On the Power Set of Quasinilpotent Operators in Banach Spaces

For a quasinilpotent operator $T$ on a Banach space $X$, Douglas and Yang defined $k_x=\underset{\lambda \to 0}{\mathrm{lim\hspace{0.17em}sup}}\frac{\ln \mathrm{\parallel }(\lambda -T{)}^{-1}x\mathrm{\parallel }}{\ln \mathrm{\parallel }(\lambda -T{)}^{-1}\mathrm{\parallel }}$ for each non-zero vector $x$, and called $\mathrm{\Lambda }(T)=\{k_x:x\mathrm{\ne }0\}$ the power set of $T$. In this paper, we prove that $\mathrm{\Lambda }(T)$ 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 $\mathrm{\Lambda }(T)=\sigma$.