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 $(A^p\sigma B)^{1/p}$ 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.