On the non-hypercyclicity of normal operators, collections of their exponentials, and symmetric operators

Abstract

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary normal operator $A$ (bounded or unbounded) in a complex Hilbert 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 $A$, is the $C_0$-semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.

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