Symmetric cone monotone functions and symmetric cone convex functions

Nonlinear and Convex Analysis, vol. 17, no.3, pp. , 2016

Symmetric cone monotone functions and symmetric cone convex functions

Yu-Lin Chang

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: ylchang@math.ntnu.edu.tw

Jein-Shan Chen ¹ Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: jschen@math.ntnu.edu.tw

Shaohua Pan²

School of Mathematical Sciences South China University of Technology

Guangzhou 510640, China E-mail: shhpan@scut.edu.cn

March 18, 2013

Abstract. Symmetric cone (SC) monotone functions and SC-convex functions are real scalar valued functions which induce L¨owner operators associated with a simple Eu- clidean Jordan algebra to preserve the monotone order and convex order, respectively.

In this paper, for a general simple Euclidean Jordan algebra except for octonion case, we show that the SC-monotonicity (respectively, SC-convexity) of order r is implied by the matrix monotonicity (respectively, matrix convexity) of some fixed order r⁰ (≥ r).

As a consequence, we draw the conclusion that (except for octonion case) a function is

1Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan.

2The author’s work is supported by National Young Natural Science Foundation (No. 10901058), Guangdong Natural Science Foundation (No. 9251802902000001) and the Fundamental Research Funds for the Central Universities (SCUT).

SC-monotone (respectively, SC-convex) if and only if it is matrix monotone (respectively, matrix convex).

Key words: Euclidean Jordan algebra, Symmetric cone, matrix-monotone, L¨owner operator, SC-monotone, SC-convex.

1 Introduction

A Euclidean Jordan algebra is a triple (V, ◦, h·, ·i) where (V, h·, ·i) is a finite dimensional inner product space over the real field R, and (x, y) 7→ x ◦ y : V × V → V is a bilinear mapping satisfying the following conditions: for all x, y, z ∈ V, (i) x ◦ y = y ◦ x; (ii) x ◦ (x²◦ y) = x²◦ (x ◦ y) with x² = x ◦ x; (iii) hx ◦ y, zi = hy, x ◦ zi, in which x ◦ y is called the Jordan product of x and y. We assume that there exists an element e ∈ V (called the unit element) such that x ◦ e = x for all x ∈ V. A Euclidean Jordan algebra is said to be simple if it is not the direct sum of two Euclidean Jordan algebras. For details regarding Euclidean Jordan algebras, we refer to the lecture note [14] and the monograph [9].

Let A = (V, ◦, h·, ·i) be a Euclidean Jordan algebra. For any x ∈ V, define ζ(x) := mink : {e, x, x², · · · , x^k} are linearly dependent .

Then, the rank of A is well defined by r := max{ζ(x) : x ∈ V}. Recall that an element c ∈ V is said to be idempotent if c² = c; and an idempotent is said to be primitive if it is nonzero and can not be written as the sum of two other nonzero idempotents. A finite set {c₁, c₂, · · · , c_r} of primitive idempotents in V is said to be a Jordan frame if

c_i ◦ c_j = 0 when i 6= j and c₁+ c₂+ · · · + c_r= e.

Then, we have the following important spectral decomposition theorem.

Theorem 1.1 [9, Theorem III.1.2] Suppose (V, ◦, h·, ·i) is a Euclidean Jordan algebra of rank r. Then, for every x ∈ V, there exist a Jordan frame {c1, · · · , c_r} and real numbers λ₁(x), · · · , λ_r(x), arranged in the decreasing order λ₁(x) ≥ λ₂(x) ≥ · · · ≥ λ_r(x), such that

x = λ₁(x)c₁+ λ₂(x)c₂+ · · · + λ_r(x)c_r.

The numbers λ₁(x), . . . , λ_r(x) (counting multiplicities), uniquely determined by x, are called the spectral values of x and Pr

j=1λ_j(x)c_j the spectral decomposition of x.

Suppose that φ : J ⊆ R → R is a scalar valued function. Let VJ be a subset in V such that all x ∈ VJ have the spectral in J . Then, by the spectral decompositionPr

j=1λ_j(x)c_j of x ∈ V^J, it is natural to define a vector valued function [4, 14] φ_V : V^J → V by

φV(x) := φ(λ₁(x))c₁+ φ(λ₂(x))c₂+ · · · + φ(λ_r(x))c_r. (1)

In a seminal paper [19], L¨owner initiated the study for φ

V in the setting of V = Sⁿ, where Sⁿ denotes the space of n × n real symmetric matrices, and for X ∈ SⁿJ, which is a subset of Sⁿ such that all eigenvalues of X ∈ SⁿJ belong to J , φ

Sn(X) has the expression φSn(X) := P diag(φ(λ₁(X)), · · · , φ(λ_n(X)))P^T,

where P is an n × n orthogonal matrix and λ₁(X), λ₂(X), . . . , λ_n(X) are real numbers arranged in the decreasing order, such that

X = P diag(λ₁(X), . . . , λ_n(X))P^T. The result of [19] on the monotonicity of φ

Sn was later extended to φ

V by Kor´anyi [15].

