A new variation of Weyl type theorem and perturbations

Abstract

In  this  paper  we introduce  the new property (aR), which extends  property (R) introduced by Aiena and his collaborators.  We investigate the property (aR) in connection with Weyl type theorem, and establish  sufficient  and  necessary conditions for which  property (aR) holds. We also study the stability of property (aR) under perturbations by finite rank operators, by nilpotent operators, by quasi-nilpotent operators and by algebraic operators commuting with T.