A hypercyclic operator whose adjoint is also hypercyclic

Salas, Héctor

doi:10.1090/S0002-9939-1991-1049848-8

A hypercyclic operator whose adjoint is also hypercyclic
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by Héctor Salas

Proc. Amer. Math. Soc. 112 (1991), 765-770

DOI: https://doi.org/10.1090/S0002-9939-1991-1049848-8

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Abstract:

An operator $T$ acting on a Hilbert space $H$ is hypercyclic if, for some vector $x$ in $H$, the orbit $\{ {T^n}x:n \geq 0\}$ is dense in $H$. We show the existence of a hypercyclic operator—in fact, a bilateral weighted shift—whose adjoint is also hypercyclic. This answers positively a question of D. A. Herrero.

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Invariant closed sets for linear operators