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 ${\ell }_p\left(\mathfrak{L}\right),$ $c_0\left(\mathfrak{L}\right),$ $c\left(\mathfrak{L}\right),$ and ${\ell }_{\infty }\left(\mathfrak{L}\right),$ where $\mathfrak{L}=\left({\mathfrak{l}}_{v,k}\right)$ is a regular matrix defined by ${\mathfrak{l}}_{v,k}=\left\{\begin{array}{c}\left({\mathfrak{l}}_k/{\mathfrak{l}}_{v+2}-\left(v+2\right)\right),0\le k\le v,\hfill \\ 0,k>v,\hfill \end{array}\right.$ for all $v,k=0,1,2,\cdots$ , where $\mathfrak{l}=\left({\mathfrak{l}}_k\right)$ is a sequence of Leonardo numbers. We study their topological and inclusion relations and construct Schauder bases of the sequence spaces ${\ell }_p\left(\mathfrak{L}\right),$ $c_0\left(\mathfrak{L}\right),$ and $c\left(\mathfrak{L}\right).$ Besides, $\alpha$ -, $\beta$ - and $\gamma$ -duals of the aforementioned spaces are computed. We state and prove results of the characterization of the matrix classes between the sequence spaces ${\ell }_p\left(\mathfrak{L}\right),$ $c_0\left(\mathfrak{L}\right),$ $c\left(\mathfrak{L}\right),$ and ${\ell }_{\infty }\left(\mathfrak{L}\right)$ to any one of the spaces ${\ell }_1,$ $c_0,$ $c$ , and ${\ell }_{\infty }.$ Finally, under a definite functional $\rho$ and a weighted sequence of positive reals $r$ , we introduce new sequence spaces ${\left(c_0\left(\mathfrak{L},r\right)\right)}_{\rho }$ and ${\left({\ell }_p\left(\mathfrak{L},r\right)\right)}_{\rho }$ . 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.