-
Previous Article
Strichartz estimates for $N$-body Schrödinger operators with small potential interactions
- DCDS Home
- This Issue
-
Next Article
Relations between Bohl and general exponents
On the Bonsall cone spectral radius and the approximate point spectrum
1. | Institute of Mathematics, Czech Academy of Sciences, Žitna 25, 115 67 Prague, Czech Republic |
2. | Faculty of Mechanical Engineering, University of Ljubljana Aškerčeva 6, SI-1000 Ljubljana, Slovenia |
3. | Institute of Mathematics, Physics and Mechanics Jadranska 19, SI-1000 Ljubljana, Slovenia |
We study the Bonsall cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone spectral radius of such maps is always included in its approximate point spectrum. Moreover, the approximate point spectrum always contains a (possibly trivial) interval. Our results apply to a large class of (nonlinear) max-type operators.
We also generalize a known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum. Under additional generalized compactness type assumptions our results imply Krein-Rutman type results.
References:
[1] |
Y. A. Abramovich and C. D. Aliprantis,
An Invitation to Operator Theory American Mathematical Society, Providence, 2002.
doi: 10.1090/gsm/050. |
[2] |
M. Akian and S. Gaubert, Policy iteration for perfect information stochastic mean payoff games with bounded first return times is strongly polynomial, preprint, arXiv: 1310.4953 |
[3] |
M. Akian, S. Gaubert and A. Hochart,
Ergodicity conditions for zero-sum games, Discrete and Continuous Dynamical Systems -A, 35 (2015), 3901-3931.
doi: 10.3934/dcds.2015.35.3901. |
[4] |
M. Akian, S. Gaubert and C. Walsh, Discrete max-plus spectral theory, in Idempotent Mathematics and Mathematical Physics, G. L. Litvinov and V. P. Maslov, Eds, Contemporary Mathematics, AMS, 377 (2005), 53–77, arXiv: math.SP/0405225
doi: 10.1090/conm/377/6982. |
[5] |
M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, preprint, arXiv: 1112.5968 |
[6] |
C. D. Aliprantis and K. C. Border,
Infinite Dimensional Analysis, A Hitchhiker's Guide Third Edition, Springer, 2006. |
[7] |
C. D. Aliprantis, D. J. Brown and O. Burkinshaw,
Existence and Optimality of Competitive Equilibria Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-61521-4. |
[8] |
C. D. Aliprantis and O. Burkinshaw,
Positive Operators Reprint of the 1985 original, Springer, Dordrecht, 2006.
doi: 10.1007/978-1-4020-5008-4. |
[9] |
C. D. Aliprantis and O. Burkinshaw,
Locally Solid Riesz Spaces with Applications to Economics Second edition, Mathematical Surveys and Monographs, 105, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/surv/105. |
[10] |
C. D. Aliprantis and R. Tourky,
Cones and Duality American Mathematical Society, Providence, 2007.
doi: 10.1090/gsm/084. |
[11] |
J. Appell, E. De Pascale and A. Vignoli,
A comparison of different spectra for nonlinear operators, Nonlinear Anal., 40 (2000), 73-90.
doi: 10.1016/S0362-546X(00)85005-1. |
[12] |
J. Appell, E. De Pascale and A. Vignoli,
Nonlinear Spectral Theory Walter de Gruyter GmbH and Co. KG, Berlin, 2004.
doi: 10.1515/9783110199260. |
[13] |
J. Appell, E. Giorgieri and M. Väth,
Nonlinear spectral theory for homogeneous operators, Nonlinear Funct. Anal. Appl., 7 (2002), 589-618.
|
[14] |
F. L. Baccelli, G. Cohen, G. -J. Olsder and J. -P. Quadrat,
Synchronization and Linearity John Wiley, Chichester, New York, 1992. |
[15] |
R. B. Bapat,
A max version of the Perron-Frobenius theorem, Linear Algebra Appl., 275/276 (1998), 3-18.
doi: 10.1016/S0024-3795(97)10057-X. |
[16] |
P. Butkovič,
Max-linear Systems: Theory and Algorithms Springer-Verlag, London, 2010.
