October  2017, 37(10): 5337-5354. doi: 10.3934/dcds.2017232

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

* Corresponding author: Aljoša Peperko

Received  March 2017 Revised  May 2017 Published  June 2017

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.

Citation: Vladimir Müller, Aljoša Peperko. On the Bonsall cone spectral radius and the approximate point spectrum. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5337-5354. doi: 10.3934/dcds.2017232
References:
[1]

Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory American Mathematical Society, Providence, 2002. doi: 10.1090/gsm/050.  Google Scholar

[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 Google Scholar

[3]

M. AkianS. 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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[6]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, A Hitchhiker's Guide Third Edition, Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. D. Aliprantis and R. Tourky, Cones and Duality American Mathematical Society, Providence, 2007. doi: 10.1090/gsm/084.  Google Scholar

[11]

J. AppellE. 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.  Google Scholar

[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.  Google Scholar

[13]

J. AppellE. Giorgieri and M. Väth, Nonlinear spectral theory for homogeneous operators, Nonlinear Funct. Anal. Appl., 7 (2002), 589-618.   Google Scholar

[14]

F. L. Baccelli, G. Cohen, G. -J. Olsder and J. -P. Quadrat, Synchronization and Linearity John Wiley, Chichester, New York, 1992.  Google Scholar

[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.  Google Scholar

[16]

P. Butkovič, Max-linear Systems: Theory and Algorithms Springer-Verlag, London, 2010. doi: 10.1007/978-1-84996-299-5.  Google Scholar

[17]

P. ButkovičS. 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.  Google Scholar

[18]

W. Feng, A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 2 (1997), 163-183.  doi: 10.1155/S1085337597000328.  Google Scholar

[19]

M. FuriM. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory Cambridge University Press, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[26]

J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II A reprint of the 1977 and 1979 editions, Springer, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

G. B. Shpiz, An eigenvector existence theorem in idempotent analysis, Mathematical Notes, 82 (2007), 410-417.  doi: 10.1134/S0001434607090131.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[3]

M. AkianS. 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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[6]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, A Hitchhiker's Guide Third Edition, Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. D. Aliprantis and R. Tourky, Cones and Duality American Mathematical Society, Providence, 2007. doi: 10.1090/gsm/084.  Google Scholar

[11]

J. AppellE. 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.  Google Scholar

[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.  Google Scholar

[13]

J. AppellE. Giorgieri and M. Väth, Nonlinear spectral theory for homogeneous operators, Nonlinear Funct. Anal. Appl., 7 (2002), 589-618.   Google Scholar

[14]

F. L. Baccelli, G. Cohen, G. -J. Olsder and J. -P. Quadrat, Synchronization and Linearity John Wiley, Chichester, New York, 1992.  Google Scholar

[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.  Google Scholar

[16]

P. Butkovič, Max-linear Systems: Theory and Algorithms Springer-Verlag, London, 2010. doi: 10.1007/978-1-84996-299-5.  Google Scholar

[17]

P. ButkovičS. 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.  Google Scholar

[18]

W. Feng, A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal., 2 (1997), 163-183.  doi: 10.1155/S1085337597000328.  Google Scholar

[19]

M. FuriM. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory Cambridge University Press, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[26]

J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II A reprint of the 1977 and 1979 editions, Springer, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

G. B. Shpiz, An eigenvector existence theorem in idempotent analysis, Mathematical Notes, 82 (2007), 410-417.  doi: 10.1134/S0001434607090131.  Google Scholar

[42]

W. Wnuk, Banach Lattices with Order Continuous Norms Polish Scientific Publ., PWN, Warszawa, 1999. Google Scholar

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