Revisiting Some Relationships Between the Weighted Spectral Mean and the Wasserstein Mean

Abstract

In this paper, we introduce the 2-geometric mean and explore its connections with the spectral geometric mean and the Wasserstein mean for positive definite matrices. Additionally, we revisit and establish several inequalities involving these means in the context of the near order relation.

1. Introduction

The study of matrix means plays a crucial role in various mathematical and applied fields, particularly in matrix analysis, optimization, and quantum information theory. Among these means, the geometric mean [1] is one of the most extensively studied. It is well known [2] that the geometric mean serves as the barycenter of two given positive definite matrices under the Riemannian metric. The weighted matrix geometric mean, which serves as the geodesic curve connecting two given matrices in quantum information in the form of quantum entropic [3,4] and fidelity [5,6,7,8] measures. In 1980, Kubo and Ando [9] introduced an axiomatic framework for matrix means on the set of positive definite matrices. These matrix means play a central role in defining various quantum divergences in quantum information theory ([10,11,12,13,14,15]).

The spectral geometric mean of positive definite matrices was introduced by Fiedler and Pták in 1997 [16]. Later, in 2015, Kim and Lee [17] defined the weighted spectral mean and studied several of its properties. Although the spectral geometric mean has received less attention compared with the geometric mean, it remains an object of interest among researchers.

In [18], Bhatia, Jain, and Lim investigated the Bures–Wasserstein distance between positive definite matrices and solved the least squares problem to determine the barycenter of two positive definite matrices, known as the weighted Wasserstein mean. Notably, the weighted spectral geometric mean and the weighted Wasserstein mean are, in some sense, generated by the functions $x^t$ and $1-t+tx$, respectively. This suggests that the relationship between these means may exhibit similarities to that between the weighted geometric mean and the weighted arithmetic mean.

Recently, there has been a growing body of research on these means. Franco and Dumitru [19] introduced a near-order relation. Building on this, researchers such as J. Kim, H. Huang, and L. Gan further explored various inequalities associated with different order relations [20,21]. More recently, Furuichi discovered connections between the spectral geometric mean and Tsallis entropy in quantum information theory [22].

In this paper, we introduce the concept of the 2-geometric mean and investigate its relationships with the spectral geometric mean and the Wasserstein mean. These formulations provide a foundation for revisiting and further exploring their properties. Additionally, we establish several inequalities satisfied by these means within the context of the near-order relation.

The paper is organized as follows: In Section 2, we recall fundamental definitions and properties of matrix means. In Section 3, we introduce new representations of the spectral geometric mean and the Wasserstein mean, along with concise proofs of their fundamental properties. In Section 4, we reexamine the near-order relation and derive general inequalities for these means. Furthermore, we establish norm inequalities and identify conditions under which equality holds.

2. Preliminaries

Let ${\mathbb{P}}_n$ be the set of all $n\times n$ positive definite matrices, ${\mathbb{H}}_n$ be the space of all $n\times n$ Hermitian matrices, and $\mathrm{U}\left(n\right)$ be the group of $n\times n$ unitary matrices. For $A,B\in {\mathbb{P}}_n$, the geometric mean $A{♯}_{t}B$ was firstly defined by Puzs and Woronowicz [1] in 1975. They showed that it is the unique positive definite solution to the Riccati equation

$$X{A}^{-1}X=B.$$

It is well-known [2] that the geometric mean $A♯B$ is the midpoint of the geodesic

$$A{♯}_{t}B=A^{1/2}{\left(A^{-1/2}B{A}^{-1/2}\right)}^t{A}^{1/2},t\in [0,1],$$

joining A and B under the Riemannian metric ${\delta }_R(A,B)={\parallel log\left(A^{-1/2}B{A}^{-1/2}\right)\parallel }_F$, where ${\parallel ·\parallel }_F$ denotes the Frobenius norm.

The spectral geometric mean of $A,B\in {\mathbb{P}}_n$ was introduced by Fiedler and Pták in 1997 [16], and one of its formulations is

$$A♮B:={(A^{-1}♯B)}^{1/2}A{(A^{-1}♯B)}^{1/2}.$$

(1)

It is called the spectral geometric mean because ${(A♮B)}^2$ is similar to $AB$, and the eigenvalues of their spectral mean are the positive square roots of the corresponding eigenvalues of $AB$ ([Theorem 3.2] in [16]).

