We define some new sets of sequences the
mth-order differences of which are
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-bounded, convergent and convergent to zero, and apply the general methods in [E. Malkowsky, V. Rako
10d;evi
107;, 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
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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. Rako
10d;evi
107;, The measure of non-compactness of linear operators between spaces of
mth-order difference sequences, Stud. Sci. Math. Hungar. 33 (1999) 3
81–391].