Some $B$-Difference Sequence Spaces Derived by Using Generalized Means and Compact Operators

Abstract

This paper presents new sequence spaces $X(r,s,t,p;B)$ for $X\in \{l_{\infty }(p),c(p),c_0(p),l(p)\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete paranormed spaces and the spaces $X(r,s,t,p;B)$ for $X\in \{c(p),c_0(p),l(p)\}$ have Schauder basis. Furthermore, the $\alpha$-, $\beta$-, $\gamma$- duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r,s,t,p;B)$ to $X$. Finally, some classes of compact operators on the space $l_p(r,s,t;B)$ are characterized by using the Hausdorff measure of noncompactness.

arXiv comment23 pages. arXiv admin note: substantial text overlap with arXiv:1307.5883, arXiv:1307.5884, arXiv:1307.6208