#### Abstract

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 inequalities, operator monotone functions, Borel measures, and certain operator equations.

#### 1. Introduction

The concept of means, a natural notion in mathematics, plays important roles in mathematics itself, computer science, statistics, various branches in science, engineering, and economics. This concept was developed since the ancient Greeks until the last century by many mathematicians (see ). Nowadays, according to the definition of a mean for positive real numbers in , a mean is defined to be satisfied by the following properties:(i)Positivity: and .(ii)Betweenness: .

A mean is said to be(i)strict at the left if for each and , (ii)strict at the right if for each and , (iii)strict if it is strict at both the right and the left.

This paper focuses on means for positive operators on a Hilbert space. Let be the algebra of bounded linear operators on a Hilbert space . The set of positive operators on is denoted by . Denote the spectrum of an operator by . For self-adjoint operators, , the partial order indicates that . If is invertible, then we write .

A starting point for the theory of operator means is the presence of the notion of parallel sum in electrical network analysis (see ). A connection is a binary operation assigned to each pair of operators in such that the following conditions are satisfied for all :(M1)monotonicity: ;(M2)transformer inequality: ;(M3)continuity from above: for , if and , then . Here, indicates that is a decreasing sequence (with respect to the partial order) and converges strongly to .

This definition was modelled from significant properties of the parallel sum by Kubo and Ando in . Two trivial examples are the left-trivial mean and the right-trivial mean . See [4, Section 3] and  for more information about operator connections. From the transformer inequality, every connection is congruence invariant in the sense that for each and we have

A mean in Kubo-Ando sense is a connection with fixed-point property for all . The class of Kubo-Ando means cover many well-known means in practice, for example,(i)-weighted arithmetic means: ;(ii)-weighted geometric means:(iii)-weighted harmonic means:(iv)logarithmic mean: where the function is given by for each , , and .

A summary of Kubo-Ando theory is given in terms of one-to-one correspondences between operator connections on , operator monotone functions from to , and finite Borel measures on . Recall that a continuous function is said to be operator monotone iffor all positive operators and for all Hilbert spaces . This concept was introduced in ; see also [7, Chapter ], [4, Section 2], and . A connection on can be characterized via operator monotone functions as follows.

Theorem 1 (see [3, Theorem 3.2]). Given a connection , there is a unique operator monotone function satisfyingMoreover, the map is a bijection.

We call the representing function of . A connection also has a canonical characterization with respect to a Borel measure via a meaningful integral representation as follows.

Theorem 2. Given a finite Borel measure on , the binary operationis a connection on . Moreover, the map is bijective, in which case the representing function of is given by

Theorem 2 is a modification of Kubo-Ando theorem ([3, Theorem 3.4]). We call the associated measure of .

Theorem 3 (see [3, Theorem 3.3]). Let be a connection on with representing function and associated measure . Then the following statements are equivalent:(1);(2) for all ;(3) is normalized; that is, ; (4) is normalized; that is, is a probability measure.

Hence every mean can be regarded as an average of weighted harmonic means. From (8) and (9) in Theorem 2, and are related by

In this paper, we provide various characterizations for the concepts of positivity, betweenness, and strictness of operator means in terms of operator inequalities, 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, resp.).

#### 2. Positivity

We say that a connection satisfies the positivity property ifRecall that the transpose of a connection is the connectionIf is the representing function of , then the representing function of its transpose is given byand is defined by continuity (see [3, Corollary 4.2]).

Theorem 4. Let be a connection on with representing function and associated measure . Then the following statements are equivalent:(1) satisfies the positivity property;(2);(3) (here, is the zero connection ); (4)for all , (positive definiteness); (5)for all , ; (6)for all , ; (7)for all and , ; (8)for all and , ; (9) (here, is the function ); (10); (11).

Proof. Implications , , , , and are clear. Using the integral representations in Theorem 2, it is straightforward to verify that the representing function of the zero connection is the constant function and its associated measure is the zero measure. Hence, we have the equivalences .
. Assume . Suppose that there is such that . Then for all . The concavity of implies that for all . Hence , a contradiction.
. Assume . Let and be such that . Thenand . Now, (5) yields ; that is, .
. It is similar to .
. Assume that for all . Since by spectral mapping theorem, we have for all . Hence, for each and ,. Assume (10). Let be such that . Note thathere . Since , we have .
. Assume . Let be such that . Then where is the representing function of the transpose of . We see that for . The injectivity of functional calculus implies that for all . We conclude that ; that is, .
. Assume (10). Let be such that . Then . By the injectivity of functional calculus, we have for all . Assumption implies that . Thus, .

Remark 5. It is not true that implies the condition that for all and , implies or . Indeed, take to be the geometric mean and

#### 3. Betweenness

We say that a connection satisfies the betweenness property if for each and ,

By Theorem 4, 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 6. The following statements are equivalent for a connection with representing function :(1) is a mean;(2) satisfies the betweenness property;(3)for all , ; (4)for all , ; (5)for all , ; (6)for all , ; (7)for all and , ; (8)for all , ; (9)for all , ; (10)the only solution to the equation is ; (11)for all , the only solution to the equation is .

