Cesàro \(\mathfrak {q}\)-Difference Sequence Spaces and Spectrum of Weighted \(\mathfrak {q}\)-Difference Operator

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