Nagel, Rainer; Räbiger, Frank Superstable operators on Banach spaces. (English) Zbl 0788.47003 Isr. J. Math. 81, No. 1-2, 213-226 (1993). The paper is devoted to study the spectrum of bounded linear operators on Banach spaces. The main result given in Theorem 3.7 is that the spectrum of a power bounded linear operator on a superreflexive Banach space, situated on the unit circle, is countable if and only if this operator is superstable. The latter notion is introduced by using ultrapower technique, see J. Stern [Trans. Amer. Math. Soc. 240, 231-252 (1978; Zbl 0402.03025), S. Heinrich, J. Reine Angew. Math. 313, 72- 104 (1980; Zbl 0412.46017) and C. W. Henson and L. C. Moore, Lect. Notes Math. 283, 27-112 (1983; Zbl 0511.46070)]. To obtain their main result, the authors have proved new interesting assertions in the theory of ultrapowers, in the geometry of Banach spaces and in the theory of operators on Banach lattices. Reviewer: A.A.Kilbas (Minsk) Cited in 1 ReviewCited in 6 Documents MSC: 47A10 Spectrum, resolvent 47A65 Structure theory of linear operators 46M07 Ultraproducts in functional analysis 03C20 Ultraproducts and related constructions Keywords:superstable operator; spectrum of bounded linear operators on Banach spaces; spectrum of a power bounded linear operator on a superreflexive Banach space; ultrapowers; geometry of Banach spaces; operators on Banach lattices Citations:Zbl 0402.03025; Zbl 0412.46017; Zbl 0511.46070 PDFBibTeX XMLCite \textit{R. Nagel} and \textit{F. Räbiger}, Isr. J. Math. 81, No. 1--2, 213--226 (1993; Zbl 0788.47003) Full Text: DOI References: [1] Arendt, W.; Batty, C. J.K., Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306, 837-852 (1988) · Zbl 0652.47022 · doi:10.2307/2000826 [2] Dowson, H. R., Spectral Theory of Linear Operators (1978), London-New York-San Francisco: Academic Press, London-New York-San Francisco · Zbl 0384.47001 [3] Heinrich, S., Ultraproducts in Banach space theory, J. Reine Angew. Math., 313, 72-104 (1980) · Zbl 0412.46017 [4] Henson, C. W.; Moore, L. C.; Hurd, A. E., Nonstandard analysis and the theory of Banach spaces, Nonstandard Analysis-Recent Developments, 27-112 (1983), Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, Berlin-Heidelberg-New York-Tokyo · Zbl 0511.46070 · doi:10.1007/BFb0065334 [5] Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis I (1979), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0416.43001 [6] Krengel, U., Ergodic Theorems (1985), Berlin-New York: De Gruyter, Berlin-New York · Zbl 0575.28009 [7] Lyubich, Yu. I.; Vu Quoc, Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88, 37-42 (1988) · Zbl 0639.34050 [8] Lyubich, Yu. I.; Phong, Vu Quoc, A spectral criterion of asymptotic almost periodicity for uniformly continuous representations of abelian semigroups, J. Soviet Math., 49, 1263-1266 (1990) · Zbl 0699.22006 · doi:10.1007/BF02209170 [9] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces I: Sequence Spaces (1977), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0362.46013 [10] Nagel, R., On the linear operator approach to dynamical systems, Semesterbericht Funktionalanalysis, Tübingen, 1990/91, 19, 121-140 (1991) [11] Schaefer, H. H., Banach Lattices and Positive Operators (1974), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0296.47023 [12] Semadeni, Z., Banach Spaces of Continuous Functions (1971), Warszawa: Polish Scientific Publishers, Warszawa · Zbl 0225.46030 [13] Sims, B., Ultra-Techniques in Banach Space Theory (1982), Kingston, Ontario: Queen’s University, Kingston, Ontario · Zbl 0611.46019 [14] Stern, J., Ultrapowers and local properties of Banach spaces, Trans. Amer. Math. Soc., 240, 231-252 (1978) · Zbl 0402.03025 · doi:10.2307/1998816 [15] Tomczak-Jaegermann, N., Banach-Mazur Distances and Finite-Dimensional Operator Ideals (1989), Essex: Longman Scientific & Technical, Essex · Zbl 0721.46004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.