Classes of operators satisfying $a$-Weyl's theorem

Pietro Aiena¹

¹ Dipartimento di Metodi e Modelli Matematici Università di Palermo Viale delle Scienze I-90128 Palermo, Italy

Studia Mathematica, Tome 169 (2005) no. 2, pp. 105-122

Résumé

In this article Weyl's theorem and $a$-Weyl's theorem on Banach spaces are related to an important property which has a leading role in local spectral theory: the single-valued extension theory.We show that if $T$ has SVEP then Weyl's theorem and $a$-Weyl's theorem for $T^\ast$ are equivalent, and analogously, if $T^\ast$ has SVEP then Weyl's theorem and $a$-Weyl's theorem for $T$ are equivalent. From this result we deduce that $a$-Weyl's theorem holds for classes of operators for which the quasi-nilpotent part $H_0(\lambda I-T)$ is equal to $\ker\, (\lambda I-T)^p$ for some $p\in \mathbb N$ and every $\lambda \in \mathbb C$, and for algebraically paranormal operators on Hilbert spaces. We also improve recent results established by Curto and Han, Han and Lee, and Oudghiri.

Détail

DOI : 10.4064/sm169-2-1

Keywords: article weyls theorem a weyls theorem banach spaces related important property which has leading role local spectral theory single valued extension theory has svep weyls theorem a weyls theorem ast equivalent analogously ast has svep weyls theorem a weyls theorem equivalent result deduce a weyls theorem holds classes operators which quasi nilpotent part lambda i t equal ker lambda i t mathbb every lambda mathbb algebraically paranormal operators hilbert spaces improve recent results established curto han han lee oudghiri