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Ji, Youqing; Zhang, Yuanhang
Integral equations and operator theory, 12/2023, Letnik: 95, Številka: 4Journal Article
For a quasinilpotent operator T on a separable Hilbert space H , Douglas and Yang define k x = lim sup λ → 0 ln ‖ ( λ - T ) - 1 x ‖ 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 L 2 0 , 1 , is (0, 1.
