The power set of a quasinilpotent backward weighted shift

For a quasinilpotent operator $T$ on a Banach space $X$, R. Douglas and R. Yang associated with each nonzero vector $x$ the local resolvent-growth exponent $k_x$, and introduced the power set $\mathrm{\Lambda }(T)=\{k_x:x\mathrm{\ne }0\}$. We prove that $1\in \mathrm{\Lambda }(T)$ for every quasinilpotent operator on an arbitrary Banach space, which answers a question of Ji and Zhang. We further show that $\mathrm{\Lambda }(T)=[0,1]$ for every backward unilateral weighted shift on ${\mathrm{\ell }}^p$ whose weight sequence is strictly decreasing and $p^{\mathrm{\prime }}$-summable for some $p^{\mathrm{\prime }}>0$, thereby weakening the hypotheses imposed by Hu and Ji.