Property (gb) through local spectral theory (Q2923250)

In this paper, the authors give several characterizations of operators \(T\) for which property (gb) holds. These characterizations are obtained by using typical tools from local spectral theory. They also show that property (gb) holds for large classes of operators and prove the stability of property (gb) under some commuting perturbations. We recall that for a bounded linear operator \(T\) on a complex Banach space \(X\) to satisfy property (gb) means that the points of the approximate point spectrum for which \(aI - T\) is upper semi B-Weyl are exactly the poles of the resolvent.