Isolated points of spectrum of $k$-quasi-$*$-class $A$ operators
Volume 208 / 2012
Studia Mathematica 208 (2012), 87-96
MSC: Primary 47B47, 47A30, 47B20; Secondary 47B10.
DOI: 10.4064/sm208-1-6
Abstract
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.
