Generalized \(a\)-Weyl’s theorem and the single-valued extension property. (English) Zbl 1123.47005

Amouch, Mohamed

Extr. Math. 21, No. 1, 51-65 (2006).

The author investigates the generalized \(a\)-Weyl theorem and the generalized \(a\)-Browder theorem for Banach space operators \(T\) for which \(T\) or \(T^*\) has the single-valued extension property. He proves that the spectral mapping theorem holds for the semi-essential approximate point spectrum of such operators. Moreover, he gives some applications for bounded linear operators \(T\) on a complex Banach space \(X\) for which there is a function \(p:\mathbb{C}\to\mathbb{Z}^+\) such that \[ H_0(T-\lambda):=\{x\in X:\lim_{n\to+\infty}\| (T-\lambda)^nx\| ^{\frac{1}{n}}=0\}=\ker(T-\lambda)^{p(\lambda)} \] for all \(\lambda\in\mathbb{C}\).

Reviewer: Abdellatif Bourhim (Québec)

MSC:

47A10	Spectrum, resolvent
47A11	Local spectral properties of linear operators
47A20	Dilations, extensions, compressions of linear operators
47A53	(Semi-) Fredholm operators; index theories

Keywords:

generalised Weyl’s theorem; generalised Browder’s theorem; generalised \(a\)-Weyl’s theorem; generalised \(a\)-Browder’s theorem; semi-essential approximate point spectrum; \(B\)-Fredholm and semi-\(B\)-Fredholm operators; SVEP