Localized SVEP, Property (b) and Property (ab)

Property (b) for a bounded linear operator \({T \in L(X)}\) on a Banach space X means that the points λ of the approximate point spectrum for which λI – T is upper semi-Weyl are exactly the spectral points λ for which λI – T is Browder. In this paper we shall give several characterizations of operators T for which T, or its dual T*, has property (b). We also investigate the property (ab) which is closely related to property (b).