Superposition operator on the space of sequences almost converging to zero

Egor Alekhno

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 619-645
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Egor Alekhno. "Superposition operator on the space of sequences almost converging to zero." Open Mathematics 10.2 (2012): 619-645. <http://eudml.org/doc/269129>.

@article{EgorAlekhno2012,
abstract = {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.},
author = {Egor Alekhno},
journal = {Open Mathematics},
keywords = {Superposition operator; Sequence space; Almost convergence; Banach-Mazur limit; superposition operator; almost convergence},
language = {eng},
number = {2},
pages = {619-645},
title = {Superposition operator on the space of sequences almost converging to zero},
url = {http://eudml.org/doc/269129},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Egor Alekhno
TI - Superposition operator on the space of sequences almost converging to zero
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 619
EP - 645
AB - 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.
LA - eng
KW - Superposition operator; Sequence space; Almost convergence; Banach-Mazur limit; superposition operator; almost convergence
UR - http://eudml.org/doc/269129
ER -

References

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  1. [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 Zbl1100.47051
  2. [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. [3] Alekhno E.A., Weak continuity of a superposition operator in sequence spaces, Vladikavkaz. Mat. Zh., 2009, 11(2), 6–18 (in Russian) Zbl1324.46064
  4. [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. [5] Aliprantis C.D., Burkinshaw O., Positive Operators, Pure Appl. Math., 119, Academic Press, Orlando, 1985 
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  7. [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 Zbl0301.46005
  8. [8] Dunford N., Schwartz J.T., Linear Operators I. General Theory, Pure Appl. Math., 7, Interscience, New York, 1958 Zbl0084.10402
  9. [9] Gillman L., Jerison M., Rings of Continuous Functions, The University Series in Higher Mathematics, Van Nostrand, Princeton, 1960 Zbl0093.30001
  10. [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 Zbl0077.31004
  11. [11] Luxemburg W.A.J., Nonstandard hulls, generalized limits and almost convergence, In: Analysis and Geometry, Bibliographisches Inst., Mannheim, 1992, 19–45 Zbl0765.26012
  12. [12] Sucheston L., Banach limits, Amer. Math. Monthly, 1967, 74(3), 308–311 http://dx.doi.org/10.2307/2316038 Zbl0148.12202

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