We point out that the Schatten p-norm is different from the operator norm (matrix norm) induced by lp-norm, which is given by ∥A∥p := sup∥x∥p=1∥Ax∥p, although these two norms share the same notation. The readers can distinguish them from the contexts, so we do not denote them by different symbols. It is also known that the Schatten p-norm satisfies
• ∥A1A2∥1≤ ∥A1∥p·∥A2∥qwhere 1p+1q = 1 and p∈ [1, ∞] (Holder inequality).
• ∥A∥1 ≥ ∥A∥p1 ≥ ∥A∥p2 ≥ ∥A∥∞ where 1≤ p1≤ p2 ≤ ∞ (Monotonicity).
For more details and applications of the Schatten p-norm, please refer to [1, 2, 8, 14, 15] and references therein.
The second one is the so-called Ky Fan k-norm defined by
∥A∥(k):=
∑k i=1
(si(A)), 1≤ k ≤ n.
In other words, ∥A∥(k) is exactly the sum of the k largest singular values of A. In addition, the Ky Fan 1-norm is the operator norm induced by the Euclidean norm;
and hence it is also called the operator 2-norm. For more details and applications of the Ky Fan k-norm, please refer to [1, 2, 6, 8, 15] and references therein.
Now, we consider the space Sn of n × n real symmetric matrices. Under the Jordan product X◦ Y = 12(XY + Y X) and the bilinear form ⟨X, Y ⟩ := tr(XY ), (Sn,◦, ⟨·, ·⟩) forms a Euclidean Jordan algebra whose definition will be elaborated in Section 2. Based on Spectral Decomposition Theorem [7, Theorem III.1.2], we also note that the eigenvalues of A coincide with the spectral values of A∈ Sn. It is also known thatRn can be viewed as a Euclidean Jordan algebra under appropriate Jordan product and inner product. This motivates us to study whether the Schatten p-norms can be defined onRn or not.
In this short paper, we shall define two types of Schatten p-norm on Rn and investigate some inequalities about these two norms. The paper is organized as below. In Section 2, we recall some basic definitions and properties about Euclidean Jordan algebra. Under the standard inner product, two types of Euclidean Jordan algebra overRnare also established in Section 3. Moreover, some relationship about these two norms are deduced as well.
2. Preliminary
In this section, we review the basic concepts and properties concerning Jordan algebras and symmetric cones from the book [7] which are needed in the subsequent analysis.
A Euclidean Jordan algebra is a finite dimensional inner product space (V, ⟨·, ·⟩) (V for short) over the field of real numbers R equipped with a bilinear map (x, y) 7→
x◦ y : V × V → V, which satisfies the following conditions:
(i) x◦ y = y ◦ x for all x, y ∈ V;
THE SCHATTEN p-NORM ONR 23
(ii) x◦ (x2◦ y) = x2◦ (x ◦ y) for all x, y ∈ V;
(iii) ⟨x ◦ y, z⟩ = ⟨x, y ◦ z⟩ for all x, y, z ∈ V,
where x2 := x◦ x, and x ◦ y is called the Jordan product of x and y. If a Jordan product only satisfies the conditions (i) and (ii) in the above definition, the algebra V is said to be a Jordan algebra. Moreover, if there is an (unique) element e ∈ V such that x ◦ e = x for all x ∈ V, the element e is called the Jordan identity in V. Note that a Jordan algebra does not necessarily have an identity element.
Throughout this paper, we assume that V is a Euclidean Jordan algebra with an identity element e.
In the Euclidean Jordan algebra V, the set of squares K := {x2 : x ∈ V} is called a symmetric cone [7, Theorem III.2.1], which means K is a self-dual closed convex cone and, for any two elements x, y ∈ int(K), there exists an invertible linear transformation Γ : V → V such that Γ(x) = y and Γ(K) = K. An element c ∈ V is called an idempotent if c2 = c, and it is a primitive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempotents. Two idempotents c, d are said to be orthogonal if c◦ d = 0. In addition, we say that a finite set {e(1), e(2),· · · , e(r)} of primitive idempotents in V is a Jordan frame if
e(i)◦ e(j)= 0 for i̸= j, and
∑r i=1
e(i)= e.
Note that ⟨e(i), e(j)⟩ = ⟨e(i)◦ e(j), e⟩ whenever i ̸= j. With the above, there has the spectral decomposition of an element x in V.
Theorem 2.1 (Spectral Decomposition Theorem ([7, Theorem III.1.2])). Let V be a Euclidean Jordan algebra. Then there is a number r such that, for every x∈ V, there exists a Jordan frame {e(1),· · · , e(r)} and real numbers λ1(x),· · · , λr(x) with
x = λ1(x)e(1)+· · · + λr(x)e(r).
