On Some Generalized B-m-Difference Riesz Sequence Spaces and Uniform Opial Property

Abstract

We define the new generalized difference Riesz sequence spaces r(infinity)(q)(p, B-m), r(c)(q)(p, B-m), and r(0)(q)(p, B-m) which consist of all the sequences whose B-m-transforms are in the Riesz sequence spaces r(infinity)(q)(p), r(c)(q)(p), and r(0)(q)(p), respectively, introduced by Altay and Basar (2006). We examine some topological properties and compute the alpha-, beta-, and gamma-duals of the spaces r(infinity)(q)(p, B-m), r(c)(q)(p, B-m), and r(0)(q)(p, B-m). Finally, we determine the necessary and sufficient conditions on the matrix transformation from the spaces r(infinity)(q)(p, B-m), r(c)(q)(p, B-m), and r(0)(q)(p, B-m) to the spaces l(infinity) and c and prove that sequence spaces r(0)(q)(p, B-m) and r(c)(q)(p, B-m) have the uniform Opial property for p(k) <= 1 for all k is an element of N.