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Automatic sequences as good weights for ergodic theorems
Lower spectral radius and spectral mapping theorem for suprema preserving mappings
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 Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max-type kernel operators.
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, 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. |
[3] |
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/377/06984. |
[4] |
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 Google Scholar |
[5] |
C. D. Aliprantis and K. C. Border,
Infinite Dimensional Analysis, A Hitchhiker's Guide, Third Edition, Springer, 2006. |
[6] |
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. |
[7] |
C. D. Aliprantis and O. Burkinshaw,
Positive Operators, Reprint of the 1985 original, Springer, Dordrecht, 2006.
doi: 10.1007/978-1-4020-5008-4. |
[8] |
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. |
[9] |
C. D. Aliprantis and R. Tourky,
Cones and Duality, American Mathematical Society, Providence, 2007.
doi: 10.1090/gsm/084. |
[10] |
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. |
[11] |
J. Appell, E. De Pascale and A. Vignoli,
Nonlinear Spectral Theory, Walter de Gruyter GmbH and Co. KG, Berlin, 2004.
doi: 10.1515/9783110199260. |
[12] |
J. Appell, E. Giorgieri and M. Väth,
Nonlinear spectral theory for homogeneous operators, Nonlinear Funct. Anal. Appl., 7 (2002), 589-618.
|
[13] |
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. |
[14] |
P. Butkovič,
Max-linear Systems: Theory and Algorithms, Springer-Verlag, London, 2010.
doi: 10.1007/978-1-84996-299-5. |
[15] |
R. Drnovšek and A. Peperko,
Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces, Banach J. Math. Anal., 10 (2016), 800-814.
doi: 10.1215/17358787-3649524. |
[16] |
W. Feng,
A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 2 (1997), 163-183.
doi: 10.1155/S1085337597000328. |
[17] |
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. |
[18] |
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. |
[19] |
R. D. Katz, H. Schneider and S. Sergeev,
On commuting matrices in max algebra and in nonnegative matrix algebra, Linear Algebra Appl., 436 (2012), 276-292.
doi: 10.1016/j.laa.2010.08.027. |
[20] |
V. N. Kolokoltsov and V. P. Maslov,
Idempotent Analysis and Its Applications, Kluwer Acad. Publ., 1997.
doi: 10.1007/978-94-015-8901-7. |
[21] |
B. Lemmens and R. D. Nussbaum,
Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754.
doi: 10.1090/S0002-9939-2013-11520-0. |
[22] |
B. Lemmens and R. D. Nussbaum,
Nonlinear Perron-Frobenius Theory, Cambridge University Press, 2012.
doi: 10.1017/CBO9781139026079. |
[23] |
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, I and II, A reprint of the 1977 and 1979 editions, Springer, 1996. Google Scholar |
[24] |
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. |
[25] |
G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: A brief introduction,
J. Math. Sci. (N. Y.), 140 (2007), 426-444, arXiv: math/0507014
doi: 10.1007/s10958-007-0450-5. |
[26] |
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. |
[27] |
G. L. Litvinov and V. P. Maslov (eds.),
Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005.
doi: 10.1090/conm/377/6982. |
[28] |
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. |
[29] |
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. |
[30] |
V. Müller and A. Peperko,
Generalized spectral radius and its max algebra version, Linear Algebra Appl., 439 (2013), 1006-1016.
doi: 10.1016/j.laa.2012.09.024. |
[31] |
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. |
[32] |
V. Müller and A. Peperko,
On the Bonsall cone spectral radius and the approximate point spectrum, Discrete and Continuous Dynamical Systems - Series A, 37 (2017), 5337-5354.
doi: 10.3934/dcds.2017232. |
[33] |
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 886, Springer-Verlag, Berlin, (1981), 309–331. |
[34] |
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. |
[35] |
L. Pachter and B. Sturmfels (eds.), Algebraic statistics for computational biology, Cambridge Univ. Press, New York, (2005), 125–159.
doi: 10.1017/CBO9780511610684.007. |
[36] |
A. Peperko,
Inequalities for the spectral radius of non-negative functions, Positivity, 13 (2009), 255-272.
doi: 10.1007/s11117-008-2188-9. |
[37] |
A. Peperko,
Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequence spaces, Linear Algebra Appl., 437 (2012), 189-201.
doi: 10.1016/j.laa.2012.02.022. |
[38] |
A. Peperko,
Bounds on the joint and generalized spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators, Linear Algebra Appl., 533 (2017), 418-427.
doi: 10.1016/j.laa.2017.07.020. |
[39] |
A. Peperko, Inequalities on the spectral radius, operator norm and numerical radius of the Hadamard weighted geometric mean of positive kernel operators,
Linear and Multilinear Algebra, (2018), arXiv: 1612.01767.
