Calculations in New Sequence Spaces and Application to Statistical Convergence
Author
De Malafosse, Bruno
Rakočević, Vladimir
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 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).