In 2015, Kim and Lee [17] defined the weighted spectral mean:

$$A{♮}_{t}B:={\left(A^{-1}♯B\right)}^{t}A{\left(A^{-1}♯B\right)}^t,t\in [0,1].$$

(2)

It is obvious that $A{♮}_{t}B$ is a curve joining A and B. They studied the relative operator entropy associated with the spectral geometric mean, as well as several properties similar to those of the Tsallis relative operator entropy defined via the matrix geometric mean.

Note that in (2), the geometric mean $A^{-1}♯B$ is a main component of the weighted spectral mean $A{♮}_{t}B$, while the middle term is A, independent of t. We define a new weighted mean below.

$$F_t(A,B):={\left(A^{-1}{♯}_{t}B\right)}^{1/2}{A}^{2-2t}{\left(A^{-1}{♯}_{t}B\right)}^{1/2},t\in [0,1].$$

(3)

It is obvious that $F_0(A,B)=A$ and $F_1(A,B)=B$; hence, $F_t(A,B)$ is a curve joining A and B. For $t={\displaystyle \frac{1}{2}}$, $F_{{\displaystyle \frac{1}{2}}}(A,B)$ is the spectral geometric mean (1). We call $F_t(A,B)$ the weighted F-mean and it is different from (2).

Although the spectral geometric mean appears to be less studied and less prominent than the geometric mean, it remains of interest to several researchers. One of the motivations for studying the spectral geometric mean is that its trace corresponds to the quantum fidelity:

$$\mathrm{Tr}\left({(A^{-1}♯B)}^{1/2}A{(A^{-1}♯B)}^{1/2}\right)=\mathrm{Tr}\left(A(A^{-1}♯B)\right)=\mathrm{Tr}\left({\left(A^{1/2}B{A}^{1/2}\right)}^{1/2}\right).$$

In 1980, Kubo and Ando developed a general theory of matrix means for positive definite matrices [9]. Let f be a positive real-valued function on $(0,\infty )$. For any positive integer n and positive definite matrices $A,B$ in ${\mathbb{P}}_n$, we define a binary operation $\sigma$ on ${\mathcal{P}}_n$ as follows:

$$A\sigma B=A^{1/2}f\left(A^{-1/2}B{A}^{-1/2}\right)A^{1/2}.$$

(4)

This binary operation is called an operator connection if the following requirements are fulfilled:

(1)

$A\le C$ and $B\le D$ imply $A\sigma B\le C\sigma D$;

(2)

$C\left(A\sigma B\right)C\le \left(CAC\right)\sigma \left(CBC\right)$;

(3)

If $A_n\searrow A$ and $B_n\searrow B$, then $A_n\sigma B_n\searrow A\sigma B$.

An operator connection $\sigma$ is called an operator mean or a matrix mean if $I\sigma I=I$, and is called symmetric if $A\sigma B=B\sigma A$ for all $A,B$ in ${\mathcal{P}}_n$. Kubo and Ando showed that there exists an affine order-isomorphism from the class of connections to the class of positive operator monotone functions on ${\mathbb{R}}^{+}$, which is given by $\sigma \mapsto f_{\sigma }\left(t\right)=1\sigma t$.

The weighted Wasserstein mean $A{\diamond }_{t}B$ [18] is the geodesic curve joining A and B in ${\mathbb{P}}_n$ with respect to the Wasserstein distance:

$$A{\diamond }_{t}B={(1-t)}^{2}A+t^{2}B+t(1-t)(A(A^{-1}♯B)+(A^{-1}♯B)A).$$

If we denote $A^{-1}♯B=X_0,$ then from the Riccati equation, $B=X_{0}A{X}_0.$ And, the weighted Wasserstein mean adopts the following form

$$A{\diamond }_{t}B=(1-t+t{X}_0)A(1-t+t{X}_0).$$

(5)

Recently, Franco and Dumitru introduced a near-order relation defined by $A^{-1}♯B\le I$. Then, T.Y. Tam, J. Kim, H. Huang, F. Furuichi, Y. Seo, and L. Gan, among others, have continued to investigate these objects and explore various inequalities within different order relations [20,21] and applications in quantum information.

