Abstract: An operator T∈B(X) defined on a Banach space X satisfies property(z), a new variant of Weyls theorem if the complement in the spectrum σ(T) of the upper semi-Weyl spectrum is the set of all isolated points of the approximate point spectrum which are eigenvalues of finite multiplicity. In this note, we first study the conditions between property(z)and other Weyl type theorem, then establish for a bounded linear operator and the calculus defined on a Banach space the sufficient and necessary conditions for which property(z)holds by means of the variant of essential approximate point spectrum. The perturbation of property(z)under finite rank operators is considered.

Key words: property(az), perturbation, spectrum, property(z)