Matrix transformations and compact operators on some new mth-order difference sequences

We define some new sets of sequences the mth-order differences of which are α-bounded, convergent and convergent to zero, and apply the general methods in [E. Malkowsky, V. Rakočević, On matrix domains of triangles, Appl. Math. Comput. 189 (2) (2007) 1146-1163] to give Schauder bases for the latter two, determine their β-duals and characterize matrix transformations on them. Our results generalize those in [B. de Malafosse, The Banach algebra Sα and applications, Acta Sci. Math. (Szeged) 70 (1-2) (2004) 125-145] and improve those in [E. Malkowsky, S.D. Parashar, Matrix transformations in spaces of bounded and convergent difference sequences of order m, Analysis 17 (1997) 87-97]. We also establish identities and estimates for the Hausdorff measure of non-compactness of matrix operators from our spaces into the spaces of bounded, convergent and null sequences, and characterize the respective classes of compact operators. Some of these results generalize those in [E. Malkowsky, V. Rakočević, The measure of non-compactness of linear operators between spaces of mth-order difference sequences, Stud. Sci. Math. Hungar. 33 (1999) 381-391].