3. The 2-Geometric Mean

In this section, we provide a new form of the spectral geometric mean and establish new simple proofs for its basic properties. The formula of the weighted geometric mean $A{♯}_{t}B$ can be extended to any real number t. For $t=2$, we notice that

$$A{♯}_{2}B=A^{1/2}\left(A^{-1/2}B{A}^{-1/2}\right)\left(A^{-1/2}B{A}^{-1/2}\right)A^{1/2}=B{A}^{-1}B.$$

Also,

$$A♯X=B\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}X=A{♯}_{2}B.$$

(6)

Indeed, $X=A{♯}_{2}B=B{A}^{-1}B$, if and only if, according to the Riccati equation, $B=A♯X.$

From (1) and (6), we have the following new form of the spectral geometric mean.

Definition 1.

For positive definite matrices A and B in ${\mathbb{P}}_n$, the spectral geometric mean A is represented as

$$A♮B=A^{-1}{♯}_2{(A^{-1}♯B)}^{1/2}=A^{-1}{♯}_2{X}_0^{1/2},\mathrm{where}{X}_0=A^{-1}♯B.$$

(7)

Similarly, we have the following representations:

$\left(F_t\right)$

The new weighted spectral geometric mean $F_t(A,B)$ [23] is represented as

$$F_t(A,B)=A^{2t-2}{♯}_2{\left(A^{-1}{♯}_{t}B\right)}^{1/2};$$

(8)

(${♯}_t$)

The weighted spectral geometric mean $A{♮}_{t}B$ is defined as

$$A{♮}_{t}B=A^{-1}{♯}_2{(A^{-1}♯B)}^t=A^{-1}{♯}_2{X}_0^t;$$

(9)

(${\diamond }_t$)

The Wasserstein mean [18] is represented as

$$A{\diamond }_{t}B=\left(I{\nabla }_t{X}_0\right)A\left(I{\nabla }_t{X}_0\right)=A^{-1}{♯}_{2}k\left(X_0\right),$$

(10)

where $k\left(x\right)=1-t+tx.$

From this point of view, one can see that the difference between the spectral geometric mean and the Wasserstein mean lies in how they depend on the function of $X_0$. This distinction is particularly evident in the form of the 2-geometric mean. Therefore, employing this notation could lead to broader generalizations.

Recall also that Kraus’ theorem (see, for example, [Theorems 8.1] in [24]) characterizes completely positive maps, which model quantum operations between quantum states. Informally, the theorem guarantees that the action of any such quantum operation $\mathrm{\Phi }$ on a quantum state $\rho$ can always be written as

$$\mathrm{\Phi }\left(\rho \right)=\sum _k{B}_k\rho B_k^{\ast },$$

for some set of matrices ${\left\{B_k\right\}}_k$ satisfying the inequality ${\sum }_k{B}_k^{\ast }{B}_k\le I,$ where I denotes the identity matrix. Therefore, if $B_i$ are positive definite matrices, then

$$\mathrm{\Phi }\left(\rho \right)=\sum _k{\rho }^{-1}{♯}_2{B}_k^{1/2}.$$

Notice that in the definition of the spectral geometric mean, Fiedler and Pták introduced a set of properties that this mean must satisfy, but they did not explain why $A^{-1}♯B$ is included in the definition. However, let us provide some reasons for its existence. The following result can be found in ([Theorem 3.19] in [25]), but for the convenience of readers, we provide a short proof here.

Proposition 1

(Theorem 3.19 in [25]). For given positive definite matrices A and B, the geometric mean $A^{-1}♯B$ is the unique minimizer of the loss function

$$F\left(X\right)=\mathrm{Tr}\left(AX\right)+\mathrm{Tr}\left(B{X}^{-1}\right)(X>0).$$

Proof. 

