Abstract
The aim of the paper is introduced the composition of the two infinite matrices $\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right).$ Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of new spaces and also construct the basis for the space $\ell_{p}^{\lambda}(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$Finally we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators by applying the Hausdorff measure of noncompactness, where $X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\},$ and $1\leq p<\infty.$
Highlights
For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞
The matrix domain XA of an infinite matrix A in a sequence space X is defined by XA = {x = ∈ ω : Ax ∈ X}
The Hausdorff measure of non-compactness of linear operators given by infinite matrices in some special classes of sequence spaces were studied in [1,6,27,29,32]
Summary
Here and in what follows, the summation without limits runs from 0 to ∞. The matrix domain XA of an infinite matrix A in a sequence space X is defined by XA = {x = (xk) ∈ ω : Ax ∈ X}. Several authors studied matrix transformations on sequence spaces that are the matrix domain of the difference operator, or of the matrices of some classical methods of summability in different sequence spaces, for instance we refer to [7,8,9,19,20,21,22,25,28,36,37,38] and references therein. The Hausdorff measure of non-compactness of linear operators given by infinite matrices in some special classes of sequence spaces were studied in [1,6,27,29,32].
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