On the power set of quasinilpotent operators

2023/05/17 by Ji, Youqing, Zhang, Yuanhang
#FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47A10. Secondary: 47B37

paper · doi:10.48550/arxiv.2305.09963

Abstract

For a quasinilpotent operator T on a separable Hilbert space H, Douglas and Yang define k_x=\limsup_{λ→ 0}\fracln‖(λ-T)^-1x‖ln‖(λ-T)^-1‖ for each nonzero vector x, and call Λ(T)=\k_x:x≠ 0\ the power set of T. In this paper, we prove that Λ(T) is right closed, that is, sup σ∈Λ(T) for each nonempty subset σ of Λ(T). Moreover, for any right closed subset σ of [0,1] containing 1, we show that there exists a quasinilpotent operator T with Λ(T)=σ. Finally, we prove that the power set of V, the Volterra operator on L²[0,1], is (0,1].

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