Localized SVEP, property \((b)\) and property \((ab)\) (Q382077)

Let \(T\in L(X)\) be a bounded linear operator on an infinite dimensional complex Banach space \(X\). Two interesting properties, property (b) and property (ab) for \(T\in L(X)\), have been recently introduced by \textit{M. Berkani} and \textit{H. Zariouh} [Math. Bohem. 134, No.~4, 369--378 (2009; Zbl 1211.47011)]. In the present paper, the authors prove that property (b) admits a characterization by means of the localized SVEP. Indeed, property (b) for \(T\) holds if and only if \(T^{*}\) has the SVEP at the points of \(\Delta_{a}(T)\). Analogously, property (ab) may be characterized by means of the localized SVEP for \(T^{*}\). The authors also show that property (b) for \(T\) is equivalent to saying that the analytic core \(K(\lambda I-T)\) has finite codimension as \(\lambda\) ranges at the points of \(\Delta_{a}(T)\), while property (b) for \(T^{*}\) is equivalent to saying that the quasi-nilpotent part \(H_{0}(\lambda I-T)\) has finite dimension as \(\lambda\) ranges at the points of \(\Delta_{a}(T^{*})\). Similar characterizations are obtained for property (ab).