  In this paper, we determine the spectrum, the point spectrum, the continuous
spectrum and the residual spectrum of the generalized difference operator
$\Delta_{a,b}$ on the sequence space $\ell_p \ (1< p < \infty)$ where the real
sequences $a=\{a_k\}$ and $b=\{b_k\}$ are not necessarily convergent. Hence our
results generalize the work given by Akhmedov and El-Shabrawy [Math. Slovaca
65~(5) (2015) 1137--1152] for the sequence space $\ell_p (1< p <\infty)$.
