WEYL’S TYPE THEOREMS AND HYPERCYCLIC OPERATORS

doi:10.1016/S0252-9602(12)60036-8

Abstract

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*∈ SC, then generalized a-Browder’s theorem holds for f(T) for every f ∈Hol(σ(T)). (ii) If T or T* ∈ HC 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 ∈ HC has topological uniform descent at all
λ∈ E(T), then T satisfies generalized Weyl’s theorem. (iv) Let T ∈HC. If T satisfies the growth condition G_d(d ≥ 1), then generalized Weyl’s theorem holds for f(T) for every f ∈ Hol(σ(T)). (v) If T ∈ SC, then, f(σ_SBF±(T)) = σ_SBF±(f(T)) for all f ∈ Hol(σ(T)). (vi) Let T be a-isoloid such that T* ∈ HC. If T − λI has finite ascent at every λ ∈ E^a(T) and if F is of finite rank on H such that TF = FT, then T +F obeys generalized a-Weyl’s theorem.

Key words: Weyl’s theorem, hypercyclic operators, supercyclic operators

CLC Number:

47A53

M.H. M. Rashid. WEYL’S TYPE THEOREMS AND HYPERCYCLIC OPERATORS[J].Acta mathematica scientia,Series B, 2012, 32(2): 539-551.

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