On the power set of quasinilpotent operators

For a quasinilpotent operator $T$ on a separable Hilbert space $\mathcal{H}$, Douglas and Yang define $k_x=\underset{\lambda \to 0}{\mathrm{lim\hspace{0.17em}sup}}\frac{\ln \mathrm{\parallel }(\lambda -T{)}^{-1}x\mathrm{\parallel }}{\ln \mathrm{\parallel }(\lambda -T{)}^{-1}\mathrm{\parallel }}$ for each nonzero vector $x$, and call $\mathrm{\Lambda }(T)=\{k_x:x\mathrm{\ne }0\}$ the power set of $T$. In this paper, we prove that $\mathrm{\Lambda }(T)$ is right closed, that is, $\sup \sigma \in \mathrm{\Lambda }(T)$ for each nonempty subset $\sigma$ of $\mathrm{\Lambda }(T)$. Moreover, for any right closed subset $\sigma$ of $[0,1]$ containing $1$, we show that there exists a quasinilpotent operator $T$ with $\mathrm{\Lambda }(T)=\sigma$. Finally, we prove that the power set of $V$, the Volterra operator on $L^2[0,1]$, is $(0,1]$.