In addition, Sun and Sun [24] studied the continuous differentiability and strong semis- moothness of φ

V, and called φ

V L¨owner operator associated with V in recognition of L¨owner’s contribution.

From [9, Theorem III.2.1] we know that the set of all squares K := {x ∈ V : x ◦ x} in V is a symmetric cone, i.e., a self-dual homogeneous closed convex cone. So, there is a natural partial order in V. We write x ^K y if x − y ∈ K, and x K y if x − y ∈ intK.

For any x, y ∈ V^J, let λ1(x) ≥ λ2(x) ≥ · · · ≥ λr(x) and λ1(y) ≥ λ2(y) ≥ · · · ≥ λr(y) be the spectral values of x and y, respectively. From [3, Prop. 4.4] or [2, Theorem 23],

r

X

i=1

(λ_i(x) − λ_i(y))² ≤

r

X

i=1

λ_i(x)²+

r

X

i=1

λ_i(y)²− 2hx, yi = kx − yk².

By this, it is easy to verify that V^J is open in V if and only if J is open on R. Also, since λ₁(αx + (1 − α)y) ≤ αλ₁(x) + (1 − α)λ₁(y)

λ_r(αx + (1 − α)y) ≥ αλ_r(x) + (1 − α)λ_r(y)

for any α ∈ [0, 1] (see [25, Lemma 14]), where λ₁(αx + (1 − α)y), . . . , λ_r(αx + (1 − α)y) are the spectral values of αx + (1 − α)y, arranged in decreasing order, the set VJ is always convex. Now we introduce the concepts of SC-monotone and SC-convex functions.

Definition 1.1 Let (V, ◦, h·, ·i) be a simple Euclidean Jordan algebra of rank r. For any given φ : J ⊆ R → R, let φ_V : VJ → V be defined as in (1). Then,

(a) φ is said to be SC-monotone of order r if for any x, y ∈ VJ, it holds that x _Ky =⇒ φ

V(x) _K φ

V(y).

(b) φ is said to be SC-convex of order r if for any x, y ∈ VJ and α ∈ (0, 1), it holds that φ_V(αx + (1 − α)y) K αφ_V(x) + (1 − α)φ_V(y).

We call φ SC-monotone (SC-convex) if it is SC-monotone (SC-convex) of all orders.

When V is the algebra Sⁿof n×n real symmetric matrices, Def. 1.1 represents the con- cepts of matrix monotone and matrix convex functions of order n; when V is the Jordan spin algebra (see Example 2.4), it gives the concepts of SOC-monotone and SOC-convex functions [6, 7]. After the seminal paper [19], there are many research works about matrix monotone and matrix convex functions (see, e.g., [16, 8, 11, 12, 20, 5, 21, 22, 17, 26]).

However, to our best of knowledge, there are few papers to study SC-monotone and SC- convex functions except that Kor´anyi [15] gave a sufficient and necessary condition for differentiable SC-monotone functions, and furthermore, this condition is the same as the one for matrix monotone functions in [13, Theorem 6.6.36].

In this paper, we establish that the SC-monotonicity (respectively, SC-convexity) of order r of φ is implied by its matrix monotonicity (respectively, matrix convexity) of some fixed order r⁰ (≥ r). For example, φ is SC-monotone (respectively, SC-convex) of order r if it is matrix monotone (respectively, matrix convex) of order 4r; see Theorem 3.1 As a consequence, we draw the conclusion that φ is SC-monotone (respectively, SC-convex) if and only if it is matrix monotone (respectively, matrix convex). These results are achieved by employing the connection between φ

V and φ

Sn, the results of SOC-monotone (SOC-convex) functions [23], and the classification of simple Euclidean Jordan algebras.

2 Preliminaries

For any given x ∈ V, we define the following linear operator L(x) of V by L(x)y := x ◦ y for every y ∈ V.

