Unexpected relations which characterize operator means

By Hiroyuki Osaka and Shuhei Wada

Abstract

We give some characterizations of self-adjointness and symmetricity of operator monotone functions by using the Barbour transform and show that there are many non-symmetric operator means between the harmonic mean ! and the arithmetic mean . Indeed, we show that there exists a non-symmetric operator mean between any two symmetric operator means.

1. Introduction

A bounded operator acting on a Hilbert space is said to be positive if for all . We denote this by . Let be the set of all positive operators on and let be the set of all positive invertible operators on .

A real-valued function on is operator monotone if whenever bounded operators , satisfy , . Functions and are typical operator monotone functions. Let be the set of positive operator monotone functions on and .

In Reference 6, Kubo and Ando developed an axiomatic theory for operator connections and operator means for pairs of positive operators. That is, a binary operation on the class of positive operators, , is called a connection if the following requirements are fulfilled:

(I)

and imply .

(II)

.

(III)

If and , then .

A mean is a connection with normalization condition

(IV) .

Kubo and Ando showed that there exists an affine order-isomorphism from the class of operator connections onto the class of positive operator monotone functions, .

This theory has found a number of applications in operator theory. In particular, Petz Reference 9 connected the theory of monotone metrics with Kubo and Ando’s operator connections. He proved that an operator monotone function satisfying the functional equation

is related to a Morozova–Chentsov function which gives a monotone metric on the Riemannian manifold of invertible density matrices.

It is well known that if is operator monotone, the transpose , the adjoint , and the dual are also operator monotone (Reference 6) and we call symmetric if and self-adjoint if . It is shown in Reference 6 that if is symmetric with , then the corresponding operator mean exists between the harmonic mean ! and the arithmetic mean , that is, .

In this note, we characterize symmetric functions and self-adjoint functions using Barbour transform defined by , and characterize the class of non-symmetric operator functions and the class of operator connections which satisfy or . We show the existence of non-symmetric operator means between any pair of symmetric means.

2. Barbour transform

In Reference 7, for any strictly positive continuous functions on , the Barbour path was introduced by

and its basic properties are elucidated. In Reference 1, Barbour examined a function which is an approximation of the power function . We will denote a Barbour path such that , , by the triple .

Proposition 2.1 (Reference 7).

For , the Barbour path exists in .

The transform defined by plays an important role in the analysis of and we call this transform the Barbour transform.

Proposition 2.2 (cf. Reference 7).
(1)

For the transform defined by is injective and .

(2)

For , , where and , and notation means that for any positive operators and .

Note that for , .

For , we can define the inverse map of the Barbour transform by

then . Moreover, for and , we can define the inverse map of the transform by .

By a simple calculation, we have the following relations.

Lemma 2.3.

For and , we have

In particular, , and .

We next show some properties of the Barbour transform concerning the usual order relation for operator means.

Lemma 2.4.

For and , the following are equivalent:

(1)

for all ;

(2)

for all ;

(3)

for all and for all .

Proof.

The proof follows from the equation

In the following section, we show that the Barbour transform is a powerful tool for finding examples of non-self-adjoint operator means.

3. Self-adjoint means

In Reference 6, Kubo and Ando asked whether there exist self-adjoint operator means besides the trivial means , , the weighted geometric means corresponding to the operator monotone functions . Later, Hansen (Reference 3) gave an integral form for a strictly positive self-adjoint operator monotone function based on an exponential map.

Using the Barbour transform, we characterize the self-adjointness in and give concrete examples in this section.

Proposition 3.1.

Let be a positive continuous function on and let . The following are equivalent:

(1)

and ;

(2)

there exists an operator monotone function such that ;

(3)

there exists an operator monotone function such that

Proof.

: Set . Then , , and .

Conversely, if for some , it is obvious that .

: Since , there exists a such that . Since and the Barbour transform is injective, we have ; that is, is symmetric.

Hence,

: Set . Then is symmetric operator monotone and . Furthermore,

The above result shows that we can construct a self-adjoint mean by using a symmetric mean.

Remark 3.2.

Using Proposition 3.1, we can construct many examples of self-adjoint means. For example, if , then the corresponding operator means of functions and are self-adjoint.

4. Symmetric means and non-symmetric means

4.1. Symmetric means

Symmetric means have been discussed many times in the literature Reference 4Reference 5Reference 6Reference 8. In contrast to self-adjoint means, many examples of symmetric means are known and appear in the quantum information literature Reference 9.

Proposition 4.1.

Let be a positive continuous function on . The following are equivalent:

(1)

and ;

(2)

there exists an operator monotone function such that

(3)

there exists an operator monotone function such that

Proof.

: This is obvious.

: Set . Then . Since , . Hence by Proposition 3.1 there exists such that . Therefore

: Set . Since implies that , .

Proposition 4.2.

Let be a positive continuous function on . The following are equivalent:

(1)

and ,

(2)

there exists an operator monotone function such that

Proof.

This follows from the same argument as for Proposition 3.1 using the formula for .

4.2. Non-symmetric means between ! and

It is well known that a symmetric operator mean must be between ! and . To show that the converse is not true, we present an algorithm for constructing a non-symmetric mean such that .

Lemma 4.3.

Let be a positive operator monotone function on with . The following are equivalent:

(1)

is non-symmetric mean;

(2)

is non-self-adjoint.

Proof.

: Since , if is non-self-adjoint operator monotone and , is non-symmetric; that is, is non-symmetric.

: If is a non-symmetric mean, then , which implies .

Lemma 4.4.

If a symmetric operator mean is self-adjoint, then .

Proof.

Let be an operator monotone function corresponding to . Then

Hence, , and .

Remark 4.5.

From Lemma 4.4, we know that all the following operator means are non-self-adjoint: arithmetic mean, logarithmic mean, harmonic mean, Heinz mean, and Lehmer mean Reference 8.

Hence, we have the following result.

Proposition 4.6.
Proof.

The first equality is clear from Lemma 4.3.

It follows from and Proposition 2.2 (2) that

Thus, we have the second equality.

From Lemma 4.4, we have

which means the last inclusion holds.

Remark 4.7.

From Proposition 4.6, a non-self-adjoint positive operator monotone function with gives a non-symmetric operator mean such that . For example, let and be the function corresponding to the power difference mean defined by

Then is symmetric and non-self-adjoint by Lemma 4.4. Hence, is non-symmetric. Moreover, the Petz-Hasegawa function which is defined by

is non-self-adjoint. Hence, is a non-symmetric operator mean between ! and .

Later, in Section 5, we shall give an algorithm for constructing non-symmetric operator means between ! and (see Lemmas 5.1 and 5.2).

4.3. Non-symmetric means between symmetric means

In this section, we show the existence of non-symmetric operator means between any pair of symmetric means.

Theorem 4.8.

Let be symmetric operator means. If and , then there exists a non-symmetric operator mean such that .

To prove the above theorem, we need the following lemma.

Lemma 4.9.

Let and be self-adjoint positive operator monotone functions on with and . If for all and for all , then there exists a non-self-adjoint positive operator monotone function such that

Proof.

Fix a non-self-adjoint mean . If is an operator monotone function corresponding to (see Reference 6, (2.9)), then satisfies the above inequalities.

It follows from the assumption that there exists such that