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- Title
General methods of convergence and summability.
- Authors
García-Pacheco, Francisco Javier; Kama, Ramazan; Listán-García, María del Carmen
- Abstract
This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of N and study the space of convergence associated with the filter. We notice that c (X) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then ℓ ∞ (X) is a space of convergence associated with any free ultrafilter of N ; and that if X is not complete, then ℓ ∞ (X) is never the space of convergence associated with any free filter of N . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that ℓ ∞ (X) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then c (X) is a space of convergence through a certain class of such operators; and that if X is not complete, then c (X) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set HB (lim) : = { T ∈ B (ℓ ∞ (X) , X) : T | c (X) = lim and ∥ T ∥ = 1 } and prove that HB (lim) is a face of B L X 0 if X has the Bade property, where L X 0 : = { T ∈ B (ℓ ∞ (X) , X) : c 0 (X) ⊆ ker (T) } . Finally, we study the multipliers associated with series for the above methods of convergence.
- Subjects
SUMMABILITY theory; FINITE, The
- Publication
Journal of Inequalities & Applications, 2021, Vol 2021, Issue 1, p1
- ISSN
1025-5834
- Publication type
Article
- DOI
10.1186/s13660-021-02587-x