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

In this research paper, we undertake an investigation into Ces & agrave;ro q-difference sequence spaces X(C delta;q 1 ), where X is an element of {infinity, c, c0}. These spaces are generated using the matrix C delta,q 1 , which is a product of the Ces & agrave;ro matrix C1 of the first-order and the second-order q-difference operator del 2 q defined by (del q(2) f)k = fk - (1 + q)fk-1 + qfk-2, (k is an element of N-0), where q is an element of (0, 1) and fk = 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 (X, Y), with X chosen from the set {infinity(C delta;q 1 ), c(C delta;q 1 ), c0(C delta;q 1 )} and Y chosen from the set {l(infinity), c, c(0), 1}. The final section of our study is dedicated to the meticulous spectral analysis of the weighted q-difference operator del q(2;z) over the space c0 of null sequences.