Some Fibonacci sequence spaces of non-absolute type derived from ${\ell }_p$ with $(1\le p\le \infty )$ and Hausdorff measure of non-compactness of composition operators

Abstract

The aim of the paper is to introduce the spaces ${\ell }_{\infty }^{\lambda }(\hat{F})$ and ${\ell }_p^{\lambda }(\hat{F})$ derived by the composition of the two infinite matrices $\Lambda =({\lambda }_{nk})$ and $\hat{F}=\left(f_{nk}\right),$ 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 ${\ell }_p^{\lambda }(\hat{F}).$ Additionally, we characterize some matrix classes on the spaces ${\ell }_{\infty }^{\lambda }(\hat{F})$ and ${\ell }_p^{\lambda }(\hat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$ Here we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators, where $X\in \{{\ell }_{\infty }^{\lambda }(\hat{F}),{\ell }_p^{\lambda }(\hat{F})\}$ and $Y\in \{c_0,c,{\ell }_{\infty },{\ell }_1,bv\}$ by applying the Hausdorff measure of non-compactness, and $1\le p<\infty .$

19 pages. arXiv admin note: text overlap with arXiv:1604.07396