Hypercyclictty and countable hypercyclicity for adjoint of θ- operators

Abstract EN

Let H be an infinite dimensional separable complex Hilbert space and let T ϵ B (H), where B (H) is the Banach algebra of all bounded linear operators on H.

In this paper we prove the following results.

If T ϵ B (H) is a ∅- operator, then 1.

T* is a hyper cyclic operator if and only if ð (T \ M) ⋂ D ≠ ∅ and ð (T \ M ) ⋂ (C \ D ) ≠ ∅ for every hyper invariant subspace M of T.

2.

If T is a pure, then T* is a countably hyper cyclic operator if and only if ð (T \ M) ⋂ (C \ D) ≠∅ and ð (T) ⋂D ≠ ∅ for every hyper invariant subspace M of T.

3.

T* has a bounded set with dense orbit if and only if for every hyper invariant subspace M of T, ð (T \ M) ⋂ (C \ D ) ≠∅.