Subdivision of the Spectrum

Abstract

Let S be the right shift operator on the space ℓ₁. Since ‖S‖ = 1 the spectrum σ(S) is contained in the closed unit disk D. We have seen that S has no eigenvalue. The adjoint of S is the left shift operator T on the space ℓ_∞. If λ is any complex number with |λ| ≤ 1, then the vector x_λ = (1, λ, λ²,…) is in ℓ_∞ and Tx_λ = λx_λ. Thus every point λ in the disk D is an eigenvalue of T. This shows also that σ(S) = σ(T) = D.