doi: 10.1007/978-1-84996-299-5. |
[17] |
P. Butkovič, S. Gaubert and R. A. Cuninghame-Green,
Reducible spectral theory with applications to the robustness of matrices in max-algebra, SIAM J. Matrix Anal. Appl., 31 (2009), 1412-1431.
doi: 10.1137/080731232. |
[18] |
W. Feng,
A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 2 (1997), 163-183.
doi: 10.1155/S1085337597000328. |
[19] |
M. Furi, M. Martelli and A. Vignoli,
Contributions to the spectral theory for nonlinear operators in Banach spaces, Ann. Pura Appl., 118 (1978), 229-294.
doi: 10.1007/BF02415132. |
[20] |
G. Gripenberg,
On the definition of the cone spectral radius, Proc. Amer. Math. Soc., 143 (2015), 1617-1625.
doi: 10.1090/S0002-9939-2014-12375-6. |
[21] |
J. Gunawardena, Cycle times and fixed points of min-max functions, In G. Cohen and J. -P.
Quadrat, editors, 11th International Conference on Analysis and Optimization of Systems,
Springer LNCIS, 199 (1994), 266–272. |
[22] |
M. de Jeu and M. Messerschmidt,
A strong open mapping theorem for surjections from cones
onto Banach spaces, Advances in Math., 259 (2014), 43-66.
doi: 10.1016/j.aim.2014.03.008. |
[23] |
V. N. Kolokoltsov and V. P. Maslov,
Idempotent Analysis and Its Applications Kluwer Acad. Publ., 1997.
doi: 10.1007/978-94-015-8901-7. |
[24] |
B. Lemmens and R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math.
Soc., 141 (2013), 2741–2754, arXiv: 1107.4532.
doi: 10.1090/S0002-9939-2013-11520-0. |
[25] |
B. Lemmens and R. D. Nussbaum,
Nonlinear Perron-Frobenius Theory Cambridge University Press, 2012.
doi: 10.1017/CBO9781139026079. |
[26] |
J. Lindenstrauss and L. Tzafriri,
Classical Banach Spaces I and II A reprint of the 1977 and 1979 editions, Springer, 1996. |
[27] |
B. Lins and R. D. Nussbaum,
Denjoy-Wolff theorems, Hilbert metric nonexpansive maps on
reproduction-decimation operators, J. Funct. Anal., 254 (2008), 2365-2386.
doi: 10.1016/j.jfa.2008.02.001. |
[28] |
G. L. Litvinov,
The Maslov dequantization, idempotent and tropical mathematics: A brief
introduction, J. Math. Sci.(N. Y.), 140 (2007), 426-444.
doi: 10.1007/s10958-007-0450-5. |
[29] |
G. L. Litvinov, V. P. Maslov and G. B. Shpiz, Idempotent functional analysis: An algebraic
approach, Math Notes, 69 (2001), 696–729, arXiv: math.FA/0009128.
doi: 10.1023/A:1010266012029. |
[30] |
G. L. Litvinov and V. P. Maslov (eds.), Idempotent mathematics and mathematical physics,
Contemp. Math., Amer. Math. Soc., Providence, RI, 377 (2005), 1–17.
doi: 10.1090/conm/377/6982. |
[31] |
J. Mallet-Paret and R. D. Nussbaum,
Eigenvalues for a class of homogeneous cone maps
arising from max-plus operators, Discrete and Continuous Dynamical Systems, 8 (2002), 519-562.
doi: 10.3934/dcds.2002.8.519. |
[32] |
J. Mallet-Paret and R. D. Nussbaum,
Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Applications, 7 (2010), 103-143.
doi: 10.1007/s11784-010-0010-3. |
[33] |
J. Mallet-Paret and R. D. Nussbaum,
Inequivalent measures of noncompactness, Ann. Mat. Pura Appl.(4), 190 (2011), 453-488.
doi: 10.1007/s10231-010-0158-x. |
[34] |
J. Mallet-Paret and R. D. Nussbaum,
Inequivalent measures of noncompactness and the radius
of the essential radius, Proc. Amer. Math. Soc., 139 (2011), 917-930.
doi: 10.1090/S0002-9939-2010-10511-7. |
[35] |
V. Müller and A. Peperko,
Generalized spectral radius and its max algebra version, Linear
Algebra Appl, Linear Algebra Appl., 439 (2013), 1006-1016.