Recall that the Fréchet derivative of a function $f\left(X\right)$ at X in the direction of a perturbation H is defined as follows:

$$Df\left(X\right)\left[H\right]=\underset{ϵ\to 0}{lim}{\displaystyle \frac{f(X+ϵH)-f\left(X\right)}{ϵ}}.$$

(11)

Applying (11) to $F\left(X\right)$, for any Hermitian matrix H, we have

$$DF\left(X\right)\left(H\right)=\mathrm{Tr}(AH-X^{-1}B{X}^{-1}H).$$

Setting $DF\left(X\right)\left(H\right)=0$ for any Hermitian H, we obtain

$$A-X^{-1}B{X}^{-1}=0,$$

or,

$$X^{-1}B{X}^{-1}=A.$$

From the Riccati equation, it follows that $X=B♯A^{-1}=A^{-1}♯B$, since the geometric mean is symmetric. □

In the case where A and B are two density matrices—which represent quantum states in quantum information theory—the function value $F\left(X\right)$ evaluate at the minimizer $A^{-1}♯B$ corresponds precicely to the quantum fidelity ([Theorem 3.19] in [25]):

$$\begin{array}{c}\hfill F(A^{-1}♯B)=\mathrm{Tr}\left(A(A^{-1}♯B)\right)+\mathrm{Tr}\left(B{(A^{-1}♯B)}^{-1}\right)2\mathrm{Tr}\left(A\right)=2\mathrm{Tr}\left({\left(A^{1/2}B{A}^{1/2}\right)}^{1/2}\right).\end{array}$$

We collect some properties of ${♯}_2$ that we often use in the paper.

Lemma 1.

For positive definite matrices $A,B$, and C,

(A1)

$A♯\left(A{♯}_{2}B\right)=A{♯}_2(A♯B)=B$

(A2)

${\left(A{♯}_{2}B\right)}^{-1}=A^{-1}{♯}_2{B}^{-1}$; hence, ${\left(A^{-1}{♯}_2{B}^{-1}\right)}^{-1}=A{♯}_{2}B;$

(A3)

$A{♯}_2\left(Cf\left(C\right)\right)=\left(C^{-1}A{C}^{-1}\right){♯}_{2}f\left(C\right)$ for some given function f.

(A4)

If $AB=BA$, then for any positive definite matrix X,

$$\left(X{♯}_{2}A\right)♯\left(X^{-1}{♯}_{2}B\right)=AB.$$

(12)

Proof. 

Properties (A1) and (A2) follow from (6). For (A3), we have

$$A{♯}_2\left(Cf\left(C\right)\right)=f\left(C\right)C{A}^{-1}Cf\left(C\right)=f\left(C\right){\left(C^{-1}A{C}^{-1}\right)}^{-1}f\left(C\right)=\left(C^{-1}A{C}^{-1}\right){♯}_{2}f\left(C\right).$$

The identity (12) can be rewritten as

$$\left(B{X}^{-1}B\right)♯\left(CXC\right)=BC.$$

Multiplying both sides by ${\left(BC\right)}^{1/2}$, we obtain

$${\left(B^{-1/2}{C}^{1/2}X{C}^{1/2}{B}^{-1/2}\right)}^{-1}♯\left(B^{-1/2}{C}^{1/2}X{C}^{1/2}{B}^{-1/2}\right)=I.$$

□

Let us use the new forms to re-establish some basic properties of the spectral geometric means mentioned above.

Proposition 2.

For positive definite matrices A and B in ${\mathbb{P}}_n$, the following are satisfied:

(i)

$A{♮}_{t}B=B{♮}_{1-t}A;$

(ii)

${\left(A{♮}_{t}B\right)}^{-1}=A^{-1}{♮}_t{B}^{-1};$

(iii)

$\left(A{♮}_{s}B\right){♮}_t\left(A{♮}_{u}B\right)=A{♮}_{(1-t)s+tu}B.$ When $s=u$, we have $\left(A{♮}_{s}B\right){♮}_t\left(A{♮}_{s}B\right)=A{♮}_{s}B;$

(iv)

$A{\diamond }_{t}B=B{\diamond }_{1-t}A;$

(v)

${\left(A{\diamond }_{t}B\right)}^{-1}=A^{-1}{\diamond }_t{B}^{-1}$, if and only if $A=B$;

(vi)

$\left(A{\diamond }_{s}B\right){\diamond }_t\left(A{\diamond }_{u}B\right)=A{\diamond }_{(1-t)s+tu}B;$

(vii)

$A{\diamond }_{t}B\le A{\nabla }_{t}B$

Proof. 

