Continuity of the cone spectral radius

Proceedings of the American Mathematical Society

Volumn 141, Issue 8, 2013, Pages 2741-2754

  Lemmens, Bas ^a   Nussbaum, Roger ^b  

^a UNIVERSITY OF KENT   (United Kingdom)

^b RUTGERS UNIVERSITY   (United States)

Author keywords

Cone spectral radius; Cone spectrum; Continuity; Fixed point index; Nonlinear cone maps

Indexed keywords

EID: 84878183063     PISSN: 00029939     EISSN: 10886826     Source Type: Journal    
DOI: 10.1090/S0002-9939-2013-11520-0     Document Type: Article

References (22)

1. F.F. Bonsall, Linear operators in complete positive cones. Proc. London Math. Soc. 8(3) (1958), 53-75. MR0092938 (19:1183c)
2. A.D. Burbanks, R.D. Nussbaum, and C. Sparrow, Extension of order-preserving maps on a cone. Proc. Royal Soc. Edinburgh Sect. A 133A (2003), 35-59. MR1960046 (2004a:47061)
3. L. Burlando, Continuity of spectrum and spectral radius in Banach algebras. In Functional Analysis and Operator Theory (Warsaw, 1992), pp. 53-100, Banach Center Publ., 30, Polish Acad. Sci., Warsaw, 1994. MR1285600 (95i:46062)
4. A. Granas and J. Dugundji, Fixed Point Theory. Springer Monogr. Math., Springer-Verlag, New York, 2003. MR1987179 (2004d:58012)
5. M.G. Krěi{dotless}n and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space. Amer. Math. Soc. Translation 1950 (1950), no. 26. MR0038008 (12:341b)
6. B. Lins, Asymptotic behavior and Denjoy-Wolff theorems for Hilbert metric nonexpansive maps, Ph.D. thesis, Rutgers University, New Brunswick, NJ, 2007. MR2711887
7. K. L̈owner, ̈Uber monotone Matrixfunktionen. Math. Z. 38 (1934), 177-216. MR1545446
8. J. Mallet-Paret and R.D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators. Discrete Contin. Dyn. Syst. 8 (2002), 519-563. MR1897866 (2003c:47088)
9. J. Mallet-Paret and R.D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index. J. Fixed Point Theory Appl. 7(1) (2010), 103-143. MR2652513 (2011j:47148)
10. S. Mouton, On spectral continuity of positive elements. Studia Math. 174(1) (2006), 75-84. MR2239814 (2007c:46047)
11. S. Mouton, A condition for spectral continuity of positive elements. Proc. Amer. Math. Soc. 137(5) (2009), 1777-1782. MR2470837 (2009m:46072)
12. G.J. Murphy, Continuity of the spectrum and spectral radius. Proc. Amer. Math. Soc. 82(4) (1981), 619-621. MR614889 (82h:46066)
13. J.D. Newburgh, The variation of spectra. Duke Math. J. 18 (1951), 165-176. MR0051441 (14:481b)
14. R.D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem. In E. Fadell and G. Fournier, editors, Fixed Point Theory. Lecture Notes in Math., 886, 309-331, Springer-Verlag, 1981. MR643014 (83b:47068)
15. R.D. Nussbaum, The fixed point index and some applications. Séminaire de Mathématiques Supérieures, 94. Presses de l'Université de Montréal, Montreal, QC, 1985. MR791918 (87a:47085)
16. R.D. Nussbaum, Convexity and log convexity for the spectral radius. Linear Algebra Appl. 73 (1986), 59-122. MR818894 (87g:15026)
17. R.D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc. 391 (1988), 1-137. MR961211 (89m:47046)
18. C.E. Rickart, General Theory of Banach Algebras. Univ. Ser. in Higher Math., D. van Nostrand Co., Inc., Princeton, NJ, 1960. MR0115101 (22:5903)
19. R.T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, Princeton, NJ, 1997. MR1451876 (97m:49001)
20. A. Schrijver, Theory of Linear and Integer Programming. John Wiley, Chichester, 1986. MR874114 (88m:90090)
21. S. Straszewicz, ̈Uber exponierte Punkte abgeschlossener Punktmengen. Fund. Math. 24 (1935), 139-143.
22. A.C. Thompson, On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. 14 (1963), 438-443. MR0149237 (26:6727)