Jensen and Minkowski inequalities for operator means and anti-norms

Abstract

Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means such as some p-means and some Kubo-Ando means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q ∈ [−∞, 1] and the Minkowski functional det 1/n A. An interpolation theorem for the Schur multiplication is given in this setting.2010 Mathematics Subject Classification: Primary 15A60, 47A30, 47A60

Unitary orbits of Hermitian operators with convex or concave functions

“…The above proof is much simpler than the original one. For more details and many results on anti-norms and derived anti-norms, often in connection with Theorem 2.1, see [10,11]. Several results in these papers are generalizations of Corollary 2.3.…”

Anti-norms on Finite von Neumann Algebras

“…Therefore, (ii) ⇒ (i) does not hold for general A, B ∈ M + . Such a subtle difference between the two conditions (i) and (ii) never occurs in the matrix case: In the matrix algebra M n , the conditions (i) and (ii) in Theorem 6.11 are equivalent; (i) ⇒ (ii) is shown in [7,Lemma 4.10], and (ii) ⇒ (i) is implicit in [7,Example 4.5], the discrete version of the anti-norms in (6.2).…”

Geometric convexity of an operator mean

“…Although there are a number of functions belonging to PMI, it is not easy to show the pmi property (1.1) of a certain operator mean because the verification of condition (1.1) or (2.1) requires considerable calculation. Bourin and Hiai [3] mention that a positive operator monotone function f on [0, ∞) belongs to GM if and only if d n dt n f (e t ) ≥ 0 for all n ≥ 0. Thus we need to determine a technique for obtaining pmi means and evaluate this technique.…”