From positive to accretive matrices

Yassine Bedrani, Fuad Kittaneh, Mohammed Sababheh

Published 2020-02-25Version 1

The main goal of this paper is to discuss the recent advancements of operator means for accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean, then we define arbitrary operator means and functional calculus for accretive matrices. Applications of this new discussion involve generalizations of known inequalities from the setting of positive matrices to that of accretive matrices. This includes the arithmetic-harmonic mean comparisons, monotony of operator means, Ando's inequality, Choi's inequality, Ando-Zhan subadditive inequality and much more. It will be noticed that, while this article treats accretive operators, the corresponding results for positive operators will be special cases of our results. This means that this article can be also viewed as an exposition for celebrated inequalities of positive matrices!