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).