The power set of a quasinilpotent backward weighted shift

Abstract

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 $Λ(T) = \{k_x : x \neq 0\}$. We prove that $1 \in Λ(T)$ for every quasinilpotent operator on an arbitrary Banach space, which answers a question of Ji and Zhang. We further show that $Λ(T) = [0,1]$ 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.

Disclosure

“rs that Theorem 4 remains valid without the monotonicity as- sumption on the weights. The corresponding generalization, or the construction of a counterexample, deserves a separate investigation. Declaration on the use of generative AI Generative AI (Claude Opus, Anthropic) was used substantially in the prepara- tion of this paper. Its contribution was decisive for the formulation and proof of Lemma 1, notably, and it assisted in drafting a number of the other argumen”

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Verified

2026-08-23T18:57:25.239984Z

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