Operators with normal Aluthge transforms

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.

RésuméDans 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.