Superposition operator on the space of sequences almost converging to zero

• Autorzy: Egor Alekhno
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Superposition operator; Sequence space; Almost convergence; Banach-Mazur limit

Kategorie tematyczne

• 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytski{\u\i}, Uryson, etc.)
• 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
• 47J05: Equations involving nonlinear operators (general)
• 46B45: Banach sequence spaces

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 619-645

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Egor Alekhno

• Belarussian State University

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)
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• [8] Dunford N., Schwartz J.T., Linear Operators I. General Theory, Pure Appl. Math., 7, Interscience, New York, 1958
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• [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
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Identyfikatory

DOI

10.2478/s11533-011-0135-7