Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators

Abstract

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 , .