Power Set of Some Quasinilpotent Weighted shifts on l^p

2023/06/01 by Chenlin Hu, Hu, Chaolong, You Qing Ji +1
Mathematics · #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2306.00387

Abstract

For a quasinilpotent operator T on a Banach space X, Douglas and Yang defined k_x=\limsup_{z→ 0}\fracln‖(z-T)^-1x‖ln‖(z-T)^-1‖ for each nonzero vector x∈ X, and call Λ(T)=\k_x: x≠ 0\ the power set of T. They proved that the power set have a close link with T's lattice of hyperinvariant subspaces. This paper computes the power set of quasinilpotent weighted shifts on l^p for 1≤ p< ∞. We obtain the following results: (1) If T is an injective quasinilpotent forward unilateral weighted shift on l^p(ℕ), then Λ(T)=\1\ when k_e0=1, where \e_n\_n=0^∞ be the canonical basis for l^p(ℕ); (2) There is a class of backward unilateral weighted shifts on l^p(ℕ) whose power set is [0,1]; (3) There exists a bilateral weighted shift on l^p(ℤ) with power set [(1)/(2),1].

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