The power set of a quasinilpotent backward weighted shift

2026/07/18 by Egor Ignatev
Mathematics · #math.FA

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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 ≠ 0\. We prove that 1 ∈ 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 ℓ^p whose weight sequence is strictly decreasing and p'-summable for some p' > 0, thereby weakening the hypotheses imposed by Hu and Ji.

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