From positive to accretive matrices

The main goal of this paper is to discuss the recent advancements of matrix means from positive matrices to 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 matrix 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, monotonicity of matrix means, Ando's inequality, Choi's inequality, Ando-Zhan subadditive inequality and much more.

Automated summary

This paper discusses recent advancements in matrix means from positive matrices to accretive matrices, presenting the general form governing the definition of geometric mean and defining arbitrary matrix means and functional calculus for accretive matrices. Applications of this discussion involve generalizations of known inequalities from positive matrices to accretive matrices.