Calculations in New Sequence Spaces and Application to Statistical Convergence
De Malafosse, Bruno
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 Wτ = Dτω∞ and characterize the sets Wτ (A) where A is either of the operators ∆, Σ, ∆(λ), or C(λ). Afterwards we consider the sets [A1, A2]Wτ of all sequences X such that A1 (λ) (ǀA2(µ)Xǀ) ∈ Wτ 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)]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/τ∆(λ)∆(µ), D1/τ∆(λ)C(µ). We also give conditions to have xk → 0(S(A)) where A is either of the operators D1/τC+ (λ)∆(µ), D1/τC+ (λ)C(µ), D1/τC+(λ)C+(µ), or D1/τ∆(λ)C+(µ).