Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NOTE ON ∗-PARANORMAL OPERATORS AND RELATED CLASSES OF OPERATORS

  • Tanahashi, Kotoro (Department of Mathematics Tohoku Pharmaceutical University) ;
  • Uchiyama, Atsushi (Department of Mathematical Science Faculty of Science Yamagata University)
  • Received : 2012.06.16
  • Published : 2014.03.31
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Abstract

We shall show that the Riesz idempotent $E_{\lambda}$ of every *-paranormal operator T on a complex Hilbert space H with respect to each isolated point ${\lambda}$ of its spectrum ${\sigma}(T)$ is self-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 $||Tx||^n{\leq}||T^nx||\,||x||^{n-1}$ 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.

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References

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