Bull. Korean Math. Soc. 2014; 51(2): 357-371
Printed March 1, 2014
https://doi.org/10.4134/BKMS.2014.51.2.357
Copyright © The Korean Mathematical Society.
Kotoro Tanahashi and Atsushi Uchiyama
Tohoku Pharmaceutical University, Yamagata University
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
2002; 39(1): 33-41
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2006; 43(3): 509-517
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