(i) Using (A3) and the fact that $B=X_{0}A{X}_0$, we have

$$B{♮}_{1-t}A=B^{-1}{♯}_2{X}_0^{t-1}=\left(X_0^{-1}{A}^{-1}{X}_0^{-1}\right){♯}_2{X}_0^{t-1}=A^{-1}{♯}_2{X}_0^t=A{♮}_{t}B.$$

(ii) Using (A2), we have

$$A^{-1}{♮}_t{B}^{-1}=A{♯}_2{X}_0^{-t}={\left(A^{-1}{♯}_2{X}_0^t\right)}^{-1}={\left(A{♮}_{t}B\right)}^{-1}.$$

(iii) Firstly, using (A1), we have the following

$$\begin{array}{c}\hfill A{♮}_s\left(A{♮}_{t}B\right)=A^{-1}{♯}_2{(A^{-1}♯\left(A^{-1}{♯}_2{X}_0^t\right))}^s=A^{-1}{♯}_2{X}_0^{st}=A{♮}_{st}B.\end{array}$$

(13)

On account of (i), from (13), we have

$$\begin{array}{c}\hfill \left(A{♮}_{s}B\right){♮}_{t}B=B{♮}_{1-t}\left(B{♮}_{1-s}A\right)=B{♮}_{(1-t)(1-s)}A=A{♮}_{(1-t)s+t}B.\end{array}$$

(14)

As a consequence of (13) and (14), we have

$$\left(A{♮}_{s}B\right){♮}_t\left(A{♮}_{u}B\right)=\left(A{♮}_{s/u}\left(A{♮}_{u}B\right)\right){♮}_t\left(A{♮}_{u}B\right)=A{♮}_{(1-t)s/u+t}\left(A{♮}_{u}B\right)=A{♮}_{(1-t)s+tu}B.$$

(iv) Using the representation (${\diamond }_t$) and (A3), we have

$$B{\diamond }_{1-t}A=\left(X_0^{-1}{A}^{-1}{X}_0^{-1}\right){♯}_2\left(X_0^{-1}k\left(X_0\right)\right)=A^{-1}{♯}_{2}k\left(X_0\right)=A{\diamond }_{t}B.$$

(v) We have

$${\left(A^{-1}{\diamond }_t{B}^{-1}\right)}^{-1}={\left(A{♯}_{2}k\left(X_0^{-1}\right)\right)}^{-1}=A^{-1}{♯}_2{k}^{-1}\left(X_0^{-1}\right)=A^{-1}{♯}_{2}k\left(X_0\right)$$

(15)

if and only if $k^{-1}\left(X_0^{-1}\right)=k\left(X_0\right).$ Since $k\left(x\right)=1-t+tx,$ it is easy to check that the identity $k\left(x\right)k\left(x^{-1}\right)=1$, if and only if $x=1.$

(vi) The proof of this identity is similar to the proof of (iii), considering that

$$\begin{array}{cc}\hfill A{\diamond }_s\left(A{\diamond }_{t}B\right)=A^{-1}{♯}_2{k}_s(A^{-1}♯\left(A^{-1}{♯}_2{k}_t\left(X_0\right)\right)))=A^{-1}{♯}_2{k}_s\left(k_t\left(X_0\right)\right)& =A^{-1}{♯}_2{k}_{st}\left(X_0\right)\hfill \\ \hfill & =A{\diamond }_{st}B,\hfill \end{array}$$

where $k_s\left(X_0\right)=1-s+s{X}_0,$ and the second identity follows from (A1).

(vii) For any $t\in (0,1)$, the inequality is equivalent to the following

$${(1-t)}^{2}A+t^{2}B+t(1-t)(A(A^{-1}♯B)+(A^{-1}♯B)A)\le (1-t)A+tB,$$

which is equivalent to

$$A+X_{0}A{X}_0\ge A{X}_0+X_{0}A,$$

where B is replaced with $X_{0}A{X}_0$. The last inequality is obvious since

$$(I-X_0)A(I-X_0)\ge 0.$$

□

Recall that the arithmetic mean and the harmonic mean are dual, which means $A!B={(A^{-1}\nabla B^{-1})}^{-1}$, and the functions satisfy the relation $f_{!}\left(x\right)=x{f}_{\nabla }^{-1}\left(x\right)$. Between the arithmetic mean, the geometric mean, and the harmonic mean, there are two main relations:

(R1)

$A{!}_{t}B\le A{♯}_{t}B\le A{\nabla }_{t}B$ for any $t\in [0,1]$;

