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

Marat V. Markin

Published 2018-07-17Version 1

We give a straightforward proof of the non-hypercyclicity of an arbitrary scalar type spectral operator $A$ (bounded or not) in a complex Banach space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of its exponentials, which, under a certain condition on the spectrum of $A$, coincides with the $C_0$-semigroup generated by it. The spectrum of $A$ lying on the imaginary axis, it is shown that non-hypercyclic is also the generated by it strongly continuous group $\left\{e^{tA}\right\}_{t\in {\mathbb R}}$ of bounded linear operators. As an important particular case, we immediately obtain that of a normal operator $A$ in a complex Hilbert space. From the general results, we infer that, in the complex Hilbert space $L_2({\mathbb R})$, the anti-self-adjoint differentiation operator $A:=\dfrac{d}{dx}$ with the domain $D(A):=W_2^1({\mathbb R})$ is not hypercyclic and neither is the left-translation group generated by it.