Let {c₁, · · · , c_r} be a Jordan frame in a Euclidean Jordan algebra (V, ◦, h·, ·i). Then, from [9, Lemma IV.1.3], the operators L(c_j), j = 1, 2, · · · , r commute and admit a simultaneous diagonalization. Besides, for i, j ∈ {1, 2, · · · , r}, we denote the eigenspaces

Vii:= {x ∈ V : x ◦ ci = x} = Rci

and when i 6= j,

V^ij :=



x ∈ V : x ◦ ci = 1

2x = x ◦ c_j

 .

Then, from [9, Theorem IV.2.1], we have the following Peirce decomposition.

Proposition 2.1 The space V is the orthogonal direct sum of spaces Vij (i ≤ j). Also, Vij ◦ Vij ⊂ Vii+ Vjj;

Vij ◦ Vjk ⊂ Vik if i 6= k;

V^ij ◦ Vkl= {0} if {i, j} ∩ {k, l} = ∅.

Let x ∈ V have the spectral decomposition x = Pr

j=1λ_j(x)c_j, where λ₁(x) ≥ λ₂(x) ≥

· · · ≥ λ_r(x) are the spectral eigenvalues of x and {c₁, c₂, . . . , c_r} is the corresponding Jordan frame. For all i, j ∈ {1, 2, . . . , r}, let C_ij(x) be the orthogonal projection operator onto V^ij, from [9, Theorem IV 2.1], it follows that for all i, j = 1, 2, . . . , r,

C_jj(x) = 2L(c_j)² − L(c_j) and C_ij(x) = 4L(c_i)L(c_j) = 4L(c_j)L(c_i) = C_ji(x). (2) Moreover, the orthogonal projection operators {C_ij(x) : i, j = 1, 2, . . . , r} satisfy

Cij(x) = C_ij^∗(x), C_ij²(x) = Cij(x), Cij(x)Ckl(x) = 0 if {i, j} 6= {k, l} (3) and

X

1≤i≤j≤r

C_ij(x) = I (4)

where C_ij^∗(x) means the adjoint of C_ij(x), and I is the identity operator from V to V.

The following lemma gives the spectral decomposition of the operator L(x), whose proof can be found in [14, Chapter V, Sec. 5 and Chapter VI, Sec. 4].

Lemma 2.1 Let x ∈ V have the spectral decomposition x = Pr

j=1λ_j(x)c_j. Then, the linear symmetric operator L(x) has the spectral decomposition

L(x) =

r

X

j=1

λ_j(x)C_jj(x) + X

1≤j<l≤r

1

2(λ_j(x) + λ_l(x)) C_jl(x) (5) with the spectrum σ(L(x)) consisting of all distinct numbers ¹₂(λ_j(x) + λ_l(x)).

Next, we introduce several examples of simple Euclidean Jordan algebras, and recall the classification theorem of simple Euclidean Jordan algebras.

Example 2.1. The algebra Hⁿ of n × n complex Hermitian matrices. A square matrix A of complex entries is said to be Hermitian if A^∗ := ¯A^T = A, where ‘bar’ denotes the complex conjugate, and the superscript ‘T’ means the transpose. Let Hⁿ be the set of all n × n complex Hermitian matrices. On Hⁿ, let define the Jordan product and inner product be X ◦ Y := ¹₂(XY + Y X) and hX, Y i := trace(XY ). Then, Hⁿ is a Euclidean Jordan algebra of rank n and dimension n², with e being the n × n identity matrix I.

There exists an embedding from Hⁿ to S²ⁿ which is one-to-one and onto, and also pre- serves the Jordan algebra structures on the both sides by matrix block multiplication.

As below, we present this embedding for H². First, we know that H² is the set which contains all

 α₁ β β¯ α2



, α₁, α₂ ∈ R and β ∈ C.

We also know that each complex number a + bi can be represented as a 2 × 2 real matrix:

a 1 0 0 1

 + b

 0 1

−1 0

 ,

where

 0 1

−1 0



satisfies

 0 1

−1 0

²

= − 1 0 0 1



. Hence, we can embed  α₁ β β¯ α₂

 into an element in S⁴:

H² 3 α₁ β β¯ α₂

 7−→









 α₁ 0 0 α₁

 

a b

−b a



 a −b b a

  α₂ 0 0 α₂









∈ S⁴

where β = a + ib.

For general n, it is also true that Hⁿis a Jordan sub-algebra of S²ⁿ. The general embed- ding map T_Hⁿ : Hⁿ,→ T (Hⁿ) ⊂ S²ⁿ is given by

Hⁿ 3











α₁ β · · · γ β¯ α2 · · · δ ... ... . .. ...

