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.

[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. 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.

[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. 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.

[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. AppellE. 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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. AppellE. 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. 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.

[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. 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.

[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.

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