Exploring the $q$-analogue of Fibonacci sequence spaces associated with $c$ and $c_0$

We have proposed a $q$-analogue $c(\mathcal{F}(q))$ and $c_0(\mathcal{F}(q))$ of Fibonacci sequence spaces, where $\mathcal{F}(q)=(f_{km}^q)$ denotes a $q$-Fibonacci matrix defined in the following manner:

$$f_{km}^q=\{\begin{array}{ll}{q}^{m+1}\frac{f_{m+1}(q)}{f_{k+3}(q)-1},& \text{if}0\le m\le k,\\ 0,& \text{if}m>k,\end{array}$$

for all $k,m\in {\mathbb{Z}}_0^{+}$, where $(f_k(q))$ denotes a sequence of $q$-Fibonacci numbers. We developed a Schauder basis and determined several important duals ( $\alpha$-, $\beta$-, $\gamma$-) of the aforesaid constructed spaces $c(\mathcal{F}(q))$ and $c_0(\mathcal{F}(q))$. Additionally, we examined certain characterization results for the matrix class $(\mathfrak{U},\mathfrak{V})$, where $\mathfrak{U}\in \{c(\mathcal{F}(q)),c_0(\mathcal{F}(q))\}$ and $\mathfrak{V}\in \{{\ell }_{\mathrm{\infty }},c,c_0,{\ell }_1\}$. Essential conditions for the compactness of the matrix operators on the space $c_0(\mathcal{F}(q))$ via the Hausdorff measure of noncompactness (Hmnc) were presented.