Geometrical significance of the Löwner-Heinz inequality

Andruchow, E.; Corach, G.; Stojanoff, D.

doi:10.1090/S0002-9939-99-05085-6

Geometrical significance of the Löwner-Heinz inequality
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by E. Andruchow, G. Corach and D. Stojanoff PDF

Proc. Amer. Math. Soc. 128 (2000), 1031-1037 Request permission

Abstract:

It is proven that the Löwner-Heinz inequality ${\|A^{t}B^{t}\|\le \|AB\|^{t}}$, valid for all positive invertible operators ${A, B}$ on the Hilbert space ${\mathcal H }$ and ${t\in [0,1]}$, has equivalent forms related to the Finsler structure of the space of positive invertible elements of ${\mathcal L (\mathcal H )}$ or, more generally, of a unital ${C^{*}}$-algebra. In particular, the Löwner-Heinz inequality is equivalent to some type of “nonpositive curvature" property of that space.

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