American Institute of Mathematical Sciences

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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

A. Peperko, Inequalities for the spectral radius of non-negative functions, Positivity, 13 (2009), 255-272.  doi: 10.1007/s11117-008-2188-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

A. Peperko, Inequalities for the spectral radius of non-negative functions, Positivity, 13 (2009), 255-272.  doi: 10.1007/s11117-008-2188-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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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