Operators with normal Aluthge transforms

Ali Oloomi; Mehdi Radjabalipour

doi:10.1016/j.crma.2012.02.003

Functional Analysis

Operators with normal Aluthge transforms

Presented by: the Editorial Board

Ali Oloomi ¹; Mehdi Radjabalipour ^2,³

¹ Department of Mathematics, Science and Research Branch, Islamic Azad University, Kerman, Iran
² SBUK Center for Linear Algebra and Optimization, University of Kerman, Iran
³ Iranian Academy of Sciences, Tehran, Iran

Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 263-266.

Abstract
Résumé

The main purpose of the Note is to show that if the second Aluthge transform of an invertible operator is normal, so it is its first Aluthge transform. This extends results due to Moslehian and Nabavi Sales [Some conditions implying normality of operators, C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 251–254] and Rose and Spitkovsky [On the stabilization of the Aluthge sequence, International Journal of Information and Systems Sciences 4 (1) (2008) 178–189]. Also, the structure of an injective operator with normal Aluthge transform is studied.

Dans cette Note on démontre que, si la deuxième transformation de Aluthge dʼun opérateur inversible est normale, alors sa première transformation de Aluthge est aussi normale, on étend ainsi les résultats de Moslehian et Nabavi Sales [Some conditions implying normality of operators, CRAS, Paris, Ser. I 349 (2011) 251–254], et Rose et Spitkovsky [On the stabilization of of the Aluthge sequence, International Journal of Information and Systems Sciences 4 (1) (2008) 178–189]. Par ailleurs on établit la structure dʼopérateur injectif avec transformation normale de Altuthge.

Received: 2011-12-31
Accepted: 2012-02-09
Published online: 2012-02-23

DOI: 10.1016/j.crma.2012.02.003

Author's affiliations:

Ali Oloomi ¹; Mehdi Radjabalipour ^{2, 3}

¹ Department of Mathematics, Science and Research Branch, Islamic Azad University, Kerman, Iran
² SBUK Center for Linear Algebra and Optimization, University of Kerman, Iran
³ Iranian Academy of Sciences, Tehran, Iran

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     title = {Operators with normal {Aluthge} transforms},
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     pages = {263--266},
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Ali Oloomi; Mehdi Radjabalipour. Operators with normal Aluthge transforms. Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 263-266. doi : 10.1016/j.crma.2012.02.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.02.003/

References
Cited by

[1] A. Aluthge On p-hyponormal operators for $0 < p < 1$ , Integral Equations and Operator Theory, Volume 13 (1990) no. 3, pp. 307-315

[2] T. Ando Aluthge transforms and the convex hull of the eigenvalues of a matrix, Linear and Multilinear Algebra, Volume 52 (2004), pp. 281-292

[3] J. Antezana; E.R. Pujlas; D. Stojanoff The iterated Aluthge transforms of a matrix converge, Advances in Mathematics, Volume 226 (2011) no. 2, pp. 1591-1620

[4] M.S. Brodskiĭ Triangular and Jordan Representations of Linear Operators, Translations of Mathematical Monographs, vol. 32, Nauka, Moscow, 1969 (English translation:, 1971, Amer. Math. Soc., Providence, RI)

[5] I.B. Jung; E. Ko; C. Pearcy Aluthge transform of operators, Integral Equations and Operator Theory, Volume 37 (2000), pp. 437-448

[6] I.B. Jung; E. Ko; C. Pearcy The iterated Aluthge transform of an operator, Integral Equations and Operator Theory, Volume 45 (2003) no. 4, pp. 375-387

[7] M.S. Moslehian; S.M.S. Nabavi Sales Some conditions implying normality of operators, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 349 (2011), pp. 251-254

[8] M.A. Naĭmark Normed Rings, GITTL, Moscow, 1956 (English translation:, 1959, Noordhoff, Groningen)

[9] D.E.V. Rose; I.M. Spitkovsky On the stabilization of the Aluthge sequence, International Journal of Information and Systems Sciences, Volume 4 (2008) no. 1, pp. 178-189

Cited by Sources:

^☆ The research is supported by the Iranian National Science Foundation.

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