On a Generalized Difference Operator and Its Fine Spectra

Abstract

In the present article, we compute the spectra of the generalized difference operator $$B_\nu ^{(m)}$$ over the sequence space $$c_0$$. In fact, the difference operator $$B_\nu ^{(m)}$$ usually generalizes several difference operators such as first-order difference operators ($$\Delta _{ab}, B(r,s),\Delta$$ and $$\Delta _\nu$$), second-order difference operators ($$\Delta ^2, B(r,s,t)$$ and $$\Delta _{uv}^2$$), third-order difference operator(B(r, s, t, u)), difference operators $$(\Delta ^m$$ and $$\Delta _\nu ^m)$$ of order $$m\in {\mathbb{N}}$$ and many others. Therefore, the results obtained are more general and interesting than those derived earlier.