Calculations in new sequence spaces and application to statistical convergence

Abstract

In this paper we recall recent results that are direct consequences of the fact that (ω_∞ (λ),ω_∞ (λ)) is a Banach algebra. Then we define the set W_Ï„ = D_τω_∞ and characterize the sets W_Ï„ (A) where A is either of the operators ∆, Σ, ∆(λ), or C(λ). Afterwards we consider the sets [A₁, A₂]_WÏ„ of all sequences X such that A₁ (λ) (Ç€A₂(µ)XÇ€) ∈ W_Ï„ where A₁ and A₂ are of the form C(ξ), C⁺ (ξ), ∆(ξ), or ∆⁺ (ξ) and it is given necessary conditions to get [A₁ (λ), A₂ (µ)]_WÏ„ in the form W_ξ. Finally we apply the previous results to statistical convergence. So we have conditions to have x_k → L(S(A)) where A is either of the infinite matrices D_1/Ï„C(λ)C(µ), D_1/τ∆(λ)∆(µ), D_1/τ∆(λ)C(µ). We also give conditions to have x_k → 0(S(A)) where A is either of the operators D_1/Ï„C⁺ (λ)∆(µ), D_1/Ï„C⁺ (λ)C(µ), D_1/Ï„C⁺(λ)C⁺(µ), or D_1/τ∆(λ)C⁺(µ).