Weyl type theorems for $p$-hyponormal and $M$-hyponormal operators

Xiaohong Cao¹ ; Maozheng Guo² ; Bin Meng²

¹ College of Mathematics and Information Science Shaanxi Normal University Xi'an, 710062, P.R. China
² LMAM, School of Mathematical Sciences Peking University Beijing, 100871, P.R. China

Studia Mathematica, Tome 163 (2004) no. 2, pp. 177-188

Résumé

“Generalized Weyl's theorem holds" for an operator when the complement in the spectrum of the B-Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues; and “generalized a-Weyl's theorem holds" for an operator when the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the isolated points of the approximate point spectrum which are eigenvalues. If $T$ or $T^*$ is $p$-hyponormal or $M$-hyponormal then for every $f\in H(\sigma (T))$, generalized Weyl's theorem holds for $f(T)$, so Weyl's theorem holds for $f(T)$, where $H(\sigma (T))$ denotes the set of all analytic functions on an open neighborhood of $\sigma (T)$. Moreover, if $T^*$ is $p$-hyponormal or $M$-hyponormal then for every $f\in H(\sigma (T))$, generalized a-Weyl's theorem holds for $f(T)$ and hence a-Weyl's theorem holds for $f(T)$.

Détail

DOI : 10.4064/sm163-2-5

Keywords: generalized weyls theorem holds operator complement spectrum b weyl spectrum coincides isolated points spectrum which eigenvalues generalized a weyls theorem holds operator complement approximate point spectrum semi b essential approximate point spectrum coincides isolated points approximate point spectrum which eigenvalues * p hyponormal m hyponormal every sigma generalized weyls theorem holds weyls theorem holds where sigma denotes set analytic functions neighborhood sigma moreover * p hyponormal m hyponormal every sigma generalized a weyls theorem holds hence a weyls theorem holds