doi: 10.1016/j.laa.2012.09.024. |
[36] |
V. Müller and A. Peperko,
On the spectrum in max-algebra, Linear Algebra Appl., 485 (2015), 250-266.
doi: 10.1016/j.laa.2015.07.013. |
[37] |
R. D. Nussbaum, Eigenvalues of nonlinear operators and the linear Krein-Rutman, in: Fixed
Point Theory (Sherbrooke, Quebec, 1980), E. Fadell and G. Fournier, editors, Lecture notes
in Mathematics, Springer-Verlag, Berlin, 886 (1981), 309–330. |
[38] |
R. D. Nussbaum,
Periodic points of positive linear operators and Perron-Frobenius operators, Integral Equations Operator Theory, 39 (2001), 41-97.
doi: 10.1007/BF01192149. |
[39] |
L. Pachter and B. Sturmfels (eds.), Algebraic statistics for computational biology, Cambridge
Univ. Press, New York, (2005), 125–159.
doi: 10.1017/CBO9780511610684. 007. |
[40] |
P. Santucci and M. Väth,
On the definition of eigenvalues of nonlinear operators, Nonlinear Anal., 40 (2000), 565-576.
doi: 10.1016/S0362-546X(00)85034-8. |
[41] |
G. B. Shpiz,
An eigenvector existence theorem in idempotent analysis, Mathematical Notes, 82 (2007), 410-417.
doi: 10.1134/S0001434607090131. |
[42] |
W. Wnuk,
Banach Lattices with Order Continuous Norms Polish Scientific Publ., PWN, Warszawa, 1999. |
show all references
References:
[1] |
Y. A. Abramovich and C. D. Aliprantis,
An Invitation to Operator Theory American Mathematical Society, Providence, 2002.
doi: 10.1090/gsm/050. |
[2] |
M. Akian and S. Gaubert, Policy iteration for perfect information stochastic mean payoff games with bounded first return times is strongly polynomial, preprint, arXiv: 1310.4953 |
[3] |
M. Akian, S. Gaubert and A. Hochart,
Ergodicity conditions for zero-sum games, Discrete and Continuous Dynamical Systems -A, 35 (2015), 3901-3931.
doi: 10.3934/dcds.2015.35.3901. |
[4] |
M. Akian, S. Gaubert and C. Walsh, Discrete max-plus spectral theory, in Idempotent Mathematics and Mathematical Physics, G. L. Litvinov and V. P. Maslov, Eds, Contemporary Mathematics, AMS, 377 (2005), 53–77, arXiv: math.SP/0405225
doi: 10.1090/conm/377/6982. |
[5] |
M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, preprint, arXiv: 1112.5968 |
[6] |
C. D. Aliprantis and K. C. Border,
Infinite Dimensional Analysis, A Hitchhiker's Guide Third Edition, Springer, 2006. |
[7] |
C. D. Aliprantis, D. J. Brown and O. Burkinshaw,
Existence and Optimality of Competitive Equilibria Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-61521-4. |
[8] |
C. D. Aliprantis and O. Burkinshaw,
Positive Operators Reprint of the 1985 original, Springer, Dordrecht, 2006.
doi: 10.1007/978-1-4020-5008-4. |
[9] |
C. D. Aliprantis and O. Burkinshaw,
Locally Solid Riesz Spaces with Applications to Economics Second edition, Mathematical Surveys and Monographs, 105, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/surv/105. |
[10] |
C. D. Aliprantis and R. Tourky,
Cones and Duality American Mathematical Society, Providence, 2007.
doi: 10.1090/gsm/084. |
[11] |
J. Appell, E. De Pascale and A. Vignoli,
A comparison of different spectra for nonlinear operators, Nonlinear Anal., 40 (2000), 73-90.
doi: 10.1016/S0362-546X(00)85005-1. |
[12] |
J. Appell, E. De Pascale and A. Vignoli,
Nonlinear Spectral Theory Walter de Gruyter GmbH and Co. KG, Berlin, 2004.
doi: 10.1515/9783110199260. |
[13] |
J. Appell, E. Giorgieri and M. Väth,
Nonlinear spectral theory for homogeneous operators, Nonlinear Funct. Anal. Appl., 7 (2002), 589-618.
