Abstract: Let T be a quasinilpotent operator on an infinite dimensional complex Banach space X and x∈X＼{0}. Let Λ(T)={k_x: x≠0}, and call it the power set of T. We prove that Λ(T) is right closed, that is, sup σ∈Λ(T) for each nonempty bounded subset σ of Λ(T). In particular, we prove that for any infinite dimensional complex Banach space X, there exists a quasinilpotent operator T on X such that Λ(T)=［0,1］.

Key words: quasinilpotent operator, power set, right closed, Schauder basis sequence