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

Abstract

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.