A survey on the spectra of the difference operators over the Banach space c

Abstract

Recently, due to its numerous applications, the spectra of the bounded operators over Banach spaces have been studied extensively. This work aims to collect some of the investigations on the spectra of difference operators or matrices on the Banach space c in the literature and provide a foundation for related problems. To the best of our investigations, the problem has been solved over the sequence space c so far up to maximum order 2. In the present work, the fine spectra of the difference operator $\Delta ^m, m\in \mathbb {N}$ on c have been computed. The generalized difference operator $\Delta ^m$ on the Banach space c is defined by $(\Delta ^mx)_k= \sum _{i=0}^m(-1)^i\left( {\begin{array}{c}m\\ i\end{array}}\right) x_{k-i},\;k=0,1,2,3,\dots$ with $x_{k} = 0$ for $k<0$. Indeed, the operator $\Delta ^m$ is represented by an $(m+1)$-th band matrix which generalizes several difference operators such as $\Delta ,\Delta ^2,B(r,s)$ and B(r, s, t) etc, under different limiting conditions. Initially, we provide some essential background results on the linearity and boundedness of the backward difference operator $\Delta ^m$. Finally, the sets for the spectrum and fine spectra such as the point spectrum, the continuous spectrum and the residual spectrum of the defined operator on the space c have been computed. The geometrical interpretation for the spectral subdivisions of the above difference operator is also incorporated.