D-recurrence of operators on banach spaces

Abstract

An operator $T$ is called recurrent if for any non empty set $U$ there exists $n\in \N$ such that $T^nU\cap U\neq \phi$. In this paper, we generalize this concept by using recurrence of operators in the closed unit disk $\D$ and we call it $\D$-recurrent operators. We extend some properties of recurrence to $\D$-recurrence ones. In particular, we show that an operator is $\D$-recurrent if and only if the set of all its recurrent vectors is dense. Also, we show that every operator $T$ and its iterates $T^n\, (n\in \N)$ shares the same $\D$-recurrent vectors. Unlike the case of recurrent operators, we show that the inverse of some $\D$-recurrent operators are not $\D$-recurrent, while others are $\D$-recurrent too. According to the strong connection between $\D$-recurrent operators and both diskcyclic and recurrent operators, we give an example to show that not every $\D$-recurrent operator is recurrent and diskcyclic. The later results rely on a nice characterization for an operator to be $\D$-recurrent which we call $\D$- recurrent criterion. Finally, we provide the relation between power boundedness and $\D$-recurrence. In particular, if $T$ is power bounded then the set of all $D$-recurrent vectors is closed. Also, $T^{-1}$ is power bounded and $\D$-recurrent whenever $T$ is.