Polaroid operators with SVEP and perturbations of property \((gw)\) (Q844001)

A bounded linear operator \(T\) on a Banach space \(X\) belongs to the class \(PS(X)\) if it is a polaroid operator with the single-valued extension property. It satisfies the \((gw)\) property if the set of its isolated eigenvalues coincides with \(\sigma_a(T)\setminus \sigma _{SBF^+_-}(T)\), where \(\sigma_a(T)=\{\lambda \in \mathbb{C} \mid T-\lambda I\) is not bounded from below\} and \(\sigma _{SBF^+_-}(T)=\{\lambda \in \mathbb{C} \mid \text{ind}( T-\lambda I)>0\}\). The author proves that, if \(T\in PS(X)\) and \(G\) satisfies some additional assumptions (e.g., it is algebraic or quasinilpotent and commutes with \(T\)), then \(f(T^*+G^*)\) satisfies the \((gw)\) property for any analytic function \(f\) on a neighbourhood of \(\sigma (T+G)\). He also adds some corollaries and examples concerning different versions of Weyl's theorem as well as other properties of operators.