On the power set of quasinilpotent operators

For a quasinilpotent operator $T$ on a separable Hilbert space $\mathcal{H}$, Douglas and Yang define $k_x=\limsup\limits_{\lambda\rightarrow 0}\frac{\ln\|(\lambda-T)^{-1}x\|}{\ln\|(\lambda-T)^{-1}\|}$ for each nonzero vector $x$, and call $\Lambda(T)=\{k_x:x\ne 0\}$ the power set of $T$. In this paper, we prove that $\Lambda(T)$ is right closed, that is, $\sup \sigma\in\Lambda(T)$ for each nonempty subset $\sigma$ of $\Lambda(T)$. Moreover, for any right closed subset $\sigma$ of $[0,1]$ containing $1$, we show that there exists a quasinilpotent operator $T$ with $\Lambda(T)=\sigma$. Finally, we prove that the power set of $V$, the Volterra operator on $L^2[0,1]$, is $(0,1]$.