The property（R）and the hypercyclic property for bounded linear operators

Abstract

Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. σ(T) denotes the spectrum of T ∈ B(H). T ∈ B(H) is said to satisfy property (R) if σa (T)\ σab (T) = π00 (T)，where σa (T) and σab (T) denote the approximate point spectrum and the Browder essential approximate spectrum of T respectively，π00 (T) = { λ ∈ iso σ(T)：0 < n(T - λI) < +∞ }. A new judgement for property (R) for bounded linear operator is given. In additional，the relations between the property (R) and the hypercyclic property are considered.