Ji-Zhang Question on the Power Set of a Quasinilpotent Operator

Douglas and Yang attach to each nonzero vector $x$ of a quasinilpotent operator $T$ a local resolvent-growth exponent $k_x$, giving the power set $\Lambda(T) = \{k_x : x \ne 0\}$. Ji and Zhang asked whether $1$ always belongs to $\Lambda(T)$. It does, for every quasinilpotent operator on every Banach space. Moreover $\Lambda(T) = [0,1]$ for every backward unilateral weighted shift on $\ell^p$ with strictly decreasing, $p'$-summable weights, weakening the hypotheses of Hu and Ji.

Result

Proved

Status

Resolved

AI contribution

AI co-developed

Method

Argument

Field

Operator theory

Posed by

Youqing Ji, Yuanhang Zhang

Year posed

2023

Years open

3y

Solved

2026-07-18

Model

Claude Opus

Vendor

Anthropic

Collaborators

Egor Ignatev

Verification

Unreviewed

Publication

Preprint

Significance

9 / 100*

Disclosed cost

—

Wikipedia

No dedicated article