In this note we introduce and study the property (gw1), which extend property (gw) introduced by Amouch and Berkani in [7]. We investigate the property (gw1) in connection with Weyl type theorems, and establish for abounded linear operator defined on a Banach space the sufficient and necessary conditions for which property (gw1) holds. We also study the property (gw1) for operators satisfying the single valued extension property (SVEP). Classes of operators are considered as illustrating examples.