When does Ando–Hiai inequality hold?

Wada, Shuhei

doi:10.1016/j.laa.2017.11.030

Cited by 12 publications

(11 citation statements)

References 4 publications

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“…This holds only when r ≤ 1. From these arguments on the special case of power means as well as the result in [28] mentioned above, we see that the condition r ≥ 1 or r ≤ 1 is essential for Ando-Hiai type inequalities in Section 3.…”

Section: )mentioning

confidence: 77%

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Ando–Hiai type inequalities for multivariate operator means

Hiai

Seo

Wada

2018

Linear and Multilinear Algebra

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We present several Ando-Hiai type inequalities for n-variable operator means for positive invertible operators. Ando-Hiai's inequalities given here are not only of the original type but also of the complementary type and of the reverse type involving the generalized Kantorovich constant. Multivariate meansThroughout the paper, H is a general Hilbert space assumed to be infinite-dimensional unless otherwise stated, B(H) is the algebra of all bounded linear operators on H, B(H) + the set of positive operators in B(H), and P = P(H) the set of positive invertible operators in B(H). For X ∈ P, X ∞ is the operator norm of X and λ min (X) is the minimum of the spectrum of X, that is, λ min (X) = X −1 −1 ∞ . Moreover, I denotes the identity operator, and SOT means the strong operator topology on B(H). The Thompson metric d T on P is defined bywhere M(A/B) := inf{α > 0 : A ≤ αB}. It is known [26] that (P, d T ) is a complete metric space and both topologies on P induced by d T and · ∞ coincide.

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Section: )mentioning

confidence: 77%

“…As for the weaker formulation in (3.5) restricted to the 2-variable case, it was shown in [28,Corollary 3.1] that if σ is a p.m.i. operator mean with σ = l, r and r > 0, then AσB ≥ I =⇒ A r σB r ≥ I hods for every A, B ∈ P if and only if r ≥ 1.…”

Section: )mentioning

confidence: 99%

Ando–Hiai type inequalities for multivariate operator means

Hiai

Seo

Wada

2018

Linear and Multilinear Algebra

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“…(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.…”

Section: Dense Domain Case and Boundednessmentioning

confidence: 99%

Pusz--Woronowicz functional calculus and extended operator convex perspectives

Hiai¹,

Ueda²,

Wada³

2021

Preprint

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In this article, we first study, in the framework of operator theory, Pusz and Woronowicz's functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on [0, ∞) 2 . Our construction has special features that functions on [0, ∞) 2 are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in (−∞, ∞]. Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on (0, ∞). We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Ando's theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.

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“…We conclude this short note by stating that a property f (x) r ≥ f (x r ) has been studied in [12,13] in details by the different approach. Such a property is named there as power monotonicity and applied to some known results in the framework of operator theory.…”

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confidence: 93%

Note on constants appearing in refined Young inequalities

Furuichi

2018

Preprint

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In this short note, we give the refined Young inequality with Specht's ratio by only elementary and direct calculations. The obtained inequality is better than one previously shown by the author in 2012. In addition, we give a new property of Specht's ratio. These imply an alternative proof of the refined Young ineqaulity shown by author in 2012. We also give a remark on the relation to Kantorovich constant. Finally, we give a proposition for a corresponding general function.

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When does Ando–Hiai inequality hold?

Cited by 12 publications

References 4 publications

Ando–Hiai type inequalities for multivariate operator means

Ando–Hiai type inequalities for multivariate operator means

Pusz--Woronowicz functional calculus and extended operator convex perspectives

Note on constants appearing in refined Young inequalities

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