Subdivision of spectra for some lower triangular double-band matrices as operators on $c_0$

Abstract

The generalized difference operator $\Delta_{a,b}$ was defined by El-Shabrawy: $\Delta{a,b}x = \Delta_{a,b} (x_n) = (a_nx_n + b_{n-1}x_{n-1})^{\infty}_{n = 0}$ with $x_1 = b_1 = 0$, where $(a_k), (b_k)$ are convergent sequences of nonzero real numbers satisfying certain conditions. We completely determine the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator $\Delta_{a,b}$ in the sequence space $c_0$.