On extended spectrum of a composition operator on sequence spaces

In this paper, we investigate the spectral properties of a composition operator Cα: ℓp → ℓp, where 1 ≤ p < ∞ and a null sequence α = (αk)k ∈ ℕ, defined by Cαx = (0, α1x1, α2x2, …). In particular, we prove that the point spectrum, the approximate point spectrum, as well as the spectrum of Cα all coincide and form the singleton set containing 0. However, the extended spectrum of Cα turns out to be the complement of the closed unit disc of the complex plane.