Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

Streszczenie

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.