Positivity, Betweenness and Strictness of Operator Means

An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness and strictness of operator means in terms of operator monotone functions, Borel measures and certain operator equations.


Introduction
According to the definition of a mean for positive real numbers in [10], a mean M is defined to be satisfied the following properties • strict if it is both strict at the right and the left.
A general theory of operator means was given by Kubo and Ando [8]. Let B(H) be the algebra of bounded linear operators on a Hilbert space H. The set of positive operators on H is denoted by B(H) + . Denote the spectrum of an operator X by Sp(X). For Hermitian operators A, B ∈ B(H), the partial order A B indicates that B − A ∈ B(H) + . The notation A > 0 suggests that A is a strictly positive operator. A connection is a binary operation σ on B(H) + such that for all positive operators A, B, C, D: In fact, the parallel sum, introduced by Anderson and Duffin [1] for analyzing electrical networks, is a model for general connections. From the transformer inequality, every connection is congruence invariant in the sense that for each A, B 0 and C > 0 we have A mean in Kubo-Ando sense is a connection σ with fixed-point property A σ A = A for all A 0. The class of Kubo-Ando means cover many wellknown means in practice, e.g.
• α-weighted arithmetic means: It is a fundamental that there are one-to-one correspondences between the following objects: (1) operator connections on B(H) + (2) operator monotone functions from R + to R + Recall that a continuous function f : for all positive operators A, B ∈ B(H) and for all Hilbert spaces H. This concept was introduced in [9]; see also [2,6,7]. Every operator monotone function from R + to R + is always differentiable (see e.g. [6]) and concave in usual sense (see [5]).
A connection σ on B(H) + can be characterized via operator monotone functions as follows: . Given a connection σ, there is a unique operator monotone function f : R + → R + satisfying f (x)I = I σ (xI), x 0.

Moreover, the map σ → f is a bijection.
We call f the representing function of σ. A connection also has a canonical characterization with respect to a Borel measure via a meaningful integral representation as follows.

Theorem 1.2 ([3]). Given a finite Borel measure µ on [0, 1], the binary operation
is a connection on B(H) + . Moreover, the map µ → σ is bijective, in which case the representing function of σ is given by We call µ the associated measure of σ.
Theorem 1.3. Let σ be a connection on B(H) + with representing function f and representing measure µ. Then the following statements are equivalent.
Hence every mean can be regarded as an average of weighted harmonic means. From (1) and (2) in Theorem 1.2, σ and f are related by In this paper, we provide various characterizations of the concepts of positivity, betweenness and strictness of operator means in terms of operator monotone functions, Borel measures and certain operator equations. It turns out that every mean satisfies the positivity property. The betweenness is a necessary and sufficient condition for a connection to be a mean. A mean is strict at the left (right) if and only if it is not the left-trivial mean (the right-trivial mean, respectively).

Positivity
We say that a connection σ satisfies the positivity property if Recall that the transpose of a connection σ is the connection If f is the representing function of σ, then the representing function of its transpose is given by and g(0) is defined by continuity. (1) σ satisfies the positivity property ; (2) I σ I > 0 ; (10) ⇒ (4): Assume (10). Let A 0 be such that A σ A = 0. Then

Betweenness
We say that a connection σ satisfies the betweenness property if for each A, B 0, By Theorem 2.1, every mean enjoys the positivity property. In fact, the betweenness property is a necessary and sufficient condition for a connection to be a mean: Theorem 3.1. The following statements are equivalent for a connection σ with representing function f : (1) σ is a mean ; (2) σ satisfies the betweenness property ; (3) for all A 0, A I =⇒ A A σ I I ; (4) for all A 0, I A =⇒ I I σ A A ; (10) the only solution X > 0 to the equation X σ X = I is X = I ; (11) for all A > 0, the only solution X > 0 to the equation X σ X = A is X = A.  Therefore, σ is a mean by Theorem 1.3.
(2) ⇒ (5): If t 1, then I I σ (tI) tI which is I f (t)I tI, i.e. 1 f (t) t. (1) ⇒ (11): Let A > 0. Consider X > 0 such that X σ X = A. Then by the congruence invariance of σ, we have X = X 1/2 (I σ I)X 1/2 = X σ X = A. For a connection σ and A, B 0, the operators A, B and A σ B need not be comparable. The previous theorem tells us that if σ is a mean, then the condition 0 A B guarantees the comparability between A, B and AσB.

Strictness
We consider the strictness of Kubo-Ando means as that for scalar means in [10]: • strict if it is both strict at the right and the left.
The following lemmas are easy consequences of the fact that an operator monotone function f : R + → R + is always monotone, concave and differentiable.   (2) ⇒ (4): Let A > 0 be such that A σ I = A. Then g(A) = I where g is the representing function of the transpose of σ. Hence, g(λ) = λ for all λ ∈ Sp(A). Suppose that α ≡ inf Sp(A) < r(A). Then g(x) = x for all x ∈ [α, r(A)]. It follows that g(x) = x on R + by Lemma 4.3. Hence, the transpose of σ is the right-trivial mean. This contradicts the assumption (2). We conclude α = r(A), i.e. Sp(A) = {λ} for some λ 0. Suppose now that λ < 1. Since g(1) = 1, we have that g(x) = x on the interval [λ, 1]. Lemma 4.3 forces g(x) = x on R + , a contradiction. Similarly, λ > 1 gives a contradiction. Thus λ = 1, which implies A = I.
(2) ⇒ (6): Assume that σ is not the left-trivial mean. Let A 0 be such that I σ A I. Then f (A) I. The spectral mapping theorem implies that f (λ) 1 for all λ ∈ Sp(A). Suppose there exists a t ∈ Sp(A) such that t < 1. Since f (t) f (1) = 1, we have f (t) = 1. It follows that f (x) = 1 for t x 1. By Lemma 4.2, f ≡ 1 on R + , a contradiction. We conclude λ 1 for all λ ∈ Sp(A), i.e. A I.
Theorem 4.5. Let σ be a mean with representing function f and associated measure µ. Then the following statements are equivalent: (1) σ is strict at the right ; (2) σ is not the right-trivial mean ;