Telephone sequence spaces and associated theory of matrix transformation and compactness

Given a non-negative integer k, T_k be the kᵗʰ telephone number. Let us define a matrix T=( Tk,m ) with entries for k,m=0,1,2,⋯. Based on T, the matrix domains T_0 and T_c are defined. Schauder bases are constructed for these spaces, and their α-, β-, and γ-duals are characterized. Additionally, we examine matrix mappings from T_c to classical sequence spaces l_∞ , c, c_0 , and l_1 . Moreover, the Hausdorff measure of non-compactness (Hmnc) is applied to derive necessary and sufficient conditions for the compactness of matrix operators given on T_0 . Tk,m={ [ ( (m+1 ) T_m / Tₖ₊₂−1 ) , ,, 0≤m≤k, ; 0 , ,, m>k. ]