¯

γ δ¯ · · · α_n









 7−→























 α₁ 0 0 α₁

 

a b

−b a



· · ·

 c d

−d c



 a −b b a

  α₂ 0 0 α₂



· · ·

 e f

−f e



... ... . .. ...

 c −d d c

 

e −f

f e



. . .  α_n 0 0 α_n

























∈ S²ⁿ

where β = a + ib, γ = c + id, δ = e + if . By matrix block multiplication, it can be seen the embedding T_Hⁿ preserves the Jordan algebra structures

T_Hⁿ(x ◦_Hⁿ y) = T_Hⁿ(x) ◦_S²ⁿT_Hⁿ(y) ∀ x, y ∈ Hⁿ.

Example 2.2. The algebra Qⁿ of n × n quaternion Hermitian matrices. The linear space of quaternions over R, denoted by Q, is 4-dimensional vector space [27] with a basis {1, i, j, k}. This space becomes an associated algebra via the multiplication table:

1 i j k

1 1 i j k

i i −1 k −j

j j −k −1 i

k k j −i −1

For any x = x₀1 + x₁i + x₂j + x₃k ∈ Q, we define its real part by <(x) := x0, its conjugate by ¯x := x₀1 − x₁i − x₂j − x₃k, and its norm by |x| = √

x¯x. A square matrix A with quaternion entries is called Hermitian if A coincides with its conjugate transpose. Let Qⁿ be the set of all n × n quaternion Hermitian matrices. For any X, Y ∈ Qⁿ, let

X ◦ Y := 1

2(XY + Y X) and hX, Y i := <(trace(XY )).

Then, Qⁿ is a Euclidean Jordan algebra of rank n and dimension n(2n − 1) with e being the n × n identity matrix I. Analogous to complex number, each quaternion

x = a1 + bi + cj + dk ∈ Q can be represented as a 4× 4 real matrix









a b c d

−b a −d c

−c d a −b

−d −c b a







 which is also equivalent to

a









1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1







 + b









0 1 0 0

−1 0 0 0 0 0 0 −1

0 0 1 0







 + c









0 0 1 0

0 0 0 1

−1 0 0 0

0 −1 0 0







 + d









0 0 0 1

0 0 −1 0

0 1 0 0

−1 0 0 0







 .

Following the same lines for Hⁿ, we can embed Qⁿ into S⁴ⁿ such that Qⁿ can be viewed as a Jordan sub-algebra of S⁴ⁿ. Again, the embedding map under the case for Q² is

Q² 3 α₁ x

¯ x α₂

 7−→



































α₁ 0 0 0

0 α1 0 0 0 0 α₁ 0 0 0 0 α₁

















a b c d

−b a −d c

−c d a −b

−d −c b a

















a −b −c −d

b a d −c

c −d a b

d c −b a

















α₂ 0 0 0

0 α2 0 0 0 0 α₂ 0 0 0 0 α₂



































∈ S⁸

where x = a1 + bi + cj + dk.

Moreover, the general embedding map T_Qⁿ : Qⁿ,→ T (Qⁿ) ⊂ S⁴ⁿ under this case is given by

Qⁿ3











α1 x · · · y

¯

x α₂ · · · z ... ... . .. ...

¯

y z¯ · · · αn









 7−→

























































α₁ 0 0 0

0 α₁ 0 0 0 0 α₁ 0 0 0 0 α₁

















a b c d

−b a −d c

−c d a −b

−d −c b a









· · ·









e f g h

−f e −h g

−g h e −f

−h −g f e

















a −b −c −d

b a d −c

c −d a b

d c −b a

















α₂ 0 0 0

0 α₂ 0 0 0 0 α₂ 0 0 0 0 α₂









· · ·









p q r s

−q p −s r

−r s p −q

−s −r q p









... ... . .. ...









e −f −g −h

f e h −g

g −h e f

h g −f e

















p −q −r −s

q p s −r

r −s p q

s r −q p









· · ·









α_n 0 0 0

0 α_n 0 0

0 0 α_n 0

0 0 0 α_n

























































∈ S⁴ⁿ

where x = a1 + bi + cj + dk, y = e1 + f i + gj + hk and z = p1 + qi + rj + sk.

