Extended Weyl theorems and perturbations. (English) Zbl 1289.47027

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, M. Berkani and 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 \(T\).


47A53 (Semi-) Fredholm operators; index theories
47A55 Perturbation theory of linear operators
47A10 Spectrum, resolvent


Zbl 1258.47020