Abstract:

Let $X$ be a topological space and let $T$ be a bounded operator on $X$. We say that $T$ is hypercyclic if $T$ admits a dense orbit, namely if there exists a vector $x\in X$, called a hypercyclic vector for $T$, such that $\{T^n x;\ n\geq 0\}$ is dense in $X$. We shall denote by $HC(T)$ the set of hypercyclic vectors for $T$. It is known that, provided $HC(T)$ is nonempty, then it has some nice topological and algebraic properties. For instance, $HC(T)\cup\{0\}$ always contains a dense subspace, and there are nice criteria for the existence of a closed infinite-dimensional subspace in it.

When moreover $X$ is an algebra, it is natural to study whether $HC(T)$ contains a nontrivial algebra. In this talk, we will explain some recent (negative and positive) results on this problem.