Weyl's type theorems and hypercyclic operators

Abstract

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, f(σSBF+−(T)) = σSBF+−(f(T)) 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.