Abstract
The Ando-Hiai inequality says that if A# α B ≤ I for a fixed α ∈ [0, 1] and positive invertible operators A, B on a Hilbert space, then A r # α B r ≤ I for r ≥ 1, where # α is the α-geometric mean defined by \(A \#_\alpha B=A^{\frac 12}(A^{-\frac 12}BA^{-\frac 12})^\alpha A^{\frac 12}\). This chapter is devoted by extensions and applications of Ando-Hiai inequality. It is closely related to Furuta inequality, Bebiano-Lemos-Providência inequality and grand Furuta inequality. Consequently they are given useful extensions.
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Fujii, M., Nakamoto, R. (2022). Ando-Hiai Inequality: Extensions and Applications. In: Aron, R.M., Moslehian, M.S., Spitkovsky, I.M., Woerdeman, H.J. (eds) Operator and Norm Inequalities and Related Topics. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-02104-6_2
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