We characterize chaotic linear operators on reflexive Banach spaces in terms of the existence long arithmetic progressions sets return times. also show that this characterization does not hold for arbitrary spaces. To achieve this, we study $$\mathcal {F}$$ -hypercyclicity a family subsets natural numbers associated to arbitrarily progressions.

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably often appear as symmetric with respect to a group symmetries, are interesting for possible applications. Some conditions existence topological isomorphisms between such obtained. We construct class algebras, where all normalized algebraic bases equ...