Abstract:
The judgement of equivalence for property (ω) between an operator and its conjugate on a Hilbert space is discussed. By the relation between the two new spectrum sets σCI(T) and σCFI(T) defined in view of the consistency in invertibility and in the Fredholm index respectively, the following two conclusions are proved. First, both T and its conjugate T* satisfy property (ω1) if and only if σCI(T)=σCFI(T) and Browders theorem holds for T; second, both T and its conjugate T* satisfy property (ω) and are all isoloid if and only if σCI(T)=σCFI(T) and σb(T)=σ3(T)∪σCFI(T)∪{λ∈C:n(T-λT)=n(T*-λI)=∞}, where σ3(T) is a variant of semi-Fredholm spectrum. Also, the equivalence for the perturbation of property (ω) is considered.
