*(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\le 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\le p<\infty $, are characterized. Further, the authors deduce a characterization of some other classes by means of a given basic lemma.