λ− Core of a Sequence and Related Inequalities

Abstract The sequence spaces c and cs have recently been introduced in [13] and [9], respectively, as the sets of all sequences whose Λ− transforms are in the spaces c and cs, respectively. The main purpose of this study is to introduce the new type cores, KΛ− core and SΛ− core, of a real valued sequence and also determine necessary and sufficient conditions for a matrix A to satisfy KΛ − core(Ax) ⊆ K− core(x), KΛ− core(Ax) ⊆ σ− core(x), KΛ− core(Ax) ⊆ st− core(x), and SΛ− core(Ax) ⊆ K− core(x), SΛ− core(Ax) ⊆ σ− core(x), SΛ− core(Ax) ⊆ st− core(x), for all x ∈ `∞.