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 ℓpL,c0L,cL, and ℓ∞L, where L=lv,k is a regular matrix defined by lv,k=lk/lv+2−v+2, 0≤k≤v,0, k>v, for all v,k=0,1,2,⋯, where l=lk is a sequence of Leonardo numbers. We study their topological and inclusion relations and construct Schauder bases of the sequence spaces ℓpL,c0L, and cL. 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 ℓpL,c0L,cL, and ℓ∞L to any one of the spaces ℓ1,c0,c, and ℓ∞. Finally, under a definite functional ρ and a weighted sequence of positive reals r, we introduce new sequence spaces c0L,rρ and ℓpL,rρ. 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.