Here, the numbers λi(x) (i = 1,· · · , r) are called the spectral values of x, the expres- sion λ1(x)e(1)+· · · + λr(x)e(r) is called the spectral decomposition of x. Moreover, tr(x) :=∑r
i=1λi(x) is called the trace of x, det(x) := λ1(x)λ2(x)· · · λr(x) is called the determinant of x, and r is called the rank of V.
3. Main results
In this section, we introduce two types of Euclidean Jordan algebra overRnunder the standard inner product
⟨x, y⟩ := x1y1+ x2y2+· · · + xnyn, for any x = (x1, x2, . . . , xn)∈ Rnand y = (y1, y2, . . . , yn)∈ Rn.
(I). The first type of Jordan product is defined as x• y = (x1y1, x2y2, . . . , xnyn).
It is easy to check (Rn,•, ⟨·, ·⟩) forms a Euclidean Jordan algebra with the identity element e = (1, 1, . . . , 1). Under this case, it is not a simple Euclidean Jordan algebra. Indeed, it is a Cartesian product of simple Euclidean Jordan algebras. In view of this, for all i = 1, 2, . . . , n, we denote ei the vector with the i-th component is 1 and the others are all zeros. Then, each ei is a primitive idempotent inRnand the set {e1, e2, . . . , en} forms a Jordan frame. The induced symmetric cone is
Rn+:={(x1, x2, . . . , xn)∈ Rn | xi≥ 0, i = 1, 2, . . . , n} . Moreover, it is clear that for any x∈ Rn,
x = x1e1+ x2e2+ . . . + xnen.
On the other hand, from the Spectral Decomposition Theorem, we know the spectral values of x are x1, x2, . . . , xn. In light of all the above observations, the first type of Schatten p-norm ∥ · ∥p is defined as
(3.1) ∥x∥p :=
[ n
∑
i=1
|xi|p ]1/p
, 1≤ p < ∞,
which coincides with the well-known lp-norm. Moreover, for p = ∞, the Schatten p-norm becomes
∥x∥∞= max
1≤i≤n|xi|,
which coincides with the well-known supremum norm. From the coincidence, the properties of the Schatten p-norm on Rn space can be easily obtained. As below, we only state two of them, and the proofs are omitted because in this case, they are the lp-norm exactly. In the literature, there are various nice proofs for these two properties, see [11, 12].
Proposition 3.1. For any fixed x∈ Rn, let ∥x∥p denote the first type of Schatten p-norm on Rn given as in (3.1). Then, the function p 7→ ∥x∥p is a decreasing function on [1,∞).
Proposition 3.2. For any fixed x∈ Rn, let ∥x∥p denote the first type of Schatten p-norm on Rn given as in (3.1). Then, for 1≤ p ≤ q ≤ ∞, there holds
(3.2) ∥x∥q ≤ ∥x∥p≤ n1p−1q · ∥x∥q.
(II). Now, we consider the second type of Jordan product which the induced symmetric cone is the so-called second-order cone (or Lorentz cone). For the sim- plicity of notation, we denote x = (x1, ¯x) ∈ R × Rn−1 the vector in Rn, i.e., ¯x :=
(x2, x3, . . . , xn) is a vector in Rn−1. For any x = (x1, ¯x), y = (y1, ¯y)∈ R × Rn−1, the second type of Jordan product is defined as
x◦ y = (⟨x, y⟩, y1¯x + x1¯y).
We note that e = (1, 0) ∈ R × Rn−1 acts as the Jordan identity. Besides, this Jordan product is not associative and Kn is not closed under this Jordan product.
THE SCHATTEN p-NORM ONR 25
The induced symmetric cone is an important example of symmetric cones, which is defined as follows:
Kn={
x = (x1, ¯x)∈ R × Rn−1| x1≥ ∥¯x∥2
}. For any vector x = (x1, ¯x)∈ R × Rn−1, it can be decomposed as
x = λ1(x)u(1)x + λ2(x)u(2)x ,
where λ1(x), λ2(x) and u(1)x , u(2)x are the spectral values and the associated spectral vectors of x, respectively, given by
λi(x) = x1+ (−1)i∥¯x∥2, u(i)x =
{ 1 2
(
1, (−1)i∥¯x∥¯x2)
if ¯x̸= 0,
1 2
(1, (−1)iw¯)
if ¯x = 0.
for i = 1, 2 with ¯w being any vector in Rn−1 satisfying ∥ ¯w∥2 = 1. If ¯x ̸= 0, the decomposition is unique. Here, we remark that for n = 1, the second-order cone K1 reduces to the nonnegative real numberR+ and the Jordan product is the basic multiplication on R.
