Bulletin of the
Korean Mathematical Society BKMS

We shall show that the Riesz idempotent $E_{\lambda}$ of every $*$-paranormal operator $T$ on a complex Hilbert space $\mathcal H$ with respect to each isolated point $\lambda $ of its spectrum $\sigma(T)$ isself-adjoint and satisfies $E_{\lambda}\mathcal H = \ker (T-\lambda) = \ker (T-\lambda )^*$. Moreover, Weyl's theorem holds for $*$-paranormal operators and more general for operators $T$ satisfying the norm condition $\| Tx\|^n \leq \| T^n x\| \| x\|^{n-1}$ for all $x\in \mathcal H$. Finally, for this more general class of operators we find a sufficient condition such that $E_{\lambda}\mathcal H = \ker (T-\lambda) = \ker (T-\lambda )^*$ holds.

Keywords: $*$-paranormal, $k$-paranormal, normaloid, the single valued extension property, Weyl's theorem

MSC numbers: 47B20