Cesàro 𝑞-Difference Sequence Spaces and Spectrum of Weighted 𝑞-Difference Operator

In this research paper, we undertake an investigation into Ces\`aro $\mathfrak q$-difference sequence spaces $\mathfrak X(\mathfrak{C}_1^{\delta;\mathfrak q}),$ where $\mathfrak X \in \{\ell_{\infty},c,c_0\}.$ These spaces are generated by using the matrix $\mathfrak{C}_1^{\delta,\mathfrak q},$ which is a product of the Ces\`aro matrix $\mathfrak C_1$ of the first order and the second order $\mathfrak q$-difference operator $\nabla^2_\mathfrak q$ defined by \[ (\nabla^2_\mathfrak q \mathfrak f)_k=\mathfrak{f}_k-(1+\mathfrak q)\mathfrak{f}_{k-1}+\mathfrak q\mathfrak{f}_{k-2},~(k\in \mathbb{N}_0), \] where $\mathfrak q\in (0,1)$ and $\mathfrak{f}_k=0$ for $k 0.$ Our endeavor includes the establishment of significant inclusion relationships, the determination of bases for these spaces, the investigation of their $\alpha$-, $\beta$- and $\gamma$-duals, and the formulation of characterization results pertaining to matrix classes $(\mathfrak{X},\mathfrak Y),$ with $\mathfrak X$ chosen from the set $\{\ell_{\infty}(\mathfrak{C}_1^{\delta;\mathfrak q}), c(\mathfrak{C_1^{\delta;\mathfrak q}}), c_0(\mathfrak{C}_1^{\delta;\mathfrak q})\}$ and $\mathfrak Y$ chosen from the set $\{\ell_{\infty},c,c_0,\ell_{1}\}.$ The final section of our study is dedicated to the meticulous spectral analysis of the weighted $\mathfrak q$-difference operator $\nabla^{2;\mathfrak z}_{\mathfrak q}$ over the space $c_0$ of null sequences.