On the spectra of generalized Fibonacci and Fibonacci-like operators (Q2885134)

For a positive integer \(n\), the generalized Fibonacci operator on \(\ell^1\) is defined by \( F_n:(x_1, x_2, x_3, \dots) \mapsto \left( \sum_{k=n+1}^{\infty} x_k, x_1, x_2, \dots\right)\). It is a bounded operator. In this paper, the point, the residual, and the continuous spectrum of \(F_n\) are completely determined. Fibonacci-like operators are given by \( \Gamma_n:(x_1, x_2, x_3, \dots) \mapsto \left( \rho\sum_{k=n+1}^{\infty} (k-n)x_k, x_1, x_2, \dots\right)\), where \(\rho>0\). These are not bounded operators, moreover, the resolvent set of \(\Gamma_n\) is empty. However, the spectral picture of \(G_n=D_n \Gamma_n D_{n}^{-1}\), where \(D_n=\mathrm{diag}(1, \dots, 1, \rho, 2\rho, 3\rho, \dots)\), with \(1\) on the first \(n\) places, is more interesting. It is mentioned that some of the obtained results have applications in population dynamics.