifts, J. Operator Theory, 51 (2004), 71-88. |
[34] |
K. Palmer, A diagonal dominance criterion for exponential dichotomy, Bull. Austral. Math. Soc., 17 (1977), 363-374.
doi: 10.1017/S0004972700010649. |
[35] |
G. Papaschinopoulos, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109.
doi: 10.1080/00036819108839996. |
[36] |
C. Pötzsche, A note on the dichotomy spectrum, J. Difference Equ. Appl., 15 (2009), 1021-1025, (see also the corrigendum in J. Difference Equ. Appl., 18 (2009), 1257-1261 (2012)).
doi: 10.1080/10236190802320147. |
[37] |
C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Oper. Theory, 73 (2012), 107-151.
doi: 10.1007/s00020-012-1959-7. |
[38] |
C. Pötzsche, Continuity of the dichotomy spectrum on the half line, Submitted, 2014. |
[39] |
C. Pötzsche and E. Russ, Continuity and invariance of the dichotomy spectrum, Submitted, 2014. |
[40] |
W. Ridge, Approximate point spectrum of a weighted shift, Trans. Am. Math. Soc., 147 (1970), 349-356.
doi: 10.1090/S0002-9947-1970-0254635-5. |
[41] |
R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differ. Equations, 27 (1978), 320-358.
doi: 10.1016/0022-0396(78)90057-8. |
[42] |
S. Sánchez-Perales and S. Djordjević, Continuity of spectrum and approximate point spectrum on operator matrices, J. Math. Anal. Appl., 378 (2011), 289-294.
doi: 10.1016/j.jmaa.2011.01.062. |
[43] |
S. Siegmund, Normal forms for nonautonomous differential equations, J. Differ. Equations, 178 (2002), 541-573.
doi: 10.1006/jdeq.2000.4008. |
[44] |
S. Siegmund, Normal forms for nonautonomous difference equations, Comput. Math. Appl., 45 (2003), 1059-1073.
doi: 10.1016/S0898-1221(03)00085-3. |
[45] |
E. Zerouali and H. Zguitti, Perturbation of spectra of operator matrices and local spectral theory, J. Math. Anal. Appl., 324 (2006), 992-1005.
doi: 10.1016/j.jmaa.2005.12.065. |
[46] |
Y. Zhang, H. Zhong and L. Lin, Browder spectra and essential spectra of operator matrices, Acta Mathematica Sinica, 24 (2008), 947-955.
doi: 10.1007/s10114-007-6339-x. |
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