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Spectral and asymptotic properties of resolvent-dominated operators

Part of: General theory of linear operators Groups and semigroups of linear operators, their generalizations and applications Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

Frank Räbiger  and
Manfred P. H. Wolff
Manfred P. H. Wolff
Affiliation:
Mathematisches Institut Universität Tübigen Auf der Morgenstelle 10 D-72076 Tübingen Germany e-mail: [email protected] e-mail: [email protected]
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Abstract

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Let A and B be (not necessarily bounded) linear operators on a Banach lattice E such that |(s – B)-1x|≤ (s – A)-1|x| for all x in E and sufficiently large s ∈ R. The main purpose of this paper is to investigate the relation between the spectra σ(B) and σ(A) of B and A, respectively. We apply our results to study asymptotic properties of dominated C0-semigroups.

Keywords

Banach latticedominated pseudo-resolventsdominated C0-semigroupsperipheral spectrumessential spectrumalmost periodicityuniform ergodicity.

MSC classification

Secondary: 47A10: Spectrum, resolvent 47B65: Positive operators and order-bounded operators 47D03: Groups and semigroups of linear operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 2 , April 2000 , pp. 181 - 201
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Aliprantis, C. D. and Burkinshaw, O., Positive operators (Academic Press, Orlando, 1985).Google Scholar
[2]Andreu, F. and Mazon, J. M., ‘On the boundary spectrum of dominated C 0-semigroups’, Semigroup Forum 38 (1989), 129–139.CrossRefGoogle Scholar
[3]Arendt, W., ‘Kato's inequality. A characterization of generators of positive semigroups’, Proc. Roy. Irish Acad. 84 (1984), 155–174.Google Scholar
[4]Arendt, W., ‘Resolvent positive operators’, Proc. London Math. Soc. (3) 54 (1987), 321–349.CrossRefGoogle Scholar
[5]Arendt, W. and Batty, C. J. K., ‘Domination and ergodicity for positive semigroups’, Proc. Amer. Math. Soc. 114 (1992), 743–747.CrossRefGoogle Scholar
[6]Arendt, W., ‘Absorption semigroups and Dirichlet boundary conditions’, Math. Ann. 292 (1993), 427–448.CrossRefGoogle Scholar
[7]Arendt, W. and Rhandi, A., ‘Perturbation of positive semigroups’, Arch. Math. 56 (1991), 107–119.CrossRefGoogle Scholar
[8]Batty, C. J. K. and Phóng, Vũ Quoc, ‘Stability of individual elements under one-parameter semigroups’, Trans. Amer. Math. Soc. 322 (1990), 805–818.CrossRefGoogle Scholar
[9]Clément, Ph., Heijmans, H. J. A. M., Angenent, S., Van Duijn, C. J. and de Pagter, B., One-parameter semigroups (North-Holland, Amsterdam, 1987).Google Scholar
[10]Emel'yanov, E. Yu., Kohler, U., Räbiger, F. and Wolff, M. P. H., ‘Stability and almost periodicity of dominated semigroups of operators’, preprint.Google Scholar
[11]Esterle, J., Strouse, E. and Zouakia, F., ‘Stabilité asymptotique de certains semigroups d'opérateurs et ideaux primaires’, J. Operator Theory 28 (1992), 203–228.Google Scholar
[12]Greiner, G., ‘Zur Perron-Frobenius Theorie stark stetiger Halbgruppen’, Math. Z. 117 (1981), 401–423.CrossRefGoogle Scholar
[13]Greiner, G., ‘Spektrum und Asymptotik stark stetiger Halbgruppen positiver Operatoren’, Sitzungsber. Heidelb. Akad. Wiss. Math.-Naturwiss. Klasse (1982), 55–80CrossRefGoogle Scholar
[14]Hille, E. and Phillips, R. S., Functional analysis and semi-groups, Colloquium Publications 31 (Amer. Math. Soc., Providence, 1957).Google Scholar
[15]Krengel, U., Ergodic theorems (deGruyter, Berlin, 1985).CrossRefGoogle Scholar
[16]Loomis, L. H., An introduction to abstract harmonic analysis (Van Nostrand, Princeton, 1953).Google Scholar
[17]Martinez, J. and Mazon, J. M., ‘Quasi-compactness of dominated positive operators and C 0-semigroups’, Math. Z. 207 (1991), 109–120.CrossRefGoogle Scholar
[18]Meyer-Nieberg, P., Banach lattices, (Springer, Berlin, 1991).CrossRefGoogle Scholar
[19]Nagel, R. (ed.), One-parameter semigroups of positive operators (Springer, Berlin, 1986).Google Scholar
[20]Räbiger, F., ‘Stability and ergodicity of dominated semigroups, I. The uniform case’, Math. Z. 214 (1993), 43–54.CrossRefGoogle Scholar
[21]Räbiger, F. and Wolff, M. P. H., ‘On the approximation of positive operators and the behaviour of the spectra of the approximants’, Integral Equations Operator Theory 28 (1997), 72–86.CrossRefGoogle Scholar
[22]Räbiger, F., ‘Spectral and asymptotic properties of dominated operators’, J. Austral. Math. Soc. (Ser. A) 63 (1997), 16–31.CrossRefGoogle Scholar
[23]Schaefer, H. H., Banach lattices and positive operators (Springer, Berlin, 1974).CrossRefGoogle Scholar
[24]Schep, A. R., ‘Weak Kato-inequalities and positive semigroups’, Math. Z. 190 (1985), 303–314.CrossRefGoogle Scholar
[25]Shaw, S.-Y., ‘Uniform ergodic theorems for locally integrable semigroups and pseudo-resolvents’, Proc. Amer. Math. Soc. 98 (1986), 61–67.Google Scholar
[26]Voigt, J., ‘Absorption semigroups, their generators, and Schrödinger semigroups’, J. Funct. Anal. 67 (1982), 167–205.CrossRefGoogle Scholar
[27]Vũ, Quôc Phóng, ‘Theorems of Katznelson-Tzafriri type for semigroups of operators’, J. Funct. Anal. 103 (1992), 74–84.Google Scholar
[28]Phóng, Vũ Quôc and Lyubich, Yu. I., ‘A spectral criterion for almost periodicity of one-parameter semigroups’, J. Soviet. Math. 48 (1990), 644–647.Google Scholar
[29]Yosida, K., Functional analysis, 6th edition (Springer, Berlin, 1980).Google Scholar