Proof. Implications , , , and are clear.
. Let be such that . The fixed-point property and the monotonicity of yield. Since , we have ; that is, . Hence is a mean by Theorem 3.
. It is similar to .
. We have . Hence,Therefore, is a mean by Theorem 3.
. It is similar to .
. Let . Consider such that . Then by the congruence invariance of , we have. If , then which is ; that is, .
. We have .
. It is similar to .

Remark 7. For a connection and , the operators and need not be comparable. The previous theorem tells us that if is a mean, then the condition guarantees the comparability between and .

#### 4. Strictness

We consider the strictness of Kubo-Ando means as that for scalar means in .

Definition 8. A mean on is said to be (i)strict at the left if for each and ,(ii)strict at the right if for each and ,(iii)strict if it is strict at both the right and the left.

In order to prove the next two lemmas, recall the following facts: if is operator monotone, then(i) is operator concave and hence concave in usual sense (see  or [4, Corollary ]);(ii) is convex in usual sense (see [3, Lemma 5.2]);(iii) is operator monotone on (see  or [4, Corollary ]).

Lemma 9. If is an operator monotone function such that is a constant on an interval with , then is a constant on .

Proof. Assume that for all . The case is done by using the monotonicity and concavity of . Consider the case . The monotonicity and concavity of imply that for all . If , then on by the monotonicity of . Consider the case and suppose there is an such that . Then the slope of the line segment joining the point and the point is greater than . This contradicts the convexity of the function .

Lemma 10. If is an operator monotone function such that for some and on an interval with , then on .

Proof. If there is such that , then by Lemma 9. Suppose that for all . For simplicity, assume that for all . Then the function is operator monotone on and hence on by continuity. Note that on . Lemma 9 implies that on ; that is, on .

Theorem 11. Let be a mean with representing function and associated measure . Then the following statements are equivalent:(1) is strict at the left;(2) is not the left-trivial mean;(3)for all , ; (4)for all , ; (5)for all and , ; (6)for all , ; (7)for all , ; (8)for all , ; (9)for all , ; (10)for all and , ; (11)for all and , ; (12) is not the constant function ; (13)for all , ; (14)for all , ; (15)for all , ; (16) is not the Dirac measure at .

Proof. It is clear that and each of implies . Also, each of implies (12).
. Let be such that . Then and hence for all . Suppose that where is the spectral radius of . Then for all . It follows that on by Lemma 9. This contradicts assumption . We conclude that ; that is, for some . Suppose now that . Since , we have that is a constant on the interval . Again, Lemma 9 implies that on , a contradiction. Similarly, gives a contradiction. Thus , which implies .
. Let be such that . Then where is the representing function of the transpose of . Hence, for all . Suppose that . Then for all . It follows that on by Lemma 10. Hence, the transpose of is the right-trivial mean. This contradicts assumption . We conclude that ; that is, for some . The same argument as in yields .
. Use the congruence invariance of .
. Assume that is not the left-trivial mean. Let be such that . Then . The spectral mapping theorem implies that for all . Suppose that there exists such that . Since , we have . It follows that for . By Lemma 9, on , a contradiction. We conclude that for all ; that is, .
. It is similar to .
. Assume . Let be such that . Thenwhich implies . By (6), we have or .
. It is similar to .
. Use the congruence invariance of .
. Use the congruence invariance of .
. Note that the representing function of the left-trivial mean is the constant function . Its associated measure is the Dirac measure at .
. Assume . Let be such that . Suppose that . It follows that for all lying between and . Lemma 9 implies that on , contradicting assumption .
, . Modify the argument in the proof .

Theorem 12. Let be a mean with representing function and associated measure . Then the following statements are equivalent:(1) is strict at the right;(2) is not the right-trivial mean;(3)for all , ; (4)for all , ; (5)for all and , ; (6)for all , ; (7)for all , ; (8)for all , ; (9)for all , ; (10)for all and , ; (11)for all and , ; (12) is not the identity function ; (13) is not the associated measure at .

Proof. Replace by its transpose in the previous theorem.

We immediately get the following corollaries.

Corollary 13. A mean is strict if and only if it is nontrivial.

Corollary 14. Let be a nontrivial mean. For each and , the following statements are equivalent:(i);(ii);(iii);(iv);(v).

The next result is a generalization of [10, Theorem 4.7] in which the mean is the geometric mean.

Corollary 15. Let be a nontrivial mean. For each and , the following statements are equivalent:(i);(ii);(iii);(iv);(v).

Remark 16. (i) It is not true that if is not the left-trivial mean then, for all and , . Indeed, take to be the geometric mean, , andThe case of right-trivial mean is just the same.
(ii) The assumption of invertibility of or in Corollary 14 cannot be omitted, as a counter example in (i) shows. Also, the invertibility of or in Corollary 15 cannot be omitted. Consider the geometric mean and

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The author is supported by King Mongkut’s Institute of Technology Ladkrabang Research Fund Grant no. KREF045710.