Power Set of Some Quasinilpotent Weighted Shifts

Abstract

For a quasinilpotent bounded linear operator T, we write $k_x=\limsup \limits _{z\rightarrow 0}\frac{\log \Vert (z-T)^{-1}x\Vert }{\log \Vert (z-T)^{-1}\Vert }$ for each nonzero vector x. Set $\Lambda (T)=\{k_x:x\ne 0\}$, and call it the power set $\Lambda (T)$ of T. This notation was introduced by Douglas and Yang (see Hermitian geometry on resolvent set. arXiv:1608.05990, 2016; Oper Theory Adv Appl 267, Springer, Cham, 2018; Sci Cina Math. https://doi.org/10.1007/s11425-000-0000-0). They showed that for $\tau \in \Lambda (T)$, $M_\tau :=\{0, x:k_x\le \tau \}$ is a linear subspace invariant under each A commuting with T; hence, if there are two different points $\tau _j\in \Lambda (T)$ such that $M_{\tau _j}$’s are closed, then T has a nontrivial hyperinvariant subspace. In this paper, we show that if a quasinilpotent unilateral weighted shift T is strongly strictly cyclic, then $\Lambda (T)=\{1\}$. Moreover, we construct a quasinilpotent operator T such that $\Lambda (T)=[0, 1]$ and $M_\tau$ is not closed for all $\tau$ in [0, 1). Even so, we still find a subset ${\mathcal {N}}$ of ${\text {Lat}}T$, the lattice of invariant subspaces of T, such that ${\mathcal {N}}$ is order isomorphic to $\Lambda (T)$.