|
[14] |
F. L. Baccelli, G. Cohen, G. -J. Olsder and J. -P. Quadrat,
Synchronization and Linearity John Wiley, Chichester, New York, 1992. |
[15] |
R. B. Bapat,
A max version of the Perron-Frobenius theorem, Linear Algebra Appl., 275/276 (1998), 3-18.
doi: 10.1016/S0024-3795(97)10057-X. |
[16] |
P. Butkovič,
Max-linear Systems: Theory and Algorithms Springer-Verlag, London, 2010.
doi: 10.1007/978-1-84996-299-5. |
[17] |
P. Butkovič, S. Gaubert and R. A. Cuninghame-Green,
Reducible spectral theory with applications to the robustness of matrices in max-algebra, SIAM J. Matrix Anal. Appl., 31 (2009), 1412-1431.
doi: 10.1137/080731232. |
[18] |
W. Feng,
A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 2 (1997), 163-183.
doi: 10.1155/S1085337597000328. |
[19] |
M. Furi, M. Martelli and A. Vignoli,
Contributions to the spectral theory for nonlinear operators in Banach spaces, Ann. Pura Appl., 118 (1978), 229-294.
doi: 10.1007/BF02415132. |
[20] |
G. Gripenberg,
On the definition of the cone spectral radius, Proc. Amer. Math. Soc., 143 (2015), 1617-1625.
doi: 10.1090/S0002-9939-2014-12375-6. |
[21] |
J. Gunawardena, Cycle times and fixed points of min-max functions, In G. Cohen and J. -P.
Quadrat, editors, 11th International Conference on Analysis and Optimization of Systems,
Springer LNCIS, 199 (1994), 266–272. |
[22] |
M. de Jeu and M. Messerschmidt,
A strong open mapping theorem for surjections from cones
onto Banach spaces, Advances in Math., 259 (2014), 43-66.
doi: 10.1016/j.aim.2014.03.008. |
[23] |
V. N. Kolokoltsov and V. P. Maslov,
Idempotent Analysis and Its Applications Kluwer Acad. Publ., 1997.
doi: 10.1007/978-94-015-8901-7. |
[24] |
B. Lemmens and R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math.
Soc., 141 (2013), 2741–2754, arXiv: 1107.4532.
doi: 10.1090/S0002-9939-2013-11520-0. |
[25] |
B. Lemmens and R. D. Nussbaum,
Nonlinear Perron-Frobenius Theory Cambridge University Press, 2012.
doi: 10.1017/CBO9781139026079. |
[26] |
J. Lindenstrauss and L. Tzafriri,
Classical Banach Spaces I and II A reprint of the 1977 and 1979 editions, Springer, 1996. |
[27] |
B. Lins and R. D. Nussbaum,
Denjoy-Wolff theorems, Hilbert metric nonexpansive maps on
reproduction-decimation operators, J. Funct. Anal., 254 (2008), 2365-2386.
doi: 10.1016/j.jfa.2008.02.001. |
[28] |
G. L. Litvinov,
The Maslov dequantization, idempotent and tropical mathematics: A brief
introduction, J. Math. Sci.(N. Y.), 140 (2007), 426-444.
doi: 10.1007/s10958-007-0450-5. |
[29] |
G. L. Litvinov, V. P. Maslov and G. B. Shpiz, Idempotent functional analysis: An algebraic
approach, Math Notes, 69 (2001), 696–729, arXiv: math.FA/0009128.
doi: 10.1023/A:1010266012029. |
[30] |
G. L. Litvinov and V. P. Maslov (eds.), Idempotent mathematics and mathematical physics,
Contemp. Math., Amer. Math. Soc., Providence, RI, 377 (2005), 1–17.
doi: 10.1090/conm/377/6982. |
[31] |
J. Mallet-Paret and R. D. Nussbaum,
Eigenvalues for a class of homogeneous cone maps
arising from max-plus operators, Discrete and Continuous Dynamical Systems, 8 (2002), 519-562.
doi: 10.3934/dcds.2002.8.519. |
[32] |
J. Mallet-Paret and R. D. Nussbaum,
Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Applications, 7 (2010), 103-143.
