Eigenvalue inequalities for convex and log-convex functions

Aujla, Jaspal Singh; Bourin, Jean-Christophe

doi:10.1016/j.laa.2006.02.027

Cited by 44 publications

(31 citation statements)

References 8 publications

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“…Lemma 2.9 is a part of Stinespring's theory of positive and completely positive linear maps in the influential 1955 paper [26]. The proof given here is somewhat original and is taken from [4]. Note that in the course of the proof, we prove Naimark's dilation theorem: if {A i } n i=1 are positive operators on a space S such that n i=1 A i I, then there exist some mutually orthogonal projections {P i } n i=1 on a larger space H ⊃ S such that (P i ) S = A i , (1 i n).…”

Section: Comments and Referencesmentioning

confidence: 99%

Unitary orbits of Hermitian operators with convex or concave functions

Bourin

Lee

2012

Bulletin of the London Mathematical Society

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This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen-, sub-or superadditivity-type inequalities are considered. Some of them are substitutes to classical inequalities (Choi, Davis, Hansen-Pedersen) for operator convex or concave functions. Various trace, norm and determinantal inequalities are derived. Combined with an interesting decomposition for positive semi-definite matrices, several results for partitioned matrices are also obtained.

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Section: Comments and Referencesmentioning

confidence: 99%

Unitary orbits of Hermitian operators with convex or concave functions

Bourin

Lee

2012

Bulletin of the London Mathematical Society

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“…One such substitute allows for conjugation by unitary matrices, as initiated in [1]. A satisfactory recent result [6] along these lines asserts that if f : [0, ∞) → R is nondecreasing, concave, and f For q ∈ (0, 1), when f (t) = t q and · is the Schatten 1 norm, the resulting inequality goes back to [57] and it corresponds to (165) with C = I (and K = 1).…”

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confidence: 99%

Metric Inequalities

Schechtman

2016

Forum of Mathematics, Pi

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For every p ∈ (0, ∞) we associate to every metric space (X, d X ) a numerical invariantThus, our new bi-Lipschitz invariant certifies that L q does not admit a bi-Lipschitz embedding into L p when 2 < q < p < ∞. This completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of L p spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of L q into L p when 2 < q < p < ∞. Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into L p of snowflakes of L q and integer grids in n q , for 2 < q < p < ∞. As a byproduct of our investigations, we also obtain results on the geometry of the Schatten p trace class S p that are new even in the linear setting.

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“…If f is a non-negative operator monotone function on [0, ∞), then the subadditivity of f does not hold in general. Aujla and Bourin [2] showed that if A, B ≥ 0 are matrices and f : [0, ∞) → [0, ∞) is a concave function, i.e. −f is convex, then there exist unitaries U, V such that…”

Section: S(a B)mentioning

confidence: 99%

Around Operator Monotone Functions

Moslehian

Najafi

2011

Integr. Equ. Oper. Theory

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Abstract. We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if f (A + B) ≤ f (A) + f (B) for all non-negative operator monotone functions f on [0, ∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f • g of an operator convex function f on [0, ∞) and a non-negative operator monotone function g on an interval (a, b) is operator monotone and give some applications.

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Eigenvalue inequalities for convex and log-convex functions

Cited by 44 publications

References 8 publications

Unitary orbits of Hermitian operators with convex or concave functions

Unitary orbits of Hermitian operators with convex or concave functions

Metric Inequalities

Around Operator Monotone Functions

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