Some conditions implying normality of operators

Moslehian, M.S.; Nabavi Sales, S.M.S.

doi:10.1016/j.crma.2011.01.018

Mathematical Analysis/Functional Analysis

Some conditions implying normality of operators
[Quelques conditions entraînant la normalité dʼopérateurs]

Présenté par : the Editorial Board

Moslehian, M.S.¹ ; Nabavi Sales, S.M.S.^2,³

¹ Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
² Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
³ Tusi Mathematical Research Group (TMRG), Mashhad, Iran

Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 251-254.

Résumé
Abstract

Soit $T \in B (H)$ et $T = U | T |$ sa décomposition polaire. Nous montrons que : (i) si T est log-hyponormal ou p-hyponormal et $U^{n} = U^{⁎}$ pour un certain n, alors T est normal ; (ii) si le spectre de U est contenu dans un arc de cercle, alors T est normal si et seulement sʼil en est de même de son transformé de Aluthge $\tilde{T} = {| T |}^{1 / 2} U {| T |}^{1 / 2}$ .

Let $T \in B (H)$ and $T = U | T |$ be its polar decomposition. We prove that (i) if T is log-hyponormal or p-hyponormal and $U^{n} = U^{⁎}$ for some n, then T is normal; (ii) if the spectrum of U is contained in some open semicircle, then T is normal if and only if so is its Aluthge transform $\tilde{T} = {| T |}^{\frac{1}{2}} U {| T |}^{\frac{1}{2}}$ .

Reçu le : 2010-11-25
Accepté le : 2011-01-18
Publié le : 2011-02-02

DOI : 10.1016/j.crma.2011.01.018

Affiliations des auteurs :

Moslehian, M.S. ¹ ; Nabavi Sales, S.M.S. ^{2, 3}

@article{CRMATH_2011__349_5-6_251_0,
     author = {Moslehian, M.S. and Nabavi Sales, S.M.S.},
     title = {Some conditions implying normality of operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {251--254},
     publisher = {Elsevier},
     volume = {349},
     number = {5-6},
     year = {2011},
     doi = {10.1016/j.crma.2011.01.018},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.01.018/}
}

TY  - JOUR
AU  - Moslehian, M.S.
AU  - Nabavi Sales, S.M.S.
TI  - Some conditions implying normality of operators
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 251
EP  - 254
VL  - 349
IS  - 5-6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.01.018/
DO  - 10.1016/j.crma.2011.01.018
LA  - en
ID  - CRMATH_2011__349_5-6_251_0
ER  -

%0 Journal Article
%A Moslehian, M.S.
%A Nabavi Sales, S.M.S.
%T Some conditions implying normality of operators
%J Comptes Rendus. Mathématique
%D 2011
%P 251-254
%V 349
%N 5-6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.01.018/
%R 10.1016/j.crma.2011.01.018
%G en
%F CRMATH_2011__349_5-6_251_0

Moslehian, M.S.; Nabavi Sales, S.M.S. Some conditions implying normality of operators. Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 251-254. doi : 10.1016/j.crma.2011.01.018. http://www.numdam.org/articles/10.1016/j.crma.2011.01.018/

Bibliographie
Cité par

[1] Aluthge, A. On p-hyponormal operators for $0 < p < 1$ , Integral Equations Operator Theory, Volume 13 (1990), pp. 307-315

[2] Aluthge, A.; Wang, D. w-hyponormal operators, Integral Equations Operator Theory, Volume 36 (2000), pp. 1-10

[3] Beck, W.A.; Putnam, C.R. A note on normal operators and their adjoints, J. London Math. Soc., Volume 31 (1956), pp. 213-216

[4] Duggal, P.B. On nth roots of normal operators, Bull. London Math. Soc., Volume 25 (1993), pp. 74-80

[5] Embry, M.R. nth roots of normal contractions, Proc. Amer. Math. Soc., Volume 19 (1968), pp. 63-68

[6] Jeon, I.H.; Tanahashi, K.; Uchiyama, A. On quasisimilarity for log-hyponormal operator, Glasgow Math. J., Volume 46 (2004), pp. 169-176

[7] Furuta, T. Invitation to Linear Operator, Taylor and Francis, London, New York, 2001

[8] Okayasu, T.; Ueta, Y. A condition under which $B = A = U^{⁎} B U$ follows from $B ⩾ A ⩾ U^{⁎} B U$ , Proc. Amer. Math. Soc., Volume 135 (2007), pp. 1399-1403

[9] Putnam, C.R. On square roots of normal operators, Proc. Amer. Math. Soc., Volume 8 (1957), pp. 768-769

[10] Radjavi, H.; Rosenthal, P. On roots of normal operators, J. Math. Anal. Appl., Volume 34 (1971), pp. 653-664

[11] Stampfli, J.G. Hyponormal operators, Pacific J. Math., Volume 12 (1962), pp. 1453-1458

[12] Takahashi, K. On the converse of Fuglede–Putnam theorem, Acta Sci. Math. (Szeged), Volume 43 (1981), pp. 123-125

[13] Tanahashi, K. On log-hyponormal operators, Integral Equations Operator Theory, Volume 34 (1999), pp. 364-372

[14] Uchiyama, A.; Tanahashi, K. Fuglede–Putnamʼs theorem for p-hyponormal or log-hyponormal operators, Glasgow Math. J., Volume 44 (2002), pp. 397-410

Cité par Sources :