On New Banach Sequence Spaces Involving Leonardo Numbers and the Associated Mapping Ideal

Abstract

In the present study, we have constructed new Banach sequence spaces ℓ p L , c 0 L , c L , and ℓ ∞ L , where L = l v , k is a regular matrix defined by l v , k = l k / l v + 2 − v + 2 , 0 ≤ k ≤ v , 0 , k > v , for all v , k = 0 , 1 , 2 , ⋯ , where l = l k is a sequence of Leonardo numbers. We study their topological and inclusion relations and construct Schauder bases of the sequence spaces ℓ p L , c 0 L , and c L . Besides, α -, β - and γ -duals of the aforementioned spaces are computed. We state and prove results of the characterization of the matrix classes between the sequence spaces ℓ p L , c 0 L , c L , and ℓ ∞ L to any one of the spaces ℓ 1 , c 0 , c , and ℓ ∞ . Finally, under a definite functional ρ and a weighted sequence of positive reals r , we introduce new sequence spaces c 0 L , r ρ and ℓ p L , r ρ . We present some geometric and topological properties of these spaces, as well as the eigenvalue distribution of ideal mappings generated by these spaces and s -numbers.