Abstract
We obtain limit theorems for \(\Phi (A^p)^{1/p}\) and \((A^p\sigma B)^{1/p}\) as \(p\rightarrow \infty \) 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.
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This work was supported by JSPS KAKENHI Grant Number JP17K05266.
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Hiai, F. (2019). Matrix Limit Theorems of Kato Type Related to Positive Linear Maps and Operator Means. In: Rassias, T.M., Zagrebnov, V.A. (eds) Analysis and Operator Theory . Springer Optimization and Its Applications, vol 146. Springer, Cham. https://doi.org/10.1007/978-3-030-12661-2_9
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