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On some new sequence spaces of non-absolute type related to the spaces $\ell_p$ and $\ell_\infty$ II. (English) Zbl 1276.46004
The spaces $\ell_{p}^\lambda$ and $\ell_{\infty}^\lambda$ of non-absolute type were introduced by the same authors in Part I [Filomat 25, No. 2, 33--51 (2011; Zbl 1265.46011)] as the spaces of all sequences whose $\Lambda$-transforms are in the spaces $\ell_p$ and $\ell_\infty$, respectively, where $1\leq p<\infty$. The present paper is a natural continuation of the work done in that paper. The paper is divided mainly in two parts in connection with new results besides the introduction, a general description of the spaces $\ell_{p}^\lambda$ and $\ell_{\infty}^\lambda$, and references. In the first part, the $\alpha$-, $\beta$-, $\gamma$-duals of the spaces $\ell_{p}^\lambda$ and $\ell_{\infty}^\lambda$ are computed. In the second part, the matrix classes $(\ell_{p}^\lambda : \ell_\infty)$, $(\ell_{p}^\lambda : c)$, $(\ell_{p}^\lambda : c_0)$, $(\ell_{p}^\lambda : \ell_1)$, $(\ell_{1}^\lambda : \ell_p)$ and $(\ell_{\infty}^\lambda : \ell_p)$, where $1\leq p<\infty$, are characterized. Further, the authors deduce a characterization of some other classes by means of a given basic lemma.

46A45Sequence spaces
40C05Matrix methods in summability
40H05Functional analytic methods in summability
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