Existence of Hypercyclic Algebras in Fréchet Algebras

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 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 A= \,span\,\,Orb\,(a,T) A = span Orb ( a , T ) for any prescribed a∈ A \0\ a ∈ A \ { 0 } .