Common hypercyclic vectors for the conjugate class of a hypercyclic operator
Let \(X\) be an infinite-dimensional, separable Banach space and \(B(X)\) the algebra of all bounded linear operators on \(X\). The authors show that, if \(T\) is a continuous hypercyclic operator on \(X\), then the conjugate set \(S(T):=\{L^{-1}TL: L\in B(X)\) invertible\} contains a path \(\{F_t\in B(X)\), \(t\in [1,\infty[\}\) of operators which is SOT dense in \(B(X)\), and for which the set \(\bigcap_{t\geq 1}{\mathcal{HC}}(F_t)\) of common cyclic vectors for the whole path is a dense \(G_\delta\)-set. The following interesting corollary is obtained. Let \(g\) be a nonzero vector in \(X\). Then the set \(\mathcal A=\{T\in B(X): g\in{\mathcal{HC}}(T)\} \) is SOT-dense and SOT-connected in \(B(X)\). Furthermore, the set \(\bigcap_{T\in {\mathcal A}}{\mathcal{HC}}(T)\) of common hypercyclic vectors for \(A\) is \(\{\lambda g:\lambda\not=0\}\). It is also shown that the set of all hypercyclic operators on \(X\) is SOT-connected in \(B(X)\).
scientific article; zbMATH DE number 5835285
scientific article