Ando-Hiai Inequality: Extensions and Applications

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.