Typical operators admit common cyclic vectors (Q2839307)

Let \(H\) be an infinite--dimensional separable Hilbert space. For a linear operator \(T\) on \(H\), a vector \(x\in H\) is called \textit{cyclic} if the span of the orbit \(\text{lin}\{x,Tx,T^2x,\ldots\}\) is dense in \(H\), (\textit{weakly}) \textit{supercyclic} if \(\mathbb C\{x,Tx,T^2x,\ldots\}\) is (weakly) dense in \(X\) and (\textit{weakly}) \textit{hypercyclic} if the orbit \(\{x,Tx,T^2x,\ldots\}\) is (weakly) dense in \(X\). An operator \(T\) is then called \textit{cyclic}, resp. (\textit{weakly}) \textit{supercyclic}, resp. (\textit{weakly}) \textit{hypercyclic}, if it admits a cyclic, resp. (weakly) supercyclic, resp. (weakly) hypercyclic vector. Clearly, (weak) hypercyclicity implies (weak) supercyclicity that in turn implies cyclicity. It is well known that every hypercyclic operator admits a residual set of hypercyclic vectors. Furthermore, the vectors that are hypercyclic for an operator \(T\) are also hypercyclic vectors for every positive power \(T^n\).NEWLINENEWLINEIn the paper under review, the author shows that, given a countable dense subset \(D\) of \(H\), the set of operators for which every vector in \(D\setminus\{0\}\) is hypercyclic (resp. (weakly) supercyclic) is \(G_\delta\) dense in the ball \(B_R\), with \(R>1\), in the strong operator topology (resp., (weak) strong operator topology). Moreover, the author also proves that for every operator \(T\) in such a set, the set of (weak) supercyclic vectors is residual in \(H\) in the norm topology.