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 $\Lambda(T) = \{k_x : x \neq 0\}$.

We prove that $1 \in \Lambda(T)$ for every quasinilpotent operator on an arbitrary Banach space, which answers a question of Ji and Zhang.

We further show that $\Lambda(T) = [0,1]$ for every backward unilateral weighted shift on $\ell^p$ whose weight sequence is strictly decreasing and $p'$-summable for some $p' > 0$, thereby weakening the hypotheses imposed by Hu and Ji.