Dichotomy spectra of triangular equations

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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

Abstract
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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 and 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.
[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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