Domain of Padovan $q$-Difference Matrix in Sequence Spaces $\ell_p$ and $\ell_{\infty}$

Abstract

In this study, we construct the difference sequence spaces
$\ell_{p}\big(\mathcal{P}\nabla_q^2\big)=(\ell_p)_{\mathcal{P}\nabla_q^2},$
$1\leq p\leq \infty,$ where $\mathcal{P}=(\varrho_{rs})$ is an
infinite matrix of Padovan numbers $\varrho_s$ defined by
$$
\varrho_{rs}=
\begin{cases}
\frac{\varrho_s}{\varrho_{r+5}-2} \quad & 0\leq s\leq r,\\
0\quad & s>r.
\end{cases}
$$
and $\nabla_q^2$ is a $q$-difference operator of second order. We
obtain some inclusion relations, topological properties, Schauder
basis and $\alpha$-, $\beta$- and $\gamma$-duals of the newly
defined space. We characterize certain matrix classes from the space
$\ell_p\big(\mathcal{P}\nabla^2_q\big)$ to any one of the space
$\ell_{1},$ $c_0$, $c$ or $\ell_{\infty}$. We examine some geometric
properties and give certain estimation for von-Neumann Jordan
constant and James constant of the space $\ell_{p}(\mathcal{P})$.
Finally, we estimate upper bound for Hausdorff matrix as a mapping
from $\ell_p$ to $\ell_p(\mathcal{P}).$