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

TL;DR

This paper proves that scalar type spectral operators and their exponentials, including certain groups and semigroups, are not hypercyclic in complex Banach and Hilbert spaces, extending known results for normal operators.

Contribution

It provides a straightforward proof of non-hypercyclicity for scalar type spectral operators and their exponentials, generalizing the normal operator case to broader classes.

Findings

Scalar type spectral operators are non-hypercyclic.

Exponentials of such operators, including semigroups and groups, are non-hypercyclic.

The anti-self-adjoint differentiation operator in L2(R) is non-hypercyclic.

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 $A$.