Faber-Hypercyclic semigroups of linear operators

Abstract

Let $(F_n^\Omega)_{n\geq0}$ be the series of Faber polynomials associated with $\Omega$, a bounded domain of the complex plane whose boundary is aclosed Jordan curve. A bounded linear operator $T$ on a separable Banach space $X$ is called $\Omega$-hypercyclic ($\Omega$-transitive) if there exists a vector $x$ of $X$ such that $\{F_n^\Omega(T)x:\ n \geq 0\}$ is dense in $X$ (if for each pair $(U,V)$ of open subsets, there is $n\in \mathbb{N}$ such that $F_n^\Omega(T)U\cap V\neq \emptyset$). The $\Omega$-hypercyclic and $\Omega$-transitive behavior are studied in this research within the framework of linear strongly continuous semigroups. We give sufficient constraints on the spectrum of an operator to yield a $\Omega$-hypercyclic semigroup, and we establish necessary and sufficient conditions on the semigroup to be $\Omega$-transitive.