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

Markin Marat V.

Demonstratio Mathematica

2020

Vol. 53, nr 1

352--359

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 of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C0-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 of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L2(R), the anti-self-adjoint differentiation operator A≔ddx with the domain D(A)≔W12(R) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.