Abstract
In 1951, Heinz showed the following useful norm inequality:“If A, B≥0and X∈B(H), then ‖AXB‖ r‖X‖1−r≥‖A r XB r‖holds for r∈ [0, 1].” In this paper, we shall show the following two applications of this inequality:
Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logA≥logB) by a norm inequality.
Secondly, we shall study the condition under which\(\parallel T\parallel = \parallel \tilde T\parallel \), where\(\tilde T = |T|^{\tfrac{1}{2}} U|T|^{\tfrac{1}{2}} \) is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=‖T‖) via Aluthge transformation.
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Yamazaki, T. Characterizations of logA≥logB and normaloid operators via Heinz inequality. Integr equ oper theory 43, 237–247 (2002). https://doi.org/10.1007/BF01200255
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DOI: https://doi.org/10.1007/BF01200255