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In this paper, we determine the spectrum, the point spectrum, the continuousspectrum and the residual spectrum of the generalized difference operator$\Delta_{a,b}$ on the sequence space $\ell_p \ (1< p < \infty)$ where the realsequences $a=\{a_k\}$ and $b=\{b_k\}$ are not necessarily convergent. Hence ourresults generalize the work given by Akhmedov and El-Shabrawy [Math. Slovaca65~(5) (2015) 1137--1152] for the sequence space $\ell_p (1< p <\infty)$.

On some study of the fine spectra of generalized difference operator $\Delta_{a,b}$ on $\ell_p \ (1