An extension of the Löwner–Heinz inequality

“…It is noted in [12] that inequality (32) is sharp in the sense that when A and B are positive scalars of the identity operator, and then (32) becomes equality. Following the proof of Lemma 3.7, we realize that if Φ(A) = aI and Φ(A r ) 1/r = bI with a > 0 and b > 0, then (30) turns into equality.…”

Section: Corollary 34 Let φ Be a Unital Positive Linear Map And Letmentioning

confidence: 99%

Asymmetric Choi--Davis inequalities

Kian,

Moslehian,

Nakamoto

2020

Preprint

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Let Φ be a unital positive linear map and let A be a positive invertible operator. We prove that there exist partial isometries U and V such thathold under some mild operator convex conditions and some positive numbers r.Further, we show that if f 2 is operator concave, thenIn addition, we give some counterparts to the asymmetric Choi-Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin-Ricard and Furuta.

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“…We start this section with the following lemma. The first part borrow from [11] and the second is another variant of it.…”

Section: Estimates Of Operator Monotone Functionsmentioning

confidence: 99%

“…Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In [11], an estimation of the Löwner-Heinz inequality was proposed as follows.…”

Section: Introductionmentioning

confidence: 99%

Estimates of operator convex and operator monotone functions on bounded intervals

Fujii¹,

Moslehian²,

Najafi³

et al. 2016

Preprint

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Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if f is a nonlinear operator convex function on a bounded interval (a, b) and A, B are bounded linear operators acting on a Hilbert space with spectra in (a, b) and A − B is invertible, thenA short proof for a similar known result concerning a nonconstant operator monotone function on [0, ∞) is presented. Another purpose is to find a lower bound for f (A) − f (B), where f is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the Löwner-Heinz inequality.

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An extension of the Löwner–Heinz inequality

Abstract: We extend the celebrated Löwner-Heinz inequality by showing that if A, B are Hilbert space operators such that A > B 0, thenfor each 0 < r 1. As an application we prove that

Cited by 18 publications

References 6 publications

Reverse and improved inequalities for operator monotone functions

Reverse and improved inequalities for operator monotone functions

Asymmetric Choi--Davis inequalities

Estimates of operator convex and operator monotone functions on bounded intervals

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