Lower spectral radius and spectral mapping theorem for suprema preserving mappings

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August 2018, 38(8): 4117-4132. doi: 10.3934/dcds.2018179

Lower spectral radius and spectral mapping theorem for suprema preserving mappings

Vladimir Müller ^1, and Aljoša Peperko ^2,3,,

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 November 2017 Published May 2018

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.

Keywords: Spectral mapping theorem, approximate point spectrum, Bonsall's cone spectral radius, lower spectral radius, local spectral radii, supremum preserving maps, max-kernel operators, normed vector lattices, normed spaces, cones.

Mathematics Subject Classification: Primary: 47H07, 47J10; Secondary: 47B65, 47A10.

Citation: 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

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
[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.
[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.

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
[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.
[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.

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