Power Set of Quasinilpotent Operator on Banach Space
Let T be a quasinilpotent operator on an infinite dimensional complex Banach space X and x∈X\{0}.Define kx=limλ→0supln||(λ-T)-1x||/ln||(λ-T)-1.Let A(T)={kx:x≠0},and call it the power set of T.We prove that A(T)is right closed,that is,supσ∈A(T)for each nonempty bounded subset σ of A(T).In particular,we prove that for any infinite dimensional complex Banach space X,there exists a quasinilpotent operator T on X such that A(T)=[0,1].
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