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

[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}\}}_{t\ge 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}\}}_{t\in {\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 .

[学科分类] 外科医学

[关键词] hypercyclicity;scalar type spectral operator;normal operator;C0-semigroup;strongly continuous operator group