ON THE SPECTRUM OF SECOND ORDER $(\Tilde{p}, \Tilde{q})-$DIFFERENCE OPERATOR OVER THE SEQUENCE SPACE $\ell_p$ $(1<p<\infty)$

This article presents a detailed analysis of the spectral and fine spectral properties of the second-order $(\Tilde{p}, \Tilde{q})-$difference operator $\Delta_{\Tilde{p},\Tilde{q}}^{2}$, where $(0<\Tilde{q}<\Tilde{p}\le 1)$. The operator extends the classical $\Tilde{q}-$difference operator and is examined within the Banach space $\ell_p$~$(1<p<\infty)$. In this setting, $\Delta_{\Tilde{p},\Tilde{q}}^{2}$ admits a matrix representation characterized by a lower triangular structure with three nonzero diagonals.