A new sequence space isomorphic to the space ℓ(p) and compact operators

Abstract

As a major issue in this work, we present the paranormed sequence space ℓ(u,v,p;B) consisting of all sequences whose R-transforms are in the linear space ℓ(p) introduced by Maddox [Quart. J. Math. Oxford (2), 18(1967), 345--355], where B=B(r,s) denotes double sequential band matrix provided that (r_{n})_{n=0}^{∞} and (s_{n})_{n=0}^{∞} are given convergent sequences of positive real numbers. For this purpose, we have used the generalized weighted mean G and double sequential band matrix B. Meanwhile, we have also presented the basis of this space and computed its α-, β- and γ-duals. Then, we have characterized the classes of matrix mappings from ℓ(u,v,p;B) to ℓ_{∞}, c and c₀. In conclusion, in order to characterize some classes of compact operators given by matrices on the space ℓ_{p}(u,v,B)  (1≤p<∞), we have applied the Hausdorff measure of noncompactness.