## 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 $$T$$.

### MSC:

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

### Keywords:

Weyl operator; Weyl spectrum

Zbl 1258.47020