A note on property ( $${W_E}$$ W E )

Abstract

We investigate a new spectrum property (\({W_E}\)), which extends the generalized Weyl theorem. Using the property of consistence in Fredholm and index, we establish for a bounded linear operator T defined on a Hilbert space sufficient and necessary conditions for which the property \({(W_E)}\) holds. We also explore conditions on Hilbert operators T and S so that property \({(W_E)}\) holds for \({T\oplus S}\) . Moreover, we study the permanence of property \({(W_E)}\) under perturbations by power finite rank operators commuting with T and discuss the relation between property (\({W_E}\)) and hypercyclic operators.