The Generalized Difference Operator Δ i 3 of Order Three and Its Domain in the Sequence Spaces ℓ 1 and b v

Abstract

Most recently, the generalized difference operator Δ i 3 of order three was defined and its domain in Hahn sequence space h was calculated. In this paper, the spaces ℓ 1 Δ i 3 and b v Δ i 3 are introduced as the domain of generalized difference operator Δ i 3 of order three in the sequence spaces ℓ 1 and b v . Then, some topological properties of ℓ 1 Δ i 3 and b v Δ i 3 are given, and some inclusion relations are shown. Additionally, algebraic dual, α − , β − , and γ − dual spaces of ℓ 1 Δ i 3 and b v Δ i 3 are computed. In the last section, the classes μ Δ i 3 : λ and λ : μ Δ i 3 of matrix transformations are characterized, where μ = ℓ 1 , b v and λ = c , c 0 , ℓ 1 , ℓ ∞ , b s , c s , b v , h .