(R2)

$A!B=(A♯B)(A\nabla B)(A♯B).$

Dinh, Le, and Vo [26] proved that for an arbitrary Kubo–Ando matrix mean $\sigma$, such that $\sigma \ge ♯$, for positive definite matrices X and Y,

$$X{\sigma }^{\perp }Y=(X♯Y){\left(X\sigma Y\right)}^{-1}(X♯Y),$$

or, equivalently,

$$\left(X\sigma Y\right)♯\left(X{\sigma }^{\perp }Y\right)=X♯Y.$$

(16)

This identity (16) has an interesting geometrical interpretation: the barycenter of $X\sigma Y$ and $X{\sigma }^{\perp }Y$ coincides with the barycenter of X and Y with respect to the Riemannian distance.

Recently, Franco [27] obtained a similar identity for the spectral geometric mean and the Wasserstein mean, as

$${(A^{-1}\diamond B^{-1})}^{-1}♮(A\diamond B)=A♮B.$$

(17)

In the following theorem, we establish a more general result of (17).

Theorem 1.

Let $A,B\in {\mathbb{P}}_n$. Then

$${\left(A^{-1}{\diamond }_t{B}^{-1}\right)}^{-1}♮\left(A{\diamond }_{t}B\right)=\left(k^{1/2}\left(X_0\right)k^{-1/2}\left(X_0^{-1}\right)X_0^{-1/2}\right)(A♮B)\left(X_0^{-1/2}{k}^{-1/2}\left(X_0^{-1}\right)k^{1/2}\left(X_0\right)\right).$$

Proof. 

Let $X={\left(A^{-1}{\diamond }_t{B}^{-1}\right)}^{-1}=A^{-1}{♯}_2{k}^{-1}\left(X_0^{-1}\right)$ and $Y=A{\diamond }_{t}B=A^{-1}{♯}_{2}k\left(X_0\right)$. Then, on account of (A4), we have

$$X^{-1}♯Y=\left(A{♯}_{2}k\left(X_0^{-1}\right)\right)♯\left(A^{-1}{♯}_{2}k\left(X_0\right)\right)=k\left(X_0^{-1}\right)k\left(X_0\right).$$

Consequently,

$$\begin{array}{cc}\hfill X♮Y& =X^{-1}{♯}_2{(X^{-1}♯Y)}^{1/2}\hfill \\ \hfill & =\left(A{♯}_{2}k\left(X_0^{-1}\right)\right){♯}_2\left(k^{1/2}\left(X_0^{-1}\right)k^{1/2}\left(X_0\right)\right)\hfill \\ \hfill & =\left(k^{1/2}\left(X_0^{-1}\right)k^{1/2}\left(X_0\right)\right)\left(k^{-1}\left(X_0^{-1}\right)A{k}^{-1}\left(X_0^{-1}\right)\right)\left(k^{1/2}\left(X_0^{-1}\right)k^{1/2}\left(X_0\right)\right)\hfill \\ \hfill & =\left(k^{-1/2}\left(X_0^{-1}\right)k^{1/2}\left(X_0\right)X_0^{-1/2}\right)\left(X_0^{1/2}A{X}_0^{1/2}\right)\left(X_0^{-1/2}{k}^{-1/2}\left(X_0^{-1}\right)k^{1/2}\left(X_0\right)\right)\hfill \\ \hfill & =\left(k^{1/2}\left(X_0\right)k^{-1/2}\left(X_0^{-1}\right)X_0^{-1/2}\right)(A♮B)\left(X_0^{-1/2}{k}^{-1/2}\left(X_0^{-1}\right)k^{1/2}\left(X_0\right)\right),\hfill \end{array}$$

where the last identity follows from the fact that $X_0^{1/2}A{X}_0^{1/2}=A^{-1}{♯}_2{X}_0^{1/2}=A^{-1}{♯}_2{(A^{-1}♯B)}^{1/2}$$=A♮B$. □

4. Near Order Relation and Means

Recently, Franco and Dumitru introduced the near order relation ⪯ on ${\mathbb{P}}_n$ as follows:

$$A⪯B\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{A}^{-1}♯B\ge I.$$

This order is weaker than the traditional Loewner order, which states that for Hermitian matrices A and B, $A\le B$ if and only if $B-A\ge 0.$ From Proposition 1, $A⪯B$ if the inverse of the minimizer of $F\left(X\right)$ is a contraction.

