**Reciprocal Lie-Trotter formula.** / Audenaert, Koenraad; Hiai, F.

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Let_{ A} and _{B} be positive semidefinite matrices. The limit of the expression Z_{p} := (A^{p/2 }B^{p} A^{p/2})^{1/p} as _{p} tends to _{0} is given by the well-known Lie–Trotter formula. A similar formula holds for the limit of G_{p} := (A^{p} # B^{p})^{2/p }as _{p} tends to _{0}, where _{X # Y }is the geometric mean of _{X} and _{Y}. In this paper we study the limit of Z_{p} and G_{p} as _{p} tends to _{∞} instead of _{0}, with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie–Trotter formula. We show that the limit of Z_{p} exists and find an explicit formula in a special case. The limit of G_{p} is shown for _{2 × 2} matrices only.

Original language | English |
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Pages (from-to) | 1220-1235 |

Number of pages | 16 |

Journal | Linear and Multilinear Algebra |

Volume | 64 |

Issue number | 6 |

Early online date | 7 Sep 2015 |

DOIs | |

Publication status | Published - 2016 |

This open access research output is licenced under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

ID: 23701797