Title: A theory of alternative means of matrices

 

 

Abstract: In this talk, we introduce a theory of alternative means of positive definite matrices. We define the alternative mean associated to a normalized operator monotone function by

for positive definite matrices \(A\) and \(B\). We study the adjoint, transpose, and duals of these means and obtain a characterization of the spectral geometric means in terms of alternative means and their duals. Additionally, we show the deep connection between the alternative means and the near order of matrices. In particular, we obtain in-betweenness results of the type \(A\preceq A\hat{\sigma}_f B \preceq B.\)