On the power set of quasinilpotent operators

Abstract

For a quasinilpotent operator $T$ on a separable Hilbert space $\mathcal{H}$, Douglas and Yang define $k_x=\underset{\lambda \to 0}{\mathrm{limsup}}\frac{\ln \Vert (\lambda -T{)}^{-1}x\Vert }{\ln \Vert (\lambda -T{)}^{-1}\Vert }$ 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 $(0,1]$.

21 pages. Comments are welcome