Construction of Infinite Dimensional Vector Spaces of Hypercyclic Vectors

Juan B. Seoane Sepúlveda Kent State University Kent, Ohio, USA

Consider X = ℓ_{p} or c_{0}, and define the weighted backward shift B_{λ} (|λ| > 1) on X,

B_{λ}(x_{1}, x_{2}, x_{3}, …) = λ · (x_{2}, x_{3}, x_{4}, …).

We give a different proof that the set of hypercyclic vectors for this operator is lineable, i.e. there exists an infinite dimensional hypercyclic subspace for B_{λ}. We also give a technique to construct infinite dimensional common hypercyclic subspaces for a countable family of these operators. This is a well known result, but we won’t use the Baire category theorem, we prove it in an explicit and constructive way. Joint work with Richard M. Aron (Kent State University).