Fine spectra of upper triangular double-band matrices (Q972742)

Let \(c_0\) be the space of all null sequences. Let \(c\) be the space of all convergent sequences. Let \[ V= \begin{pmatrix} r & s & 0 & 0 &\cdots\\ 0 & r & s & 0 &\cdots\\ 0 & 0 & r & s &\cdots\\ \cdots &&\cdots&&\ddots\\ \cdots &&\cdots &&\ddots\end{pmatrix} \] be an upper triangular double-band infinite matrix. The operator on \(c_0\) corresponding to the matrix \(V\) is denoted by \((V,c_0)\). Then the point spectrum is \(\sigma_p(V,c_0)= \{\lambda\in\mathbb{C}:|\lambda- r|<|s|\}\), the residual spectrum \(\sigma_r(V,c_0)= \emptyset\), the continuous spectrum \(\sigma_c(V,c_0)= \{\lambda\in \mathbb{C}:|\lambda-r|= |s|\}\). Similarly, \(\sigma_p(V,c)\), \(\sigma_r(V,c)\) and \(\sigma_c(V,c)\) are evaluated. Some applications of these results are given.