Baire theorem and hypercyclic algebras

Linear Dynamical Systems

“…When the space X supports an algebra structure, a question that has recently attracted much research activity is whether the set of hypercyclic vectors HC (T ) admits, apart from zero, a subalgebra of X. An up-to-date overview of this topic can be found in Bayart et al [14].…”

“…Let Φ be a nonconstant polynomial with Φ(0) = 0. Then the The topic of hypercyclic algebras is rapidly evolving and research is expanding into different directions such as the investigation of frequently hypercyclic algebras [71], [14].…”

“…An example in [12] demonstrates for Φ(z) = 2 exp(−z) + sin z, that Φ(D) supports a hypercyclic algebra. The following special case was also given in [14]. Theorem 6.8 (Bayart, Costa Júnior and Papathanasiou [14]).…”

“…The following special case was also given in [14]. Theorem 6.8 (Bayart, Costa Júnior and Papathanasiou [14]). Let Φ be a nonconstant entire function of exponential type that is not a multiple of an exponential function.…”

“…We note that an interesting example from [14,Example 4.12] identifies an invertible operator T acting on a Banach algebra, such that T supports a hypercyclic algebra while its inverse T −1 doe not admit a hypercyclic algebra.…”