Geometric convexity of an operator mean

Abstract

Let σ be an operator mean in the sense of Kubo and Ando. If the representation function fσ of σ satisfies fσ(t) p ≤ fσ(t p ) for all p > 1, then σ is called a pmi mean. Our main interest is the class of pmi means (denoted by P M I). To study P M I, the operator mean σ, whereinis considered in this paper. The set of such means (denoted by GCV ) includes certain significant examples and is contained in P M I. The main result presented in this paper is that GCV is a proper subset of P M I.In addition, we investigate certain operator-mean classes, which contain

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

“…From Corollary 5.3 and Proposition 5.6 we notice that if h ∈ OM 1 + is pmd, then h(t) ≤ t α where α = h (1) (∈ [0, 1]), which was recently pointed out in [37,Section 5]. Moreover it was shown in [37] that there is an h ∈ OM 1 + such that h(t) ≤ t α for some α ∈ [0, 1] but h(t p ) ≤ h(t) p for any p > 1 (hence h is not pmd). We thus see that for h ∈ OM 1 + , the AH inequality…”

“…A study of operator means whose representing functions are geometrically convex is found in a recent paper [37]. An operator mean is called a geodesic mean if it has the representing function h(t) = with α ∈ (0, 1), note by Proposition 3.15 that…”

Operator means of probability measures

“…for σ. But in[47] Wada recently proved that the converse is not true, that is, there is a p.m.i. σ that is not g.c.v.…”