Power sets of weighted backward shifts
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For a quasinilpotent operator T, Λ(T) records the local resolvent-growth exponents k_x=limsup_{z→0} log‖(z−T)⁻¹x‖/log‖(z−T)⁻¹‖ over nonzero vectors x. For backward unilateral shifts on ℓᵖ, 1≤p<∞, when is Λ(T) the full interval [0,1]?
The studied weight class is strictly decreasing and positive, with ∑w_n^r<∞ for some r>0. The summability exponent need not be conjugate to p.
Why it matters and what counts as progress
Why it matters: The full interval extends earlier conclusions that required narrower reciprocal-weight conditions.
What counts as progress: A proof of the entire interval for this weight class, an extension to weaker weight assumptions, or a weight sequence missing a proposed exponent.
How the problem developed
- 1 Jun 2023 · Partial result [1]
Chaolong Hu and Youqing Ji Identified a class of backward weighted shifts on ℓᵖ whose power sets equal [0,1], under narrower weight hypotheses.
- 18 Jul 2026 · Theorem [2]
Egor Ignatev Proved Λ(T)=[0,1] for backward shifts with strictly decreasing weights satisfying a p′-summability condition, and proved 1∈Λ(T) for every quasinilpotent operator.
AI contribution
The power set of a quasinilpotent backward weighted shift
Ignatev uses a Claude-origin max-term asymptotic lemma twice to prove Λ(T)=[0,1] for backward shifts on ℓ^p with strictly decreasing, positive, p′-summable weights.
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