In summary, we construct an embedding from Hⁿ or Qⁿto S^m respectively for certain m. Since the embedding is linear and preserves the Jordan algebra structures on both sides, it can be seen L¨owner operator commutes with the embedding, which means that for all x ∈ Hⁿ and y ∈ Qⁿ, there have

φ_S²ⁿ(T_Hⁿ(x)) = T_Hⁿ(φ_Hⁿ(x)) and φ_S⁴ⁿ(T_Qⁿ(y)) = T_Qⁿ(φ_Qⁿ(y)). (6) In the above, we present an embedding from a Jordan algebra Hⁿ or Qⁿ to a Jordan sub-algebras of S^m respectively for certain m. Indeed, there is an alternative way to interpret this. For any A = A₁ + A₂j ∈ M_n(Q), its complex adjoint matrix, symbolized χ_A, is defined by [27] :

χ_A=

 A1 A2

− ¯A₂ A¯₁



∈ M_2n(C).

It is shown that if A ∈ Qⁿ then χ_A ∈ H²ⁿ [27, Theorem 4.2(6)]. This is an embedding and preserves operations. There is also an adjoint matrix π_B ∈ M_4n(R) associated with B ∈ M_2n(C). Then, we obtain that the composite π ◦ χ(A) ∈ S⁴ⁿ for any A ∈ Qⁿ. It is obvious to see that the composite π ◦ χ is a Jordan algebra embedding from Qⁿ to S⁴ⁿ as expected.

Example 2.3. The algebra O³ of 3 × 3 octonion Hermitian matrices. The space of octonion, denoted by O, is a 8-dimensional real vector space with basis {1, e1, . . . , e₇}.

The space becomes a nonassociative algebra via the following multiplication table [1]:

Note that O is a non-commutative and non-associative algebra. For an element x = x₀1 + x₁e₁ + x₂e₂ + x₃e₃ + x₄e₄ + x₅e₅ + x₆e₆ + x₇e₇ ∈ O, we define its real part by

1 e₁ e₂ e₃ e₄ e₅ e₆ e₇

1 1 e₁ e₂ e₃ e₄ e₅ e₆ e₇

e1 e1 −1 e4 e7 −e2 e6 −e5 −e3

e₂ e₂ −e₄ −1 e₅ e₁ −e₃ e₇ −e₆

e₃ e₃ −e₇ −e₅ −1 e₆ e₂ −e₄ e₁

e₄ e₄ e₂ −e₁ −e₆ −1 e₇ e₃ −e₅

e5 e5 −e6 e3 −e2 −e7 −1 e1 e4

e₆ e₆ e₅ −e₇ e₄ −e₃ −e₁ −1 e₂

e₇ e₇ e₃ e₆ −e₁ e₅ −e₄ −e₂ −1

<(x) := x₀, its conjugate by ¯x := x₀1 − x₁e₁− x₂e₂− x₃e₃− x₄e₄− x₅e₅− x₆e₆− x₇e₇, and its norm by |x| :=√

x¯x. As in the case of a quaternion Hermitian matrix, we may define an octonion Hermitian matrix. Suppose O³ is the set of all 3 × 3 octonion Hermitian matrices. On O³, let the Jordan product and inner product be

X ◦ Y := 1

2(XY + Y X) and hX, Y i := <(trace(XY )).

Then, O³ is a Euclidean Jordan algebra of rank 3 with e being the 3 × 3 identity matrix, and is a real vector space of dimension 27.

Example 2.4. The Jordan spin algebra Jⁿ. Consider Rⁿ endowed with the usual inner product. For any x ∈ Rⁿ, write x =  x₀

¯ x



with x₀ ∈ R and ¯x ∈ Rⁿ⁻¹. Define

x ◦ y = x₀

¯ x



◦ y₀

¯ y

 :=

 hx, yi x₀y + y¯ ₀x¯

 .

Then, (Rⁿ, ◦, h·, ·i) is an Euclidean Jordan algebra, and we denote it by Jⁿ. The rank of the Euclidean Jordan algebra Jⁿ is 2 and its unit element is given by e = 1

0



. In this algebra, the set of squares is also called the second-order cone or the Lorentz cone.

Theorem 2.1 [9, Chapter V] Every simple Euclidean Jordan algebra is isomorphic to one of the following

(i) The Jordan spin algebra Jⁿ.

(ii) The algebra Sⁿ of n × n real symmetric matrices.

(iii) The algebra Hⁿ of all n × n complex Hermitian matrices.

(iv) The algebra Qⁿ of all n × n quaternion Hermitian matrices.