In this section, we review some inequalities in this the near order relation. Fortunately, many results can be obtained easily with the 2-geometric mean form for the spectral geometric mean and the Wasserstein mean.

The following lemma is crucial when we consider inequalities with respect to the near order. During the preparation of this note, we were informed that there is a similar result in Franco’s paper [27], which is in a different form.

Lemma 2.

Let A be a positive definite matrix, and let $0<f\left(x\right)\le g\left(x\right)$ for any positive $x.$ Then, for any positive definite matrix X,

(i)

$A^{-1}{♯}_{2}f\left(X\right)⪯A^{-1}{♯}_{2}g\left(X\right);$

(ii)

$\parallel A^{-1}{♯}_{2}f\left(X\right)\parallel \le \parallel A^{-1}{♯}_{2}g\left(X\right)\parallel ,$ where $\parallel ·\parallel$ is the operator norm.

Proof. 

On account of Lemma 1, inequality (i) is equivalent to the following

$$\begin{array}{cc}\hfill {\left(A^{-1}{♯}_{2}f\left(X_0\right)\right)}^{-1}♯\left(A^{-1}{♯}_{2}g\left(X_0\right)\right)& =\left(A{♯}_2{f}^{-1}\left(X_0\right)\right)♯\left(A^{-1}{♯}_{2}g\left(X_0\right)\right)\hfill \\ \hfill & =f^{-1}\left(X_0\right)g\left(X_0\right)\ge I\hfill \end{array}$$

which is true because of the assumption.

(ii) Suppose that $A^{-1}{♯}_{2}g\left(X\right)\le I$, we have $A\le g^{-2}\left(X\right)\le f^{-2}\left(X\right)$. This means, $A^{-1}{♯}_{2}f\left(X\right)\le I.$ □

The following result was obtained by Huang and Gan [28] which is an obvious consequence of Lemma 2.

Theorem 2.

The following are equivalent for $A,B\in {\mathbb{P}}_n$, and for $s,t\in \mathbb{R}:$

(i)

$A⪯B$;

(ii)

$A{\diamond }_{s}B\le A{\diamond }_{t}B$ whenever $s\le t$;

(iii)

$A{♮}_{s}B\le A{♮}_{t}B$ whenever $s\le t$.

Proof. 

Observe that the condition $A⪯B$ is equivalent to that $X_0=A^{-1}♯B\ge I.$ Also, the representing function of ${\diamond }_t$ is $k\left(x\right)=1-t+tx.$ This function is increasing in t, if and only if $x\ge 1$. Similarly, the representing function of ${♮}_t$ is $x^t$. This function is increasing for any x. □

In the following theorem, we prove a similar relation to (R2), which was obtained in [27].

Theorem 3.

Let $A,B\in {\mathbb{P}}_n$, and $t\in [0,1]$. Then,

$${\left(A^{-1}{\diamond }_t{B}^{-1}\right)}^{-1}⪯A{♮}_{t}B⪯A{\diamond }_{t}B.$$

(18)

Proof. 

The inequalities in (18) are equivalent to

$$A^{-1}{♯}_2{k}^{-1}\left(X_0^{-1}\right)⪯A^{-1}{♯}_{2}f\left(X_0\right)⪯A^{-1}{♯}_{2}k\left(X_0\right).$$

According to Lemma 2, it is enough to show that $k^{-1}\left(x^{-1}\right)\le x^t\le k\left(x\right)$ for any $x>0$ and for any $t\in (0,1).$ We show, for example, the second inequality. It is not difficult to see that the function $k\left(x\right)-x^t=1-t+tx-x^t$ attains a minimum at $x=1$ for any $t\in (0,1)$. This means, $k\left(x\right)\ge x^t$. □

Remark 1.

Notice that the second inequality in (18) was proven by Huang and Gan in [28]. Our proof is essentially shorter. Also, Huang and Gan showed that if $A⪯B,$ then $A{\diamond }_{t}B⪯A{♮}_{t}B$, and if $B⪯A$, then for any $t\in (-\infty ,0)$, $A{\diamond }_{t}B⪯A{♮}_{t}B.$ Again, these statements can be proved by analyzing the sign of the function $k\left(x\right)-x^t$, in term of x or t on $\mathbb{R}$.

