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

Dichotomy spectra of triangular equations

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.
Citation: Christian Pötzsche. Dichotomy spectra of triangular equations. Discrete & Continuous Dynamical Systems - A, 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, (2002).  doi: 10.1090/gsm/050.  Google Scholar

[2]

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

[3]

P. Aiena, Semi-Fredholm Operators, Perturbation Theory and Localized SVEP,, XX Escuela Venezolana de Matemáticas, (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.  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.  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., (2000), 45.   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.  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, (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.  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.  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.  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.  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, (2002), 197.   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.  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.  doi: 10.1090/S0002-9939-02-06808-9.  Google Scholar

[16]

B. Duggal, Upper triangular operators with SVEP: Spectral properties,, Filomat, 21 (2007), 25.  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.  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.  doi: 10.1090/S0002-9939-99-04965-5.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes Math., (1981).   Google Scholar

[20]

D. Hinrichsen and A. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness,, Texts in Applied Mathematics, (2005).  doi: 10.1007/b137541.  Google Scholar

[21]

R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (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.  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.  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.  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.  doi: 10.1137/140955434.  Google Scholar

[26]

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

[27]

R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems,, SIAM J. Math. Anal., 18 (1987), 1.  doi: 10.1137/0518001.  Google Scholar

[28]

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

[29]

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

[30]

W. Lee, Weyl spectra of operator matrices,, Proc. Am. Math. Soc., 129 (2000), 131.  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.   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.  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.   Google Scholar

[34]

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

[35]

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

[36]

C. Pötzsche, A note on the dichotomy spectrum,, J. Difference Equ. Appl., 15 (2009), 1021.  doi: 10.1080/10236190802320147.  Google Scholar

[37]

C. Pötzsche, Fine structure of the dichotomy spectrum,, Integral Equations Oper. Theory, 73 (2012), 107.  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.  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.  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.  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.  doi: 10.1006/jdeq.2000.4008.  Google Scholar

[44]

S. Siegmund, Normal forms for nonautonomous difference equations,, Comput. Math. Appl., 45 (2003), 1059.  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.  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.  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, (2002).  doi: 10.1090/gsm/050.  Google Scholar

[2]

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

[3]

P. Aiena, Semi-Fredholm Operators, Perturbation Theory and Localized SVEP,, XX Escuela Venezolana de Matemáticas, (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.  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.  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., (2000), 45.   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.  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, (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.  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.  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.  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.  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, (2002), 197.   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.  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.  doi: 10.1090/S0002-9939-02-06808-9.  Google Scholar

[16]

B. Duggal, Upper triangular operators with SVEP: Spectral properties,, Filomat, 21 (2007), 25.  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.  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.  doi: 10.1090/S0002-9939-99-04965-5.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes Math., (1981).   Google Scholar

[20]

D. Hinrichsen and A. Pritchard, Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness,, Texts in Applied Mathematics, (2005).  doi: 10.1007/b137541.  Google Scholar

[21]

R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (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.  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.  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.  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.  doi: 10.1137/140955434.  Google Scholar

[26]

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

[27]

R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems,, SIAM J. Math. Anal., 18 (1987), 1.  doi: 10.1137/0518001.  Google Scholar

[28]

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

[29]

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

[30]

W. Lee, Weyl spectra of operator matrices,, Proc. Am. Math. Soc., 129 (2000), 131.  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.   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.  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.   Google Scholar

[34]

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

[35]

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

[36]

C. Pötzsche, A note on the dichotomy spectrum,, J. Difference Equ. Appl., 15 (2009), 1021.  doi: 10.1080/10236190802320147.  Google Scholar

[37]

C. Pötzsche, Fine structure of the dichotomy spectrum,, Integral Equations Oper. Theory, 73 (2012), 107.  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.  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.  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.  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.  doi: 10.1006/jdeq.2000.4008.  Google Scholar

[44]

S. Siegmund, Normal forms for nonautonomous difference equations,, Comput. Math. Appl., 45 (2003), 1059.  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.  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.  doi: 10.1007/s10114-007-6339-x.  Google Scholar

[1]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[2]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[3]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012

[4]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[5]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[6]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[7]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[8]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[9]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[10]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[11]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[12]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[13]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[14]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[15]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[16]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[17]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[18]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[19]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[20]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]