Given a quasinilpotent bounded linear operator T on a complex Hilbert space H, we write k_x = limsup_z→0log||(z−T)^-1x||/ log||(z−T)^-1|| for each nonzero vector x. Set Λ(T)={k_x:x≠0}, and call it the power set of T denoted by Λ(T). This notation was introduced by Douglas and Yang. They showed that for  τ∈Λ(T), Mτ:={0,x:k_x ≤ τ} is a linear subspace invariant under each A commuting with T; hence, if there are two different points τ_j∈Λ(T) such that Mτ_j’s are closed, then T has a nontrivial hyperinvariant subspace. We show that if a quasinilpotent unilateral weighted shift T is strongly strictly cyclic, then Λ(T)={1}. Moreover, we construct a quasinilpotent operator T such that Λ(T)=[0,1] and M_τ is not closed for all τ in [0,1). Even so, we still find a subset N of Lat T, the lattice of invariant subspaces of T, such that N is order isomorphic to Λ(T).

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