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2012 | 10 | 2 | 619-645
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Superposition operator on the space of sequences almost converging to zero

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We study the superposition operator f on on the space ac 0 of sequences almost converging to zero. Conditions are derived for which f has a representation of the form f x = a+bx +g x, for all x ∈ ac 0 with a = f 0, b ∈ D(ac 0), g a superposition operator from ℓ∞ into I(ac 0), D(ac 0) = {z: zx ∈ ac 0 for all x ∈ ac 0}, and I(ac 0) the maximal ideal in ac 0. If f is generated by a function f of a real variable, then f is linear. We consider the conditions for which a bounded function f generates f acting on ac 0 and the conditions for which there exists a sequence y ∈ ac 0 such that y − f y ∈ ac 0. In terms of f, criteria for the boundedness, continuity, and sequential σ(ac 0ℓ1)-continuity of f on ac 0 are given. It is shown that the continuity of f is equivalent to the weak sequential continuity. Finally, properties of spaces D(ac 0) and I(ac 0) are studied, and in particular it is established that the inclusion I(ac 0) ⊕ {λe: λ ∈ ℝ} ⊂ D(ac 0) is proper, where e = (1, 1, …). By means of D(ac 0), a number of Banach-Mazur limit properties are derived.
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Bibliografia
  • [1] Alekhno E., On weak continuity of a superposition operator on the space of all bounded sequences, Methods Funct. Anal. Topology, 2005, 11(3), 207–216
  • [2] Alekhno E.A., Some special properties of Mazur's functionals. II, In: Proceedings of AMADE-2006, 2, Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, 2006, 17–23 (in Russian)
  • [3] Alekhno E.A., Weak continuity of a superposition operator in sequence spaces, Vladikavkaz. Mat. Zh., 2009, 11(2), 6–18 (in Russian)
  • [4] Alekhno E.A., Zabreĭko P.P., On the weak continuity of the superposition operator in the space L ∞, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk, 2005, 2, 17–23 (in Russian)
  • [5] Aliprantis C.D., Burkinshaw O., Positive Operators, Pure Appl. Math., 119, Academic Press, Orlando, 1985
  • [6] Appell J., Zabrejko P.P., Nonlinear Superposition Operators, Cambridge Tracts in Math., 95, Cambridge University Press, Cambridge, 1990
  • [7] Bennett G., Kalton N.J., Consistency theorems for almost convergence, Trans. Amer. Math. Soc., 1974, 198, 23–43 http://dx.doi.org/10.1090/S0002-9947-1974-0352932-X
  • [8] Dunford N., Schwartz J.T., Linear Operators I. General Theory, Pure Appl. Math., 7, Interscience, New York, 1958
  • [9] Gillman L., Jerison M., Rings of Continuous Functions, The University Series in Higher Mathematics, Van Nostrand, Princeton, 1960
  • [10] Jerison M., The set of all generalized limits of bounded sequences, Canad. J. Math., 1957, 9(1), 79–89 http://dx.doi.org/10.4153/CJM-1957-012-x
  • [11] Luxemburg W.A.J., Nonstandard hulls, generalized limits and almost convergence, In: Analysis and Geometry, Bibliographisches Inst., Mannheim, 1992, 19–45
  • [12] Sucheston L., Banach limits, Amer. Math. Monthly, 1967, 74(3), 308–311 http://dx.doi.org/10.2307/2316038
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bwmeta1.element.doi-10_2478_s11533-011-0135-7
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