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 xλ = (1, λ, λ2,…) is in ℓ∞ and Txλ = λxλ. Thus every point λ in the disk D is an eigenvalue of T. This shows also that σ(S) = σ(T) = D.
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