The (p, q)-Cesáro matrix for \(p,q\in (0,1]\) with \(q<p\) is a lower triangular matrix with non zero entries \(\begin{aligned} c_{\nu \kappa }^{(p,q)}=\frac{p^{\kappa }q^{\nu -\kappa }}{[\nu +1]_{p,q}}, 0\le \kappa \le \nu . \end{aligned}\) Motivated by this construction, the present paper investigates the operator-theoretic behavior of the (p, q)-Cesáro matrix on the sequence space \(\ell _s(1<s<\infty )\) . We examine its boundedness and compactness, determine the spectrum and fine spectrum, and establish the Goldberg classification together with several spectral decompositions. To complement the analytical results, graphical illustrations of the spectrum are included for selected choices of p and q, offering geometric insight into the obtained findings.
