A study of novel telephone sequence spaces and some geometric properties

Abstract

Abstract For a nonnegative integer k, let T k $\mathcal {T}_{k}$ denote the kth telephone number. Consider the matrix T = ( T k r ) $\mathfrak{T}=(\mathfrak {T}_{kr})$ given by T k r = { r T r − 1 T k + 1 − 1 , 0 ≤ r ≤ k , 0 , r > k , $$\begin{aligned} \mathfrak {T}_{kr}= \textstyle\begin{cases} \dfrac{r\mathcal {T}_{r-1}}{\mathcal {T}_{k+1}-1}, & 0\leq r\leq k, \\ 0 , & r>k, \end{cases}\displaystyle \end{aligned}$$ where k , r = 0 , 1 , 2 , … $k,r=0,1,2,\dots $ Using the matrix T $\mathfrak{T}$ , we define matrix domains T p : = ( ℓ p ) T $\mathscr{T}_{p}:=(\ell _{p})_{\mathfrak{T}}$ for 0 < p < ∞ $0 < p < \infty $ and T ∞ : = ( ℓ ∞ ) T $\mathscr{T}_{\infty}:=(\ell _{\infty})_{\mathfrak{T}}$ . In this context, we construct a Schauder basis for 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 T p $\mathscr{T}_{p}$ and T ∞ $\mathscr{T}_{\infty}$ to traditional spaces such as ℓ ∞ $\ell _{\infty}$ , c, c 0 $c_{0}$ , and ℓ 1 $\ell _{1}$ . Additionally, we explore several geometric attributes of the spaces T p $\mathscr{T}_{p}$ and T ∞ $\mathscr{T}_{\infty}$ , including the approximation property, D-P property, Hahn–Banach extension property, and rotundity.

Keywords

• Sequence spaces
• Telephone numbers
• Duals
• Matrix transformations
• Geometric properties