Ando–Hiai-type inequalities for operator means and operator perspectives

Abstract

We improve the existing Ando-Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie-Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators.

Pusz--Woronowicz functional calculus and extended operator convex perspectives

“…The next proposition is an extended version of (ii) and (iii) of Theorem 4.9 and also extends [32,Theorem 6.7].…”

“…A positive R-valued function f on (0, ∞) is said to be power monotone increasing (pmi for short) if f (t p ) ≥ f (t) p for all t > 0 and p ≥ 1. The next proposition is a slight extension of an AH-inequality in [32]. Proposition 8.13.…”

“…(a) A 2 ≤ λB for some λ > 0, for all (A, B) in (a certain subset of) B(H) + × B(H) + and for either all p ≥ 1 or all p ∈ (0, 1]. Inequalities of this type were first shown in [7] for the weighted geometric means and further studied in, e.g., [32,50]. A positive R-valued function f on (0, ∞) is said to be power monotone increasing (pmi for short) if f (t p ) ≥ f (t) p for all t > 0 and p ≥ 1.…”

Log-majorization Type Inequalities

“…Other Ando-Hiai type inequalities have meanwhile been obtained. See the recent works by Hiai, Seo and Wada [49,50], Kian, Moslehian and Seo [57,58,59] and references therein.…”