On the Power Set of Quasinilpotent Operators in Banach Spaces

For a quasinilpotent operator $T$ on a Banach space $X$,
Douglas and Yang defined
$k_{x}=\limsup\limits_{\lambda\rightarrow 0}\frac{\ln\|(\lambda-T)^{-1}x\|}{\ln\|(\lambda-T)^{-1}\|}$
for each non-zero vector $x$, and called $\Lambda(T)=\{k_x:x\neq 0\}$ the
power set of $T$.
In this paper, we prove that $\Lambda(T)$ always contains $1$ for every quasinilpotent operator $T$ on $X$.
Moreover, we introduce the concept of a Banach space $X$ having uniform multiplicity infinity and prove that some classical Banach spaces possess this property.
As an application,
we show that if $\sigma\subset [0,1]$ is right closed and contains $1$, then there exists a quasinilpotent operator $T$ on a class of Banach spaces with uniform multiplicity infinity such that $\Lambda(T)=\sigma$.