A study of novel telephone sequence spaces and some geometric properties

For a nonnegative integer k, let ${\mathcal{T}}_k$ $\mathcal {T}_{k}$ denote the kth telephone number. Consider the matrix $\mathfrak{T}=({\mathfrak{T}}_{kr})$ $\mathfrak{T}=(\mathfrak {T}_{kr})$ given by ${\mathfrak{T}}_{kr}=\{\begin{array}{cc}\frac{r{\mathcal{T}}_{r-1}}{{\mathcal{T}}_{k+1}-1},\hfill & 0\le r\le k,\hfill \\ 0,\hfill & r>k,\hfill \end{array}$ \(\begin{aligned} \mathfrak {T}_{kr}= \textstyle\begin{cases} \dfrac{r\mathcal {T}_{r-1}}{\mathcal {T}_{k+1}-1}, & 0\leq r\leq k, \\ 0 , & r>k, \end{cases}\displaystyle \end{aligned}\) where $k,r=0,1,2,\dots$ $k,r=0,1,2,\dots $ Using the matrix $\mathfrak{T}$ $\mathfrak{T}$ , we define matrix domains ${\mathcal{T}}_p:={({\ell }_p)}_{\mathfrak{T}}$ $\mathscr{T}_{p}:=(\ell _{p})_{\mathfrak{T}}$ for $0<p<\mathrm{\infty }$ $0 < p < \infty $ and ${\mathcal{T}}_{\mathrm{\infty }}:={({\ell }_{\mathrm{\infty }})}_{\mathfrak{T}}$ $\mathscr{T}_{\infty}:=(\ell _{\infty})_{\mathfrak{T}}$ . In this context, we construct a Schauder basis for ${\mathcal{T}}_p$ $\mathscr{T}_{p}$ and identify their α-, β-, and γ-duals. We also derive various results pertaining to matrix transformations from the spaces ${\mathcal{T}}_p$ $\mathscr{T}_{p}$ and ${\mathcal{T}}_{\mathrm{\infty }}$ $\mathscr{T}_{\infty}$ to traditional spaces such as ${\ell }_{\mathrm{\infty }}$ $\ell _{\infty}$ , c, $c_0$ $c_{0}$ , and ${\ell }_1$ $\ell _{1}$ . Additionally, we explore several geometric attributes of the spaces ${\mathcal{T}}_p$ $\mathscr{T}_{p}$ and ${\mathcal{T}}_{\mathrm{\infty }}$ $\mathscr{T}_{\infty}$ , including the approximation property, D-P property, Hahn–Banach extension property, and rotundity.