Abstract: By using quasinilpotent part, analytic core and singlevalued extension property (SVEP) of operators, the author considered property (az) and property (z) of operator T, proved if H₀(T-λI) was closed for every λ∈σf(T), then T satisfied property (az), and gave two equivalent characterizations of T satisfying property (z).

Key words: property (z), property (az), quasinilpotent part, analytic core, singlevalued extension property (SVEP)