  The aim of the paper is to introduce the spaces
$\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F})$
derived by the composition of the two infinite matrices
$\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right),$ which are the
$BK$-spaces of non-absolute type and also derive some inclusion relations.
Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of those spaces
and also construct the basis for $\ell_{p}^{\lambda}(\widehat{F}).$
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 $p.$
Here we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators,
where
$X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$
and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\}$ by applying the Hausdorff
measure of non-compactness, and $1\leq p<\infty.$
