Matrix Limit Theorems of Kato Type Related to Positive Linear Maps and Operator Means

Abstract

We obtain limit theorems for Φ(Ap)1/p and (ApσB) 1/p as p→ ∞ for positive matrices A, B, where Φ is a positive linear map between matrix algebras (in particular, Φ (A) = KAK∗) and σ 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∨ B with respect to the spectral order.