Residuality of sets of hypercyclic operators (Q434285)

Let \(X\) be a separable metrizable topological vector space and let \(L(X)\) denote the set of continuous and linear operators from \(X\) to \(X\). An operator \(T\in L(X)\) is called \textit{hypercyclic} if there exists a vector \(x\in X\) such that \(\{T^n x:n\in\mathbb{N}\}\) is dense in \(X\), and it is said to be \textit{supercyclic} if there exists a vector \(x\in X\) such that \(\{\lambda T^n x:\lambda\in\mathbb{C},\,n\in\mathbb{N}\}\) is dense in \(X\). In this paper, a criterion is stated for the existence of residual sets (in the Baire category sense) of hypercyclic operators in \(L(X)\). This criterion is applied to prove the existence of a \(G_\delta\)-set of hypercyclic operators in the set \(I+\mathcal{N}\), where \(\mathcal{N}\) is the operator norm closure of the finite rank nilpotent operators on a separable infinite dimensional Banach space. Moreover, this criterion also yields the existence of a \(G_\delta\)-set of hypercyclic operators \(T\) in \(I+\mathcal{Q}\), where \(\mathcal{Q}\) is the set of compact quasinilpotent operators on a separable infinite dimensional Hilbert space, such that its adjoint \(T^*\) is also hypercyclic. This shows the abundance of this type of operators, who were first studied by \textit{H. N. Salas} [Glasg. Math. J. 49, No. 2, 281--290 (2007; Zbl 1129.47009)] and by \textit{H. Petersson} [Proc. Am. Math. Soc. 134, No. 6, 1671--1676 (2006; Zbl 1082.47005)]. In this flavour, it is shown the abundance of quasinilpotent composition Volterra operators \(V_\varphi\) such that both \(V_\varphi\) and \(V_\varphi^*\) are supercyclic and both \(I+V_\varphi\) and \(I+V_\varphi\) are hypercyclic, too. For further information regarding composition Volterra operators, we refer to [\textit{A. Montes-Rodríguez, A. Rodríguez-Martínez} and \textit{S. Shkarin}, Math. Z. 261, No. 2, 431-472 (2009; Zbl 1160.47029); Proc. Lond. Math. Soc. (3) 103, No. 3, 535-562 (2011; Zbl 1232.47008)].