# Construction of operators with prescribed behaviour

- 99 Downloads
- 8 Citations

## 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 }.

## Mathematics Subject Classification (2000):

47A15 47A16## Preview

Unable to display preview. Download preview PDF.