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

Fumio Hiai ()
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Fumio Hiai: Tohoku University (Emeritus)

A chapter in Analysis and Operator Theory, 2019, pp 167-189 from Springer

Abstract: Abstract We obtain limit theorems for $$\Phi (A^p)^{1/p}$$ Φ ( A p ) 1 / p and $$(A^p\sigma B)^{1/p}$$ ( A p σ B ) 1 / p as $$p\rightarrow \infty $$ p → ∞ for positive matrices A, B, where $$\Phi $$ Φ is a positive linear map between matrix algebras (in particular, $$\Phi (A)=KAK^*$$ Φ ( A ) = K A K ∗ ) and $$\sigma $$ 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 $$A\vee B$$ A ∨ B with respect to the spectral order.

Keywords: Positive semidefinite matrix; Lie–Trotter formula; Positive linear map; Operator mean; Operator monotone function; Geometric mean; Antisymmetric tensor power; Rényi relative entropy; Primary: 15A45; 15A42; 47A64 (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-12661-2_9

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DOI: 10.1007/978-3-030-12661-2_9

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