doi: 10.1080/03081087.2018.1465885. |
[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. Google Scholar |
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, 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. |
[3] |
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/377/06984. |
[4] |
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 Google Scholar |
[5] |
C. D. Aliprantis and K. C. Border,
Infinite Dimensional Analysis, A Hitchhiker's Guide, Third Edition, Springer, 2006. |
[6] |
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. |
[7] |
C. D. Aliprantis and O. Burkinshaw,
Positive Operators, Reprint of the 1985 original, Springer, Dordrecht, 2006.
doi: 10.1007/978-1-4020-5008-4. |
[8] |
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. |
[9] |
C. D. Aliprantis and R. Tourky,
Cones and Duality, American Mathematical Society, Providence, 2007.
doi: 10.1090/gsm/084. |
[10] |
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. |
[11] |
J. Appell, E. De Pascale and A. Vignoli,
Nonlinear Spectral Theory, Walter de Gruyter GmbH and Co. KG, Berlin, 2004.
doi: 10.1515/9783110199260. |
[12] |
J. Appell, E. Giorgieri and M. Väth,
Nonlinear spectral theory for homogeneous operators, Nonlinear Funct. Anal. Appl., 7 (2002), 589-618.
|
[13] |
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. |
[14] |
P. Butkovič,
Max-linear Systems: Theory and Algorithms, Springer-Verlag, London, 2010.
doi: 10.1007/978-1-84996-299-5. |
[15] |
R. Drnovšek and A. Peperko,
Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces, Banach J. Math. Anal., 10 (2016), 800-814.
doi: 10.1215/17358787-3649524. |
[16] |
W. Feng,
A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 2 (1997), 163-183.
doi: 10.1155/S1085337597000328. |
[17] |
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. |
[18] |
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. |
[19] |
R. D. Katz, H. Schneider and S. Sergeev,
On commuting matrices in max algebra and in nonnegative matrix algebra, Linear Algebra Appl., 436 (2012), 276-292.
doi: 10.1016/j.laa.2010.08.027. |
[20] |
V. N. Kolokoltsov and V. P. Maslov,
Idempotent Analysis and Its Applications, Kluwer Acad. Publ., 1997.
doi: 10.1007/978-94-015-8901-7. |
[21] |
B. Lemmens and R. D. Nussbaum,
Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754.
doi: 10.1090/S0002-9939-2013-11520-0. |
[22] |
B. Lemmens and R. D. Nussbaum,
Nonlinear Perron-Frobenius Theory, Cambridge University Press, 2012.
doi: 10.1017/CBO9781139026079. |
[23] |
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, I and II, A reprint of the 1977 and 1979 editions, Springer, 1996. Google Scholar |
[24] |
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. |
[25] |
G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: A brief introduction,
J. Math. Sci. (N. Y.), 140 (2007), 426-444, arXiv: math/0507014
doi: 10.1007/s10958-007-0450-5. |
[26] |
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. |
[27] |
G. L. Litvinov and V. P. Maslov (eds.),
Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005.
doi: 10.1090/conm/377/6982. |
[28] |
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. |
[29] |
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. |
[30] |
V. Müller and A. Peperko,
Generalized spectral radius and its max algebra version, Linear Algebra Appl., 439 (2013), 1006-1016.
doi: 10.1016/j.laa.2012.09.024. |
[31] |
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. |
[32] |
V. Müller and A. Peperko,
On the Bonsall cone spectral radius and the approximate point spectrum, Discrete and Continuous Dynamical Systems - Series A, 37 (2017), 5337-5354.
doi: 10.3934/dcds.2017232. |
[33] |
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 886, Springer-Verlag, Berlin, (1981), 309–331. |
[34] |
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. |
[35] |
L. Pachter and B. Sturmfels (eds.), Algebraic statistics for computational biology, Cambridge Univ. Press, New York, (2005), 125–159.
doi: 10.1017/CBO9780511610684.007. |
[36] |
A. Peperko,
Inequalities for the spectral radius of non-negative functions, Positivity, 13 (2009), 255-272.
doi: 10.1007/s11117-008-2188-9. |
[37] |
A. Peperko,
Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequence spaces, Linear Algebra Appl., 437 (2012), 189-201.
doi: 10.1016/j.laa.2012.02.022. |
[38] |
A. Peperko,
Bounds on the joint and generalized spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators, Linear Algebra Appl., 533 (2017), 418-427.
doi: 10.1016/j.laa.2017.07.020. |
[39] |
A. Peperko, Inequalities on the spectral radius, operator norm and numerical radius of the Hadamard weighted geometric mean of positive kernel operators,
Linear and Multilinear Algebra, (2018), arXiv: 1612.01767.
doi: 10.1080/03081087.2018.1465885. |
[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. Google Scholar |
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