Ando-Hiai and Golden-Thomspon inequalities

The original Ando-Hiai and Golden-Thompson inequalities present comparisons for the operator geometric mean $\sharp_v$ when $0\leq v\leq 1.$ Our main target in this article is to study these celebrated inequalities for means other than the geometric mean and for the geometric mean when $v\not\in [0,1].$

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