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

In this research paper, we undertake an investigation into Cesàro \(\mathfrak {q}\)-difference sequence spaces \(\mathfrak {X}(\mathfrak {C}_1^{\delta ;\mathfrak {q}})\), where \(\mathfrak {X} \in \{\ell _{\infty },c,c_0\}.\) These spaces are generated using the matrix \(\mathfrak {C}_1^{\delta ,\mathfrak {q}}\), which is a product of the Cesàro matrix \(\mathfrak {C}_1\) of the first-order and the second-order \(\mathfrak {q}\)-difference operator \(\nabla ^2_\mathfrak {q}\) defined by

$$\begin{aligned} (\nabla ^2_\mathfrak {q} \mathfrak {f})_k=\mathfrak {f}_k-(1+\mathfrak {q})\mathfrak {f}_{k-1}+\mathfrak {q}\mathfrak {f}_{k-2},~(k\in \mathbb {N}_0), \end{aligned}$$

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.