The spectrum of the operator \(D(r, 0, s, 0, t)\) over the sequence spaces \(c_0\) and \(c\) (Q355704)

For given complex numbers \(r,s,t\) with \((s,t)\neq (0,0)\), consider the infinite matrix NEWLINE\[NEWLINE D(r,0,s,0,t):= \begin{pmatrix} r & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & r & 0 & 0 & 0 & 0 & 0 & \dots \\ s & 0 & r & 0 & 0 & 0 & 0 & \dots \\ 0 & s & 0 & r & 0 & 0 & 0 & \dots \\ t & 0 & s & 0 & r & 0 & 0 & \dots \\ 0 & t & 0 & s & 0 & r & 0 & \dots \\ 0 & 0 & t & 0 & s & 0 & r & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}, NEWLINE\]NEWLINE which gives rise to a bounded linear operator from \(c_0\) (the space of all null sequences) resp. \(c\) (the space of all convergent sequences) into itself. The authors describe the continuous and the residual spectrum of \(D(r,0,s,0,t)\) as an operator on \(c_0\) resp. \(c\) and show that the point spectrum is empty in both cases.