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

Abstract

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a (bounded or unbounded) scalar type spectral operator $A$ in a complex Banach space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of the exponentials of such an operator, which, under a certain condition on the spectrum of the operator $A$, coincides with the $C_0$-semigroup generated by $A$. The spectrum of $A$ lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group $\left\{e^{tA}\right\}_{t\in {\mathbb R}}$ of bounded linear operators generated by $A$. 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 non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by $A$.