On the Power Set of Quasinilpotent Operators

For a quasinilpotent operator T on a separable Hilbert space ℋ , Douglas and Yang define k_x=lim sup _λ→ 0ln‖ (λ -T)^-1x‖/ln‖ (λ -T)^-1‖ for each nonzero vector x , and call Λ (T)={k_x:x 0} the power set of T . In this paper, we prove that Λ (T) is right closed, that is, supσ∈Λ (T) for each nonempty subset σ of Λ (T) . Moreover, for any right closed subset σ of [0, 1] containing 1, we show that there exists a quasinilpotent operator T with Λ (T)=σ . Finally, we prove that the power set of V , the Volterra operator on L^2[0,1] , is (0, 1].