Weyl's type theorems and hypercyclic operators

For a bounded operator T acting on an infinite dimensional separable Hilbert space H, we prove the following assertions: (i) If T or T* ∈ , then generalized a-Browder's theorem holds for f(T) for every f ∈ Hol(σ(T)). (ii) If T or T* ∈ has topological uniform descent at all λ ∈ iso(σ(T)), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (iii) If T ∈ has topological uniform descent at all λ ∈ E(T), then T satisfies generalized Weyl's theorem. (iv) Let T ∈ . If T satisfies the growth condition Gd (d ≥ 1), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (v) If T ∈ , then, for all f ∈ Hol(σ(T)). (vi) Let T be a-isoloid such that T* ∈ . If T − λI has finite ascent at every λ ∈ Ea(T) and if F is of finite rank on H such that TF = FT, then T + F obeys generalized a-Weyl's theorem.