Existence of Hypercyclic Algebras in Fréchet Algebras

Abstract

Abstract We prove that every separable infinite-dimensional Fréchet algebra X admits a continuous linear operator T supporting a dense invariant hypercyclic algebra, giving an affirmative answer to a question of Bayart, Costa Jr. and Papathanasiou. In fact, every dense countable-dimensional subalgebra $$\mathcal {A}$$ A of X can be prescribed as an invariant hypercyclic algebra. When X admits a continuous norm, the operator can additionally be chosen in the form $$T=I+K$$ T = I + K , where K is nuclear, so that $$\mathcal {A}= {{\,\textrm{span}\,}}{{\,\textrm{Orb}\,}}(a,T)$$ A = span Orb ( a , T ) for any prescribed $$a\in \mathcal {A}\backslash \{0\}$$ a ∈ A \ { 0 } .