(v) The algebra O³ of all 3 × 3 octonion Hermitian matrices.

3 Main result

For simplicity, we employ Sⁿ+, Hⁿ+ and Qⁿ+ to denote the corresponding symmetric cones in Sⁿ, Hⁿ and Qⁿ, respectively. In other words, they represent

Sⁿ+ = {x ◦ x | x ∈ Sⁿ}, Hⁿ+ = {x ◦ x | x ∈ Hⁿ} and Qⁿ+= {x ◦ x | x ∈ Qⁿ}.

To achieve our main result, we will show that the embeddings we construct in Examples 2.1-2.2 preserve their conic orders.

Lemma 3.1 Suppose that V is the algebra Hⁿof n×n complex Hermitian matrices. The embedding T_Hⁿ defined as in Example 2.1 keeps the conic order in the following sense:

x _Hⁿ

+ y ⇐⇒ T_Hⁿ(x) _S²ⁿ

+ T_Hⁿ(y) ∀x, y ∈ Hⁿ. Proof. (⇒) Suppose that x _Hⁿ

+ y. Then, there exists an a ∈ Hⁿ such that x − y = a². Since T_Hⁿ preserves Jordan algebra structure, we have

T_Hⁿ(x) − T_Hⁿ(y) = T_Hⁿ(x − y) = T_Hⁿ(a²) = (T_Hⁿ(a))² ∈ S²ⁿ+

which gives the desired result.

(⇐) Suppose that T_Hⁿ(x) _S²ⁿ

+ T_Hⁿ(y). Then, there exists X, Y ∈ S²ⁿsuch that T_Hⁿ(x) = X and T_Hⁿ(y) = Y . By assumption of X _S²ⁿ

+ Y , there exists an A ∈ S²ⁿ such that X − Y = A². Again, since T_Hⁿ preserves Jordan algebra structure, we have

x − y = T⁻¹

Hⁿ(X) − T⁻¹

Hⁿ(Y ) = T⁻¹

Hⁿ(X − Y ) = T⁻¹

Hⁿ(A²) = (T⁻¹

Hⁿ(A))² ∈ Qⁿ+

which gives the desired result. 2

Next we present three Lemmas which are needed to establish our main result.

Lemma 3.2 Suppose that V is the algebra Hⁿ of n × n complex Hermitian matrices. For any given φ : J → R, let φ_V : VJ → V be defined as in (1). Then,

(a) φ is SC-monotone of order n associated with Hⁿ if φ is matrix monotone of order 2n.

(b) φ is SC-convex of order n associated with Hⁿ if φ is matrix convex of order 2n.

Proof. (a) Suppose x _Hⁿ

+ y and φ is matrix monotone of order 2n. First, Lemma 3.1 indicates T_Hⁿ(x) _S²ⁿ

+ T_Hⁿ(y). Then, from assumption of matrix monotonicity, we have φ_S²ⁿ(T_Hⁿ(x)) _S²ⁿ

+ φ_S²ⁿ(T_Hⁿ(y)).

This together with equation (6) implies T_Hⁿ(φ_Hⁿ(x)) _S²ⁿ

+ T_Hⁿ(φ_Hⁿ(y)). Applying Lemma 3.1 again, we obtain φ_Hⁿ(x) _Hⁿ

+ φ_Hⁿ(y).

(b) Suppose φ is matrix convex of order 2n. Then, for 0 ≤ α ≤ 1, we know φ_S²ⁿ(αT_Hⁿ(x) + (1 − α)T_Hⁿ(y)) _S²ⁿ

+ αφ_S²ⁿ(T_Hⁿ(x)) + (1 − α)φ_S²ⁿ(T_Hⁿ(y)).

In addition, the linearity of T_Hⁿ and equation (6) imply φ_S²ⁿ(T_Hⁿ(αx + (1 − α)y)) _S²ⁿ

+ αT_Hⁿ(φ_Hⁿ(x)) + (1 − α)T_Hⁿ(φ_Hⁿ(y)).

Using equation (6) and linearity of T_Hⁿ again, we have T_Hⁿ(φ_Hⁿ(αx + (1 − α)y)) _S²ⁿ

+ T_Hⁿ(αφ_Hⁿ(x) + (1 − α)φ_Hⁿ(y)) . Then, applying Lemma 3.1 yields

φ_Hⁿ(αx + (1 − α)y) _Hⁿ₊ αφ_Hⁿ(x) + (1 − α)φ_Hⁿ(y) which is the desired result. 2

Analogous to Lemma 3.1, there holds x _Qⁿ

+ y ⇐⇒ T_Qⁿ(x) _S⁴ⁿ

+ T_Qⁿ(y) ∀x, y ∈ Qⁿ

which also lead to the following lemma by similar arguments as in Lemma 3.2.

