INVERSE OF FREQUENTLY HYPERCYCLIC OPERATORS

We show that there exists an invertible frequently hypercyclic operator on $\ell ^1(\mathbb {N})$ whose inverse is not frequently hypercyclic.

Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

BANACH SPACES WITH SEPARABLE DUALS SUPPORT DUAL HYPERCYCLIC OPERATORS

Let E be a Banach space such that its dual E* is separable. We show that there exists a hypercyclic bounded operator T on E such that its adjoint T* is also hypercyclic on E*. We also exhibit a new kind of dual hypercyclic operator. Thus answers affirmatively two of the questions raised by Henrik Petersson in a recent paper.