Extended Weyl theorems and perturbations (Q2850450)

A bounded linear operator \(T\) on a Banach space \(X\) is called a Weyl operator if it is a Fredholm operator of index 0. The Weyl spectrum \(\sigma_W(T)\) is the set of those \(\lambda \in \mathbb C\) for which \(T-\lambda I\) is not a Weyl operator. Let \(\sigma (T)\) be the spectrum of \(T\), \(\Delta (T)=\sigma (T)\setminus \sigma_W(T)\). Denote by \(E_0(T)\) the set of isolated points \(\lambda \in \sigma (T)\) such that \(0<\dim \ker (T-\lambda I)<\infty\). It is said that Weyl's theorem holds for \(T\) if \(\Delta (T)=E_0(T)\). In the literature, there are numerous generalizations of this notion; see, in particular, \textit{M. Berkani} and \textit{H. Zariouh} [Mat. Vesn. 62, No. 2, 145--154 (2010; Zbl 1258.47020)]. The author finds conditions on an operator \(T\) for such properties to hold and be stable under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with \(T\).