We say a few words about the discrepancy between (Rn,◦, ⟨·, ·⟩) and (Rn,•, ⟨·, ·⟩).
Note that (Rn,◦, ⟨·, ·⟩) is a simple Euclidean Jordan algebra, whereas the Euclidean Jordan algebra (Rn,•, ⟨·, ·⟩) is not simple. As mentioned earlier, (Rn,•, ⟨·, ·⟩) can be written as the direct sum of (R, ·). In fact, there are only five types of simple Euclidean Jordan algebra [7]. For more details for second-order cones, please refers to [3, 4, 5].
According to the structure of (Rn,◦, ⟨·, ·⟩) , the second type of Schatten p-norm on Rn space ought to be defined as
(3.3) |||x|||p :=
[ 2
∑
i=1
|λi(x)|p ]1/p
.
In a recent paper, Huang et. al [9, Theorem 3.6-3.7] (also see [10, 13] for Euclidean Jordan algebra) established a trace version inequality via the trace version of Young inequality:
[tr(|x + y|p)]1/p≤ [tr(|x|p)]1/p+ [tr(|y|p)]1/p,
where x, y ∈ Rnand p≥ 1. By the definition of the second type of Schatten p-norm on Rn given as in (3.3), it is not hard to verify that this inequality is equivalent to (3.4) |||x + y|||p ≤ |||x|||p+|||y|||p.
Hence, the functional x7→ |||x|||p can actually define a norm onRn space for p≥ 1.
In particular, we note
(3.5) |||x|||2=
[ 2
∑
i=1
|λi(x)|2 ]1/2
=√
2· ∥x∥2.
Moreover, this norm ||| · |||p can be viewed as a norm by applying the first type of Schatten p-norm to the vector (x1 − ∥¯x∥, x1 +∥¯x∥) ∈ R2 which consists of the
spectral values of x. Accordingly we immediately have the following inequality by applying (3.2).
Proposition 3.3. For any x ∈ Rn, let |||x|||p denote the second type of Schatten p-norm on Rn given as in (3.3). Then, for 1≤ p ≤ q ≤ ∞, there holds
|||x|||q≤ |||x|||p ≤ 21p−1q · |||x|||q.
It is natural to ask if there is any relationship between these two types of Schatten p-norm onRn. We observe that each spectral value of x includes the term∥¯x∥. This leads us to separate the discussion into two cases: 1≤ p < 2 and 2 ≤ p.
Theorem 3.4. For any x∈ Rn, let ∥x∥p denote the first type of Schatten p-norm on Rn given as in (3.1) and |||x|||p denote the second type of Schatten p-norm on Rn given as in (3.3). Then, for 1≤ p ≤ ∞, the following hold.
(a) For 1≤ p < 2, there holds
√2 n12−1p· ∥x∥p ≤ |||x|||p ≤ 21p · ∥x∥p. (b) For p≥ 2, there holds
2p1 · ∥x∥p ≤ |||x|||p ≤√
2 n12−1p · ∥x∥p. Proof. (a) For 1≤ p < 2 and x ∈ Rn, we have
|||x|||p ≥ |||x|||2 =√
2· ∥x∥2 ≥√
2n12−1p · ∥x∥p, and
|||x|||p ≤ 21p−12 · |||x|||2 = 21p · ∥x∥2≤ 21p · ∥x∥p,
where the inequalities hold by Proposition 3.3 and inequalities (3.2) and (3.5).
(b) For p≥ 2 and x ∈ Rn, we have
|||x|||p ≤ |||x|||2 =√
2· ∥x∥2≤√
2n12−1p · ∥x∥p. and
|||x|||p ≥ 21p−12 · |||x|||2 = 21p · ∥x∥2≥ 21p · ∥x∥p.
where the inequalities hold by Proposition 3.3 and inequalities (3.2) and (3.5) as
well. □
Now, we recall that a function Φ :Rn→ R is called a symmetric gauge function if
(i) Φ is a norm on the real spaceRn;
(ii) Φ(σn(x)) = Φ(x) for all x∈ Rn, where σn(x) is a permutation of the coor- dinate of x;
(iii) Φ(δ1x1, δ2x2, . . . , δnxn) = Φ(x1, x2, . . . , xn) for δj =±1;
(iv) Φ(1, 0, . . . , 0) = 1.
THE SCHATTEN p-NORM ONR 27
We note from Problem II.5.12(iv) of [2, page 53] that for any x, y ∈ Rn+, we have Φ(x)≤ Φ(y) whenever x↓ ≺ y↓,
where x↓ = (x↓1, x↓2, . . . , x↓n), y↓ = (y1↓, y2