On the fine spectrum of the upper triangle double band matrix \({\Delta}^+\) on the sequence space \(c_0\) (Q2869100)

Let \(e_j=(\delta_{ij})_{i=0}^{\infty}\in c_0\) (\(j\geq 0\)), where \(\delta_{ij}\) is the Kronecker delta, and let \(\Delta^+\) be bounded linear operator on \(c_0\) which is given by \(\Delta^+e_j=e_j-e_{j-1}\), for \(j\geq 1\) and \(\Delta^+e_0=e_0\). The structure of the spectrum of \(\Delta^+\) is studied. It is shown that the spectrum of \(\Delta^+\) is \(\{ \lambda\in {\mathbb C}: |\lambda -1|\leq 1\}\), that the interior of this disc forms the point spectrum, and that the border of the disc belongs to the continuous spectrum. The approximate point spectrum, the defect spectrum, and the compression spectrum are considered as well.