Operators with normal Aluthge transforms

Oloomi, Ali; Radjabalipour, Mehdi

doi:10.1016/j.crma.2012.02.003

Functional Analysis

Operators with normal Aluthge transforms
[Opérateurs et transformations normales de Aluthge]

Présenté par : the Editorial Board

Oloomi, Ali ¹ ; Radjabalipour, Mehdi ^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, Tome 350 (2012) no. 5-6, pp. 263-266.

Résumé
Abstract

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.

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.

Reçu le : 2011-12-31
Accepté le : 2012-02-09
Publié le : 2012-02-23

DOI : 10.1016/j.crma.2012.02.003

Affiliations des auteurs :

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

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Oloomi, Ali; Radjabalipour, Mehdi. Operators with normal Aluthge transforms. Comptes Rendus. Mathématique, Tome 350 (2012) no. 5-6, pp. 263-266. doi : 10.1016/j.crma.2012.02.003. http://www.numdam.org/articles/10.1016/j.crma.2012.02.003/

Bibliographie
Cité par

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

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

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

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

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

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

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

Cité par Sources :

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