Dual hypercyclic extension for an operator on a Hilbert subspace (Q2786839)

Let \(H\) be a separable, infinite dimensional Hilbert space and let \(M\) be a closed, non-trivial subspace of \(H\) of infinite codimension. A (bounded and linear) operator \(T\) on \(H\) is said to be dual hypercyclic if both \(T\) and its adjoint \(T^\ast\) admit a vector that has a dense orbit. Answering an open problem of \textit{D. A. Herrero} from [J. Funct. Anal. 99, No. 1, 179--190 (1991; Zbl 0758.47016)], \textit{H. Salas} constructed concrete examples of dual hypercyclic operators [Proc. Am. Math. Soc. 112, No. 3, 765--770 (1991; Zbl 0748.47023); Glasg. Math. J. 49, No. 2, 281--290 (2007; Zbl 1129.47009); J. Math. Anal. Appl. 374, No. 1, 106--117 (2011; Zbl 1210.47024)]. The authors prove that a bounded linear operator \(A\) on \(M\) has a dual hypercyclic extension \(T\) on \(H\) if and only if the adjoint \(A^\ast\) of \(A\) is hypercyclic. The necessity of the condition is easy. The proof of the sufficiency is long, technically very demanding and it uses some techniques due to the first author in [Proc. Am. Math. Soc. 140, No. 9, 3133--3143 (2012; Zbl 1284.47004)].