Power Set of Some Quasinilpotent Weighted shifts on $l^p$

Abstract

For a quasinilpotent operator $T$ on a Banach space $X$, Douglas and Yang defined $k_x=\underset{z\to 0}{\mathrm{limsup}}\frac{\ln \Vert (z-T{)}^{-1}x\Vert }{\ln \Vert (z-T{)}^{-1}\Vert }$ for each nonzero vector $x\in X$, and call $\Lambda (T)=\{k_x:x\ne 0\}$ the power set of $T$. They proved that the power set have a close link with $T$'s lattice of hyperinvariant subspaces. This paper computes the power set of quasinilpotent weighted shifts on $l^p$ for $1\le p<\infty$. We obtain the following results: (1) If $T$ is an injective quasinilpotent forward unilateral weighted shift on $l^p(\mathbb{N})$, then $\Lambda (T)=\{1\}$ when $k_{e_0}=1$, where $\{e_n{\}}_{n=0}^{\infty }$ be the canonical basis for $l^p(\mathbb{N})$; (2) There is a class of backward unilateral weighted shifts on $l^p(\mathbb{N})$ whose power set is $[0,1]$; (3) There exists a bilateral weighted shift on $l^p(\mathbb{Z})$ with power set $[\frac{1}{2},1]$.

20 pages