Telephone sequence spaces and associated theory of matrix transformation and compactness

Abstract

Given a non-negative integer k, T-k be the k(th) telephone number. Let us define a matrix T= (T-k ,T-m ) with entries T-k,T-m ={(m+1) T- m /Tk + 2- 1 ,0 <= m <= k, 0( ), m > k. 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 alpha-, beta-, and gamma-duals are characterized. Additionally, we examine matrix mappings from T-c to classical sequence spaces & ell;(infinity), c, c(0) , and & ell;(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.