Construction of operators with prescribed behaviour

Abstract.

Let X be an infinite dimensional real or complex separable Banach space, and let $\{v_{n}, n \geq 1\}$ be a dense set of linearly independent vectors of X. We prove that there exists a bounded operator T on X such that the orbit of $v_1$ under T is exactly the set $\{v_{n}, n \geq 1\}$. This answers in the affirmative a question raised by I. Halperin, C. Kitai and P. Rosenthal, who asked whether every countable set of linearly independent vectors of X was contained in the orbit of some operator on X. If M is any infinite dimensional normed space of countable algebraic dimension, we prove that there exists a bounded operator T on M with no non-trivial invariant closed set. Finally, we show that the set of operators T on X such that M is a hypercyclic linear subspace for T is a dense $G_{\delta}$ subset of the set of hypercyclic operators. If $(T_k)_{k \geq 0}$ is a sequence of hypercyclic operators on X, there exists a dense linear subspace which is hypercyclic for every operator T _k .