Matrix transformations of sequence spaces which preserve convergence, limit, summability and sums

Abstract

All of the classical sequence space are ℓ͚, c, cₒ, ℓᵨ, (l<p<∞), ℓ, cs, bs, bvₒ and bv. Among these sequence spaces, ℓ͚ is the largest one. The main objective of this research is to characterize convergence preserving matrix transformations, limit preserving matrix transformations, summability preserving matrix transformations and sum preserving matrix transformations from each of ℓ͚, c, cₒ, ℓᵨ, (l<p<∞), ℓ, cs, bs, bvₒ and bv. Into ℓ͚. In this study, all such matrices can be completely characterized in terms of their entries and the β-duals of some sequence space with the exception of the following two types of matrices: convergence preserving matrix transformations and limit preserving matrix transformations from bs into ℓ͚. However, these two types of matrices are characterized in the class of Kᵣ-matrices and the class of infinite matrices A for which ∑ [superscript ∞] [subscript k = l] | A [subscript nk] – A [subscript n (k+l)]l converges uniformly on n = 1, 2, 3,…… Included in this research, characterizations of the above four types of matrix transformations from each of the following standard sequence spaces into ℓ͚ are also given: the space of all sequences, the space of all finite sequences and the Cesaro sequence spaces.