A study of novel telephone sequence spaces and some geometric properties

For a nonnegative integer k , let Tₖ T k denote the k th telephone number. Consider the matrix T=(Tₖᵣ) T = ( T k r ) given by aligned Tₖᵣ= cases rTᵣ₋₁Tₖ₊₁-1, & 0≤ r≤ k, \\ 0 , & r>k, cases aligned T k r = { r T r − 1 T k + 1 − 1 , 0 ≤ r ≤ k , 0 , r > k , where k,r=0,1,2, k , r = 0 , 1 , 2 , … Using the matrix T T , we define matrix domains Tₚ:=( ₚ)T T p : = ( ℓ p ) T for 0 < p < ∞ 0 < p < ∞ and T∞:=( ∞)T T ∞ : = ( ℓ ∞ ) T . In this context, we construct a Schauder basis for Tₚ T p and identify their α -, β -, and γ -duals. We also derive various results pertaining to matrix transformations from the spaces Tₚ T p and T∞ T ∞ to traditional spaces such as ∞ ℓ ∞ , c , c₀ c 0 , and ₁ ℓ 1 . Additionally, we explore several geometric attributes of the spaces Tₚ T p <