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

We obtain limit theorems for $Phi(A^p)^{1/p}$ and $(A^psigma B)^{1/p}$ as $ptoinfty$ 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 Katos limit to the supremum $Avee B$ with respect to the spectral order.