On the spectrum of 2-nd order generalized difference operator $\delta^2$ over the sequence space $c_0$

Résumé

The main purpose of  this article is to  determine the spectrum and the fine spectrum  of second order  difference operator $\Delta^2$  over the sequence space $c_0$. For any sequence $(x_k)_0^\infty$ in $c_0$, the generalized second order  difference operator $\Delta^2$  over  $c_0$ is defined by $\Delta^2(x_k)= \sum_{i=0}^2(-1)^i\binom{2}{i}x_{k-i}=x_k-2x_{k-1}+x_{k-2}$, with $ x_{n}  = 0$ for $n<0$.Throughout we use the convention that a term with a negative subscript is equal to zero.