A Note on Quasi-paranormal Operators |
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Authors: | Young Min Han Won Hee Na |
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Institution: | 1. Department of Mathematics, College of Sciences, Kyung Hee University, 1 Hoegi-dong, Dongdaemun-gu, Seoul, 130-701, Republic of Korea
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Abstract: | Let T be a bounded linear operator on an infinite dimensional complex Hilbert space. In this paper, we introduce the new class, denoted ${{\mathcal{QP}}}$ , of operators satisfying ${{\|T^{2}x\|^{2}\leq \|T^{3}x\|\|Tx\|}}$ for all ${{x \in \mathcal{H}}}$ . This class includes the classes of paranormal operators and quasi-class A operators. We prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a nonzero isolated point λ0 of the spectrum of ${{T \in \mathcal{QP}}}$ , then E is self-adjoint if and only if ${{N(T-\lambda_{0}) \subseteq N(T^{*}-\overline{\lambda}_{0})}}$ . |
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