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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.3 Temuco  2010 

CUBO A Mathematical Journal Vol.12, N°03, (121-138). October 2010


Calculations in New Sequence Spaces and Application to Statistical Convergence



LMAH Université du Havre, BP 4006 IUT Le Havre, 76610 Le Havre. France email:

Department of Mathematics, University of Niš, Videgradska 33, 18000 Niš, Serbia email:


In this paper we recall recent results that are direct consequences of the fact that (w(λ) ,w(λ)) is a Banach algebra. Then we define the set Wτ = Dτw and characterize the sets Wτ (A) where A is either of the operators Δ, ∑, Δ(λ), or C(λ). Afterwardswe consider the sets [A1,A2]Wτ of all sequences X such that A1 (λ)(|A2(μ) X|) Wτ where A1 and A2 are of the form C(ξ), C+ (ξ), Δ(ξ), or Δ+ (ξ) and it is given necessary conditions to get |A1 (λ),A2(μ)| Wτ in the form Wξ. Finally we apply the previous results to statistical convergence. So we have conditions to have xk L(S(A)) where A is either of the infinite matrices D1/τC(λ)C(μ), D1/τΔ(λ)Δ(μ), D1/τΔ(λ)C(μ). We also give conditions to have xk 0(S(A)) where A is either of the operators D1/τC+ (λ)Δ(μ), D1/τC(λ)C(μ), D1/τC+ (λ)C+(μ), or D1/τΔ(λ)C+(μ).

Key words and phrases: Banach algebra, statistical convergence, A−statistical convergence, infinite matrix.


Recordamos resultados recientes que son consecuencia directa del hecho de que (w(λ), w(λ)) es una algebra de Banach. Entonces nosotros definimos el conjunto = Dτwy caracterizamos los conjuntos (A) donde A es uno de los siguientes operadores Δ, ∑, Δ(λ), o C(λ). Después consideramos los conjuntos[A1,A2] de todas las sucesiones X tal que A1 (λ)(|A2(μ) X|) dondeA1 y A2 son de la forma C(ξ), C+ (ξ), Δ(ξ), or Δ+ (ξ) y son dadas condiciones necesarias para obtener |A1 (λ),A2(μ)| en la forma Wξ. Finalmente, aplicamos los resultados previos para tener xk L(S(A)) donde A es una de las matrices infinitas D1/τC(λ)C(μ), D1/τΔ(λ)Δ(μ), D1/τΔ(λ)C(μ) . Nosotros también damos condiciones para tener xk 0(S(A)) donde A es uno de los operadores D1/τC+ (λ)Δ(μ), D1/τC(λ)C(μ), D1/τC+ (λ)C+(μ), o D1/τΔ(λ)C+(μ).

Math. Subj. Class.: 40C05, 40F05, 40J05, 46A15.


1Supported by Grant No. 144003 of the Ministry of Science, Technology and Development, Republic of Serbia


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