On the class $(W_e)$-operators

Abstract

It is well known that an hyponormal operator satisfies Weyl's theorem. A result due to Conway shows that the essential spectrum of a normal operator $N$ consists precisely of all points in its spectrum except the isolated eigenvalues of finite multiplicity, that's ${\sigma }_e(N)=\sigma (N)\setminus E^0(N).$ In this paper, we define and study a new class named $(W_e)$ of operators satisfying ${\sigma }_e(T)=\sigma (T)\setminus E^0(T),$ as a subclass of $(W).$ A countrexample shows generally that an hyponormal does not belong to the class $(W_e),$ and we give an additional hypothesis under which an hyponormal belongs to the class $(W_e).$ We also give the generalisation class $(g{W}_e)$ in the contexte of B-Fredholm theory, and we characterize $(B_e),$ as a subclass of $(B),$ in terms of localized SVEP.