On extended spectrum of a composition operator on sequence spaces

Abstract

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