doi: 10.1007/s11784-010-0010-3. |
[33] |
J. Mallet-Paret and R. D. Nussbaum,
Inequivalent measures of noncompactness, Ann. Mat. Pura Appl.(4), 190 (2011), 453-488.
doi: 10.1007/s10231-010-0158-x. |
[34] |
J. Mallet-Paret and R. D. Nussbaum,
Inequivalent measures of noncompactness and the radius
of the essential radius, Proc. Amer. Math. Soc., 139 (2011), 917-930.
doi: 10.1090/S0002-9939-2010-10511-7. |
[35] |
V. Müller and A. Peperko,
Generalized spectral radius and its max algebra version, Linear
Algebra Appl, Linear Algebra Appl., 439 (2013), 1006-1016.
doi: 10.1016/j.laa.2012.09.024. |
[36] |
V. Müller and A. Peperko,
On the spectrum in max-algebra, Linear Algebra Appl., 485 (2015), 250-266.
doi: 10.1016/j.laa.2015.07.013. |
[37] |
R. D. Nussbaum, Eigenvalues of nonlinear operators and the linear Krein-Rutman, in: Fixed
Point Theory (Sherbrooke, Quebec, 1980), E. Fadell and G. Fournier, editors, Lecture notes
in Mathematics, Springer-Verlag, Berlin, 886 (1981), 309–330. |
[38] |
R. D. Nussbaum,
Periodic points of positive linear operators and Perron-Frobenius operators, Integral Equations Operator Theory, 39 (2001), 41-97.
doi: 10.1007/BF01192149. |
[39] |
L. Pachter and B. Sturmfels (eds.), Algebraic statistics for computational biology, Cambridge
Univ. Press, New York, (2005), 125–159.
doi: 10.1017/CBO9780511610684. 007. |
[40] |
P. Santucci and M. Väth,
On the definition of eigenvalues of nonlinear operators, Nonlinear Anal., 40 (2000), 565-576.
doi: 10.1016/S0362-546X(00)85034-8. |
[41] |
G. B. Shpiz,
An eigenvector existence theorem in idempotent analysis, Mathematical Notes, 82 (2007), 410-417.
doi: 10.1134/S0001434607090131. |
[42] |
W. Wnuk,
Banach Lattices with Order Continuous Norms Polish Scientific Publ., PWN, Warszawa, 1999. |
[1] |
Vladimir Müller, Aljoša Peperko. Lower spectral radius and spectral mapping theorem for suprema preserving mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4117-4132. doi: 10.3934/dcds.2018179 |
[2] |
Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447 |
[3] |
Mickaël D. Chekroun, Jean Roux. Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3957-3980. doi: 10.3934/dcds.2013.33.3957 |
[4] |
Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651 |
[5] |
Hongfei Yang, Xiaofeng Ding, Raymond Chan, Hui Hu, Yaxin Peng, Tieyong Zeng. A new initialization method based on normed statistical spaces in deep networks. Inverse Problems and Imaging, 2021, 15 (1) : 147-158. doi: 10.3934/ipi.2020045 |
[6] |
Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97 |
[7] |
Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917 |
[8] |
John Mallet-Paret, Roger D. Nussbaum. Eigenvalues for a class of homogeneous cone maps arising from max-plus operators. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 519-562. doi: 10.3934/dcds.2002.8.519 |
[9] |
Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial and Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975 |
[10] |
Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467 |
[11] |
Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581 |
[12] |
Daria Bugajewska, Mirosława Zima. On the spectral radius of linearly bounded operators and existence results for functional-differential equations. Conference Publications, 2003, 2003 (Special) : 147-155. doi: 10.3934/proc.2003.2003.147 |
[13] |
Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77. |
[14] |
Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22 |
[15] |
Rui Zou, Yongluo Cao, Gang Liao. Continuity of spectral radius over hyperbolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3977-3991. doi: 10.3934/dcds.2018173 |
[16] |
Xu Xu, Xin Zhao. Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4777-4800. doi: 10.3934/dcds.2020201 |
[17] |
Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91 |
[18] |
Pierre Patie, Aditya Vaidyanathan. A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators. Kinetic and Related Models, 2020, 13 (3) : 479-506. doi: 10.3934/krm.2020016 |
[19] |
Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381 |
[20] |
Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]