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On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

机译：On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

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摘要

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection {etA}t≥0{{{e}^{tA}}}_{tge 0} of its exponentials, which, under a certain condition on the spectrum of the operator A , coincides with the C0{C}_{0}-semigroup generated by A . The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group {etA}t∈ℝ{{{e}^{tA}}}_{tin {mathbb{R}}} of bounded linear operators generated by A . From the general results, we infer that, in the complex Hilbert space L2(ℝ){L}_{2}({mathbb{R}}), the anti-self-adjoint differentiation operator A≔ddxA:= tfrac{text{d}}{text{d}x} with the domain D(A)≔W21(ℝ)D(A):= {W}_{2}^{1}({mathbb{R}}) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A .

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