Stability and almost periodicity of asymptotically dominated semigroups of positive operators

Emel’yanov, E.; Kohler, U.; Räbiger, F.; Wolff, M.

doi:10.1090/S0002-9939-01-05835-X

Stability and almost periodicity of asymptotically dominated semigroups of positive operators
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by E. Yu. Emel’yanov, U. Kohler, F. Räbiger and M. P. H. Wolff PDF

Proc. Amer. Math. Soc. 129 (2001), 2633-2642 Request permission

Abstract:

We discuss conditions such that strong stability and strong asymptotic compactness of a (discrete or continuous) semiflow defined on a subset in the positive cone of an ordered Banach space is preserved under asymptotic domination. This is used to show that on a Banach lattice with order continuous norm strong stability and almost periodicity of a (discrete or strongly continuous) semigroup of positive operators is preserved under asymptotic domination.

References

Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
F. Andreu and J. M. Mazón, On the boundary spectrum of dominated $C_0$-semigroups, Semigroup Forum 38 (1989), no. 2, 129–139. Semigroups and differential operators (Oberwolfach, 1988). MR 976198, DOI 10.1007/BF02573226
W. Arendt and C. J. K. Batty, Domination and ergodicity for positive semigroups, Proc. Amer. Math. Soc. 114 (1992), no. 3, 743–747. MR 1072082, DOI 10.1090/S0002-9939-1992-1072082-3
Charles J. K. Batty and Derek W. Robinson, Positive one-parameter semigroups on ordered Banach spaces, Acta Appl. Math. 2 (1984), no. 3-4, 221–296. MR 753696, DOI 10.1007/BF02280855
Vicent Caselles, On the peripheral spectrum of positive operators, Israel J. Math. 58 (1987), no. 2, 144–160. MR 901175, DOI 10.1007/BF02785673
Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
Yurii I. Lyubich, Introduction to the theory of Banach representations of groups, Operator Theory: Advances and Applications, vol. 30, Birkhäuser Verlag, Basel, 1988. Translated from the Russian by A. Jacob [Andrei Iacob]. MR 1015717, DOI 10.1007/978-3-0348-9169-1
Josep Martínez, The essential spectral radius of dominated positive operators, Proc. Amer. Math. Soc. 118 (1993), no. 2, 419–426. MR 1128728, DOI 10.1090/S0002-9939-1993-1128728-5
Josep Martínez and José M. Mazón, Quasi-compactness of dominated positive operators and $C_0$-semigroups, Math. Z. 207 (1991), no. 1, 109–120. MR 1106817, DOI 10.1007/BF02571379
Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
B. de Pagter and A. R. Schep, Measures of noncompactness of operators in Banach lattices, J. Funct. Anal. 78 (1988), no. 1, 31–55. MR 937631, DOI 10.1016/0022-1236(88)90131-0
Frank Räbiger, Ergodic Banach lattices, Indag. Math. (N.S.) 1 (1990), no. 4, 483–488. MR 1106095, DOI 10.1016/0019-3577(90)90016-G
Frank Räbiger, Stability and ergodicity of dominated semigroups. I. The uniform case, Math. Z. 214 (1993), no. 1, 43–53. MR 1234596, DOI 10.1007/BF02572389
Frank Räbiger, Stability and ergodicity of dominated semigroups. II. The strong case, Math. Ann. 297 (1993), no. 1, 103–116. MR 1238409, DOI 10.1007/BF01459490
Frank Räbiger, Attractors and asymptotic periodicity of positive operators on Banach lattices, Forum Math. 7 (1995), no. 6, 665–683. MR 1359421, DOI 10.1515/form.1995.7.665
Frank Räbiger and Manfred P. H. Wolff, Spectral and asymptotic properties of dominated operators, J. Austral. Math. Soc. Ser. A 63 (1997), no. 1, 16–31. MR 1456587, DOI 10.1017/S144678870000029X
Frank Räbiger and Manfred P. H. Wolff, On the approximation of positive operators and the behaviour of the spectra of the approximants, Integral Equations Operator Theory 28 (1997), no. 1, 72–86. MR 1446832, DOI 10.1007/BF01198797
F. Räbiger and M.P.H. Wolff, Spectral and asymptotic properties of resolvent-dominated operators, J. Austral. Math. Soc. Ser. A 68 (2000), 181–201.
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6
Yau Chuen Wong and Kung Fu Ng, Partially ordered topological vector spaces, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1973. MR 0454581
A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021, DOI 10.1016/S0924-6509(08)70234-4