Lemma 3.3 Suppose that V is the algebra Qⁿ of n × n complex Hermitian matrices. For any given φ : J → R, let φ_V : VJ → V be defined as in (1). Then,

(a) φ is SC-monotone of order n associated with Qⁿ if φ is matrix monotone of order 4n.

(b) φ is SC-convex of order n associated with Qⁿ if φ is matrix convex of order 4n.

Lemma 3.4 [23, Theorem 3.1, Theorem 4.1] Suppose that V is the Jordan spin algebra Jⁿ. For any given φ : J → R, let φ_V : VJ → V be defined as in (1). Then,

(a) φ is SOC-monotone if φ is matrix-monotone of order 2.

(b) φ is SOC-convex if φ is matrix-convex of order 2.

The main idea here is that we employ embeddings T_Hⁿ and T_Qⁿ to provide a sufficient condition for φ being SC-monotone (SC-convex) by its matrix monotonicity (matrix convexity). Now, together with some result in [23], we prsent our main result.

Theorem 3.1 Suppose that (V, ◦, h·, ·i) is a simple Euclidean Jordan algebra of rank n except for O³. For any given φ : J → R, let φ_V : VJ → V be defined as in (1). Then, (a) φ is matrix monotone (matrix convex) of order n if it is SC-monotone (SC-convex)

of order n.

(b) φ is SC-montone (SC-convex) of order n associated with V if it is matrix monotone (matrix convex) of order 4n.

Proof. (a) When n > 3, Theorem 2.1 says V is isomorphic to the algebra Sⁿ, Hⁿ, or Qⁿ. Note that a real number is a special complex number, which is also a special quaternion.

The SC-monotonicity (SC-convexity) of order n of φ implies that φ is matrix monotone (matrix convex) of order n. When n = 2, the SC-monotonicity (SC-convexity) of order 2 of φ is equivalent to the SOC-monotonicity (SOC-convexity) (see [7]). Thus, from [23], it follows that φ is matrix monotone (matrix convex) of order 2.

(b) When n > 3, Theorem 2.1 says V is isomorphic to the algebra Sⁿ, Hⁿ, or Qⁿ. Suppose φ is matrix monotone (matrix convex) of order 4n. Then, we have that φ is also matrix monotone (matrix convex) of order 2n (order n). Thus, applying Theorem 2.1 and Lemmas 3.2-3.3, φ is SC-monotone (SC-convex) of order n. When n = 2, from [23] we know that φ is SOC-monotone (SOC-convex), which is equivalent to saying that φ SC-monotone (SC-convex) of order 2 due to Theorem 2.1. 2

Remark 3.1 It should be pointed out that for the SC-monotonicity of continuously dif- ferentiable φ, Kor´anyi [15] showed that φ is SC-monotone of order n if and only if φ is matrix-monotone of order n. Thus, for the SC-monotonicity, the result of Theorem 3.1 is weaker than that of [15] obtained via direct analysis. However, for the SC-convexity, to our best knowledge, the result of Theorem 3.1 is new. For application in symmetric cone optimization it is very important to know which class of functions is SC-convex.

Theorem 3.1 has good contribution in the literature in our opinion because it tells us that all matrix convex functions must be SC-convex.

As a consequence of Theorem 3.1, we have the following corollary which builds a bridge between matrix monotonicity (matrix convexity) and SC-monotonicity (SC-convexity).

Corollary 3.1 Let (V, ◦, h·, ·i) be a simple Euclidean Jordan algebra except for O³. For any given φ : J → R, let φ_V : VJ → V be defined as in (1). Then, φ is SC-monotone (re- spectively, SC-convex) associated with V if and only if it is matrix monotone (respectively, matrix convex).

