Isolated points of spectrum of $k$-quasi-$*$-class $A$ operators

Salah Mecheri¹

¹ Department of Mathematics College of Sciences Taibah University P.O. Box 20003 Al-Madinah Al Munawarah, Saudi Arabia

Studia Mathematica, Tome 208 (2012) no. 1, pp. 87-96

Résumé

Let $T$ be a bounded linear operator on a complex Hilbert space $H$. In this paper we introduce a new class, denoted $\mathcal{KQA}^{*}$, of operators satisfying $T^{*k}(|T^{2}|-|T^{*}|^{2})T^{k}\geq 0$ where $k$ is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if $E$ is the Riesz idempotent for a non-zero isolated point $\mu$ of the spectrum of $T\in \mathcal{KQA}^{*}$, then $E$ is self-adjoint and $EH=\ker(T-\mu)=\ker\,(T-\mu)^{*}$. Some spectral properties are also presented.

Détail

DOI : 10.4064/sm208-1-6

Keywords: bounded linear operator complex hilbert space paper introduce class denoted mathcal kqa * operators satisfying *k * geq where natural number prove basic structural properties these operators using these results riesz idempotent non zero isolated point spectrum mathcal kqa * self adjoint ker t ker t * spectral properties presented