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 \({{N(T-\lambda_{0}) \subseteq N(T^{*}-\overline{\lambda}_{0})}}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, 2004.
Ando T.: Operators with a norm condition. Acta Sci. Math. (Szeged) 33, 169–178 (1972)
Duggal B.P.: Hereditarily normaloid operators. Extracta Math. 20, 203–217 (2005)
Furuta T.: On the class of paranormal operators. Proc. Japan Acad. 43, 594–598 (1967)
T. Furuta, Invitation to Linear Operators, CRC Press, 2002.
Furuta T., Ito M., Yamazaki T.: A subclass of paranormal operators including class of log-hyponormal and several related classes. Sci. Math. 1, 389–403 (1998)
In Ho Jeon and In Hyoun Kim, On operators satisfying T*|T 2|T ≥ T*|T|2 T. Linear Algebra Appl. 418 (2006), 854–862.
C. S. Kubrusly, Hilbert Space Operators A Problem Solving Approach, Birkhäuser, Boston, 2003.
K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, London Mathematical Society Monographs New Series 20, Clarendon Press, Oxford, 2000.
Sheth I.H.: Quasi-hyponormal operators. Rev. Roumaine Math. Pures Appl. 19, 1049–1053 (1974)
Stampfli J.: operators and spectral density. Trans. Amer. Math. Soc. 117, 469–476 (1965)
Uchiyama A.: On the Isolated Points of the Spectrum of Paranormal Operators. Integral Equations Operator Theory 55, 145–151 (2006)
Uchiyama A., Tanahashi K.: On the Riesz idempotent of class A operators. Math. Inequal. Appl. 5, 291–298 (2002)
Uchiyama A., Tanahashi K.: Bishop’s property (β) for paranormal operators. Oper. Matrices 3, 517–524 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Kyung Hee University Research Fund in 2009 (KHU-20090618).
Rights and permissions
About this article
Cite this article
Han, Y.M., Na, W.H. A Note on Quasi-paranormal Operators. Mediterr. J. Math. 10, 383–393 (2013). https://doi.org/10.1007/s00009-012-0176-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-012-0176-6