Power set and invariant subspaces of cyclic quasinilpotent operators

Abstract For a quasinilpotent operator T on a separable Hilbert space H , Douglas and Yang defined k x = lim sup | z | → 0 log ⁡ ‖ ( z − T ) − 1 x ‖ log ⁡ ‖ ( z − T ) − 1 ‖ for x ∈ H , x ≠ 0 , and called Λ ( T ) = { k x : x ∈ H , x ≠ 0 } the power set of T. This paper computes the power set of cyclic quasinilpotent operators and studies the connection with their invariant subspaces. We first prove that the power set of the strictly cyclic quasinilpotent unilateral weighted shift only contains 1. Then we define the strongly strictly cyclic quasinilpotent operator, characterize its invariant subspaces, and show that its power set also only contains 1. Finally, we show that the Volterra integral operator V on L 2 [ 0 , 1 ] , whose power set contains more than one point, is cyclic but not strictly cyclic.