Unfortunately our method can not be applied to the only excetional case O³. There are two reasons to explain this. First, it seems imposible to embed O³ into some S^m. Second, there exists a discrepancy between φ

Sm(L(x)) and L(φ_O³(x)). For any x ∈ O³J, suppose x has the spectral decomposition x =P3

j=1λj(x)cj, where λ1(x) ≥ λ2(x) ≥ λ3(x) are the eigenvalues of x and {c₁, c₂, c₃} (depending on x) is the corresponding Jordan frame. Let L(x), C_jl(x) be defined as in Section 2. We have

L(φ_O³(x)) =

3

X

j=1

φ(λ_j(x))C_jj(x) + X

1≤j<l≤3

φ(λ_j(x)) + φ(λ_l(x))

2 C_jl(x) ∀x ∈ VJ. (7) Note here that φ_O³(x) = P φ(λj(x))c_j. Let {u₁, u₂, . . . , u₂₇} be an orthonormal basis of O³. Let L(x), C_jl(x) be the corresponding matrix representations of L(x), C_jl(x) with respect to the basis {u₁, u₂, . . . , u₂₇}. This means that for 1 ≤ a, b ≤ 27

[L(x)]_a,b = hu_a, L(x)u_bi and [C_jl(x)]_a,b= hu_a, C_jl(x)u_bi.

Since O³ is a Euclidean Jordan algebra, L(x) and C_jl(x) are self-adjoint. Thus, L(x) and C_jl(x) are real symmetric matrices in S²⁷J . It follows that

L(φ_O³(x)) =

3

X

j=1

φ(λj(x))Cjj(x) + X

1≤j<l≤3

φ((λj(x)) + φ(λl(x))

2 Cjl(x), ∀x ∈ V^J. For any h ∈ O³, there exists a unique ˜h ∈ R²⁷ such that h = P27

i=1˜h_iu_i. Then, it is obvious to check

hh, φ_O³(x) ◦ ki_O³ = hh, L(φ_O³(x))ki_O³ = h˜h, L(φ_O³(x))˜ki_R²⁷ ∀ h, k ∈ O³, which implies

φ_O³(x) _O³

+ φ_O³(y) ⇐⇒ L(φ_O³(x)) _S²⁷

+ L(φ_O³(y)).

However, on the other hand, we know

φ_S²⁷(L(x)) =

3

X

j=1

φ(λ_j(x))C_jj(x) + X

1≤j<l≤3

φ λ_j(x) + λ_l(x) 2



C_jl(x). (8)

Note here that

L(x) =

3

X

j=1

λ_j(x)C_jj(x) + X

1≤j<l≤3

λ_j(x) + λ_l(x)

2 C_jl(x).

Thus, the discrepency between φ_S²⁷(L(x)) and L(φ_O³(x)) is φ_S²⁷(L(x)) − L(φ_O³(x)) = X

1≤j<l≤3



φ λ_j(x) + λ_l(x) 2



− φ((λ_j(x)) + φ(λ_l(x)) 2



C_jl(x),

which is complicated to handle. Therefore, we exclude this exceptional case O³ in the conclusion.

To close this section, we take a careful look at some examples of SC-monotone func- tions. By applying [26, Example 3] and Corollary 3.1, the following functions are SC- monotone.

Example 3.1 For a general simple Euclidean Jordan algebra (V, ◦, h·, ·i) except for O³, (i) φ(t) = t^q (t ≥ 0) is SC-monotone associated with V if and only if 0 ≤ q ≤ 1.

(ii) φ(t) = −t^−q (t > 0) is SC-monotone associated with V if and only if 0 ≤ q ≤ 1.

(iii) φ(t) = − cot(t) (0 < t < π) is SC-monotone associated with V.

(iv) φ(t) = ln^q(x) (t > 0) with q ∈ (0, 1] is SC-monotone associated with V.

Moreover, [26, Example 35] and Corollary 3.1 indicate that the following functions are SC-convex.

Example 3.2 For a general simple Euclidean Jordan algebra (V, ◦, h·, ·i) except for O³, (i) φ(t) = − ln t (t > 0) is SC-convex associated with V.

(ii) φ(t) = −t^r (t ≥ 0) with r ∈ [1, 2] and φ(t) = −t^r (t > 0) with r ∈ [−1, 0] are SC-convex associated with V.

(iii) the entropy function φ(t) = t ln t (t ≥ 0) is SC-convex associated with V.

From the SC-monotonicity of the function in Example 3.1(i), we readily recover the results of [18, Corollary 9] and [10, Prop. 8]. Moreover, from the SC-monotonicity of the function in Example 3.1(ii), we have that x K y K 0 if and only if y⁻¹ K x⁻¹ K 0.

On the other hand, we show the SC-convexity of some well-known barrier functions:

logarithmic barrier function − ln t (t > 0) and the power function −t^r (t > 0) with r ∈ [−1, 0), which can be employed in the interior point methods for for solving the symmetric cone optimization problems.

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