Some properties of weighted operator means and characterizations of interpolational means

Operator means deformed by a fixed point method

“…Note that the condition in particular contains m 0 = l and m 1 = r. In [5] Fujii and Kamei constructed, given a symmetric operator mean σ, a regular continuous family of operator means {m α } α∈[0,1] in the following way: Am 2k+1 2 n+1 B := (Am k 2 n B)m(Amk+1 2 n B) for n, k ∈ N with 2k + 1 < 2 n+1 . The construction was extended by Pálfia and Petz [21] to an arbitrary (not necessarily symmetric) operator mean σ ( = l, r) in such a way that m s = σ when s = f ′ σ (1); see [24] for the equivalence between the two constructions for a symmetric σ. By Propositions 3.3 and 3.8 note also that if {m α } α∈[0,1] is such a regular continuous family, then so is the deformed {(m α ) σ } α∈[0,1] by any σ = l; for instance, {(!…”

“…It was shown in [4, Theorem 1] that σ satisfies condition (ii) if and only if {m α } is an interpolation family. Then it follows from [24,Theorem 6]…”

“…It was shown in [4, Theorem 1] that σ satisfies condition (ii) if and only if {m α } is an interpolation family. Then it follows from[24, Theorem 6] that {m α } = {p α,r } α for some r ∈ [−1, 1]. Hence σ = m 1/2 = p 1/2,r .…”

“…is a solution to equation (2.13) for τ = m α and σ = m β , that is, (m α ) m β = m α for all α ∈ [0, 1] and β ∈ (0, 1]. It was proved in[24] that if {m α } α∈[0,1] is a regular continuous family of operator means, then it is an interpolation family if and only if there exists an r ∈ [−1, 1] such that m α = p α,r for all α ∈ [0, 1]. Therefore, we have, for every α ∈ [0, 1], β ∈ (0, 1] and r ∈ [−1, 1], (p α,r ) p β,r = p α,r .…”

The Ando-Hiai inequalities for the solution of the generalized Karcher equation and related results

“…Then {σ λ } λ∈[0,1] should have the interpolatinal property [28,Theorem 4]. Moreover since σ λ is an operator mean, it should be the power mean by [28,Theorem 6]. 14…”