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

## TL;DR

This paper introduces new Fibonacci-based sequence spaces derived from classical $ extit{l}_p$ spaces, explores their duals, bases, and matrix transformations, and analyzes the Hausdorff measure of non-compactness of associated operators.

## Contribution

It constructs novel Fibonacci sequence spaces of non-absolute type from $ extit{l}_p$ spaces, determines their duals and bases, and characterizes compact operators using Hausdorff measures.

## Key findings

- Established inclusion relations among the new spaces.
- Derived dual spaces and constructed bases for the spaces.
- Characterized classes of compact operators using Hausdorff measure.

## Abstract

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 $1\leq p<\infty.$