In the following, we establish some norm inequalities for the spectral geometric mean and the Wasserstein mean.

Theorem 4.

Let $A,B\in {\mathbb{P}}_n$, and $t\in (0,1)$. Then

(i)

$\parallel A♯B\parallel \le \parallel A♮B\parallel$;

(ii)

$\parallel {\left(A^{-1}{\diamond }_t{B}^{-1}\right)}^{-1}\parallel \le \parallel A{♮}_{t}B\parallel \le \parallel A{\diamond }_{t}B\parallel ,$

where $\parallel ·\parallel$ is the operator norm.

Proof. 

(i) can be found in [29]. Here, we provide a short direct proof. Suppose that $A♮B\le I,$ or, $X_0^{1/2}A{X}_0^{1/2}\le I$, we have $B=X_{0}A{X}_0\le X_0$ and $A\le X_0^{-1}$. By monotonicity of the geometric mean, we have $A\sigma B\le X_0^{-1}♯X_0=I.$ From here, $\parallel A♯B\parallel \le \parallel A♮B\parallel .$

The second inequality in (ii) was proved in [30]. However, all inequalities in (ii) are direct consequences of Lemma 2. □

To conclude this note, we establish new characterizations of central elements in the algebra ${\mathbb{M}}_n$. We begin with the following fact, which is a special case of the main result in [31]. We also note that some global characterizations of the trace and commutativity were previously studied by the fourth author and his co-authors in [32,33].

Lemma 3.

Let $A\in {\mathbb{P}}_n$. Suppose that for any Hermitian matrix X:

$$A\le X⟹A^2\le X^2.$$

Then, A is a scalar multiple of the identity.

Theorem 5.

Let A be a positive definite matrix. Then

(i)

$\parallel A♯X\parallel =\parallel A♮X\parallel$ for any positive definite matrix X, if and only if A is a scalar multiple of the identity matrix;

(ii)

$\parallel A{♮}_{t}B\parallel =\parallel A{\diamond }_{t}B\parallel$ if and only if $A=B$;

(iii)

$\parallel {\left(A^{-1}{\diamond }_t{B}^{-1}\right)}^{-1}\parallel =\parallel A{♮}_{t}B\parallel$ if and only if $A=B$.

Proof. 

(i) was obtained in [29]. Here, we provide a direct proof. Let $\parallel A♯B\parallel =\parallel A♮B\parallel$. It follows that $A♯B\le I$, if and only if $A♮B\le I.$ This means, $Y^{1/2}\le A^{-1}$, if and only if $X_0^{1/2}A{X}_0^{1/2}\le I,$ where $Y=A^{-1/2}B{A}^{-1/2}.$ Since $B=X_{0}A{X}_0\le X_0^{1/2}{X}_0^{1/2}A{X}_0^{1/2}{X}_0^{1/2}\le X_0,$ we have that $Y\le A^{-1/2}{X}_0{A}^{-1/2}\le A^{-1/2}{A}^{-1}{A}^{-1/2}=A^{-2}.$ This means, $Y^{1/2}\le A^{-1}$, if and only if $Y\le A^{-2}$. From here, it follows that A and $Y^{-1}$ are commuting; hence, $A^{1/2}$ and Y are commuting. As a consequence, we can see that B and $A^{-1/2}$ are commuting. By Lemma 3, A commutes with every matrix $B.$ Thus, A must be a scalar multiple of the identity matrix.

(ii) From the assumption, it implies that $k\left(X_0\right)Ak\left(X_0\right)\le I$, if and only if $X_0^{t}A{X}_0^t\le I.$ The first inequality is equivalent to $X_0^{t}A{X}_0^t\le X_0^{2t}{k}^{-2}\left(X_0\right).$ This occurs if and only if $X_0^{2t}{k}^{-2}\left(X_0\right)=I$, which implies that $X_0=I.$ Consequently, $A^{-1}♯B=I.$ From here, one can see that $A=B.$

(iii) can be proved using similar arguments. □

Remark 2.

We would like to obtain a similar result for the equality $\parallel A{♯}_{t}B\parallel =\parallel A{♮}_{t}B\parallel$. Unfortunately, our proof could not be transferred to the general case.