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January  2016, 36(1): 423-450. doi: 10.3934/dcds.2016.36.423

Dichotomy spectra of triangular equations

Christian Pötzsche 1,

1. 

Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt

Received  August 2014 Revised  March 2015 Published  June 2015

Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. In this context, when dealing with time-variant linear equations having triangular coefficient matrices, their dichotomy spectrum associated to the whole time axis is not fully determined by the diagonal entries. This is surprising because such a behavior differs from both the half line situation, as well as the classical autonomous and periodic cases. At the same time triangular problems occur in various applications and particularly numerical techniques.
    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.
Keywords: exponential dichotomy, nonautonomous hyperbolicity, Bohl exponent, block triangular equation, Sacker-Sell spectrum, spectral continuity., diagonal significance, Dichotomy spectrum.
Mathematics Subject Classification: Primary: 34D09; Secondary: 37C60, 39A30, 47B3.
Citation: Christian Pötzsche. Dichotomy spectra of triangular equations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 423-450. doi: 10.3934/dcds.2016.36.423
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.  Google Scholar

[2]

P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer, Dordrecht, 2004.  Google Scholar

[3]

P. Aiena, Semi-Fredholm Operators, Perturbation Theory and Localized SVEP, XX Escuela Venezolana de Matemáticas, Caracas, Venezuela, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics 148, AMS, Providence, RI, 2013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. Duggal, Upper triangular operators with SVEP: Spectral properties, Filomat, 21 (2007), 25-37. doi: 10.2298/FIL0701025D.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840. Springer, Berlin etc., 1981.  Google Scholar

[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.  Google Scholar

[21]

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

C. Jiang and Z. Wang, Structure of Hilbert Space Operators, World Scientific, New Jersey, 2006 Google Scholar

[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.  Google Scholar

[28]

K. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. AMS, Providence, RI, 2011. doi: 10.1090/surv/176.  Google Scholar

[29]

K. Laursen and M. Neumann, An Introduction to Local Spectral Theory, Oxford Science Publications, Oxford, 2000.  Google Scholar

[30]

W. Lee, Weyl spectra of operator matrices, Proc. Am. Math. Soc., 129 (2000), 131-138. doi: 10.1090/S0002-9939-00-05846-9.  Google Scholar

[31]

J. Li, The single valued extension property for operator weighted shifts, Northeast. Math. J., 10 (1994), 99-103.  Google Scholar

[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.  Google Scholar

[33]

T. Miller, V. Miller and M. Neumann, Local spectral properties of weighted shifts, J. Operator Theory, 51 (2004), 71-88.  Google Scholar

[34]

K. Palmer, A diagonal dominance criterion for exponential dichotomy, Bull. Austral. Math. Soc., 17 (1977), 363-374. doi: 10.1017/S0004972700010649.  Google Scholar

[35]

G. Papaschinopoulos, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109. doi: 10.1080/00036819108839996.  Google Scholar

[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.  Google Scholar

[37]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Oper. Theory, 73 (2012), 107-151. doi: 10.1007/s00020-012-1959-7.  Google Scholar

[38]

C. Pötzsche, Continuity of the dichotomy spectrum on the half line, Submitted, 2014. Google Scholar

[39]

C. Pötzsche and E. Russ, Continuity and invariance of the dichotomy spectrum, Submitted, 2014. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

S. Siegmund, Normal forms for nonautonomous differential equations, J. Differ. Equations, 178 (2002), 541-573. doi: 10.1006/jdeq.2000.4008.  Google Scholar

[44]

S. Siegmund, Normal forms for nonautonomous difference equations, Comput. Math. Appl., 45 (2003), 1059-1073. doi: 10.1016/S0898-1221(03)00085-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer, Dordrecht, 2004.  Google Scholar

[3]

P. Aiena, Semi-Fredholm Operators, Perturbation Theory and Localized SVEP, XX Escuela Venezolana de Matemáticas, Caracas, Venezuela, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics 148, AMS, Providence, RI, 2013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. Duggal, Upper triangular operators with SVEP: Spectral properties, Filomat, 21 (2007), 25-37. doi: 10.2298/FIL0701025D.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840. Springer, Berlin etc., 1981.  Google Scholar

[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.  Google Scholar

[21]

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

C. Jiang and Z. Wang, Structure of Hilbert Space Operators, World Scientific, New Jersey, 2006 Google Scholar

[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.  Google Scholar

[28]

K. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. AMS, Providence, RI, 2011. doi: 10.1090/surv/176.  Google Scholar

[29]

K. Laursen and M. Neumann, An Introduction to Local Spectral Theory, Oxford Science Publications, Oxford, 2000.  Google Scholar

[30]

W. Lee, Weyl spectra of operator matrices, Proc. Am. Math. Soc., 129 (2000), 131-138. doi: 10.1090/S0002-9939-00-05846-9.  Google Scholar

[31]

J. Li, The single valued extension property for operator weighted shifts, Northeast. Math. J., 10 (1994), 99-103.  Google Scholar

[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.  Google Scholar

[33]

T. Miller, V. Miller and M. Neumann, Local spectral properties of weighted shifts, J. Operator Theory, 51 (2004), 71-88.  Google Scholar

[34]

K. Palmer, A diagonal dominance criterion for exponential dichotomy, Bull. Austral. Math. Soc., 17 (1977), 363-374. doi: 10.1017/S0004972700010649.  Google Scholar

[35]

G. Papaschinopoulos, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109. doi: 10.1080/00036819108839996.  Google Scholar

[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.  Google Scholar

[37]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Oper. Theory, 73 (2012), 107-151. doi: 10.1007/s00020-012-1959-7.  Google Scholar

[38]

C. Pötzsche, Continuity of the dichotomy spectrum on the half line, Submitted, 2014. Google Scholar

[39]

C. Pötzsche and E. Russ, Continuity and invariance of the dichotomy spectrum, Submitted, 2014. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

S. Siegmund, Normal forms for nonautonomous differential equations, J. Differ. Equations, 178 (2002), 541-573. doi: 10.1006/jdeq.2000.4008.  Google Scholar

[44]

S. Siegmund, Normal forms for nonautonomous difference equations, Comput. Math. Appl., 45 (2003), 1059-1073. doi: 10.1016/S0898-1221(03)00085-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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