Geometrical significance of the Löwner-Heinz inequality

It is proven that the Lowner-Heinz inequality ‖AtBt‖ ≤ ‖AB‖t, valid for all positive invertible operators A,B on the Hilbert space H and t ∈ [0, 1], has equivalent forms related to the Finsler structure of the space of positive invertible elements of L(H) or, more generally, of a unital C∗algebra. In particular, the Lowner-Heinz inequality is equivalent to some type of “nonpositive curvature” property of that space.

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