Matrix limit theorems of Kato type related to positive linear maps and operator means

TL;DR

This paper establishes limit theorems for positive matrices and linear maps involving operator means, generalizing Kato's limit and connecting to reciprocal Lie-Trotter formulas in matrix analysis.

Contribution

It introduces new limit theorems for matrix functions involving positive linear maps and operator means, extending Kato's limit to spectral order supremums.

Findings

Limit theorems for $igl(\Phi(A^p)igr)^{1/p}$ as $p o \infty$

Limit theorems for $(A^p \sigma B)^{1/p}$ as $p o \infty$

Generalization of Kato's limit to spectral order supremum

Abstract

We obtain limit theorems for $\Phi(A^p)^{1/p}$ and $(A^p\sigma B)^{1/p}$ as $p\to\infty$ for positive matrices $A,B$, where $\Phi$ is a positive linear map between matrix algebras (in particular, $\Phi(A)=KAK^*$) and $\sigma$ is an operator mean (in particular, the weighted geometric mean), which are considered as certain reciprocal Lie-Trotter formulas and also a generalization of Kato's limit to the supremum $A\vee B$ with respect to the spectral order.