On some study of the fine spectra of generalized difference operator ${\mathrm{\Delta }}_{a,b}$ on ${\mathrm{\ell }}_p(1<p<\mathrm{\infty })$

In this paper, we determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the generalized difference operator ${\mathrm{\Delta }}_{a,b}$ on the sequence space ${\mathrm{\ell }}_p(1<p<\mathrm{\infty })$ 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 ${\mathrm{\ell }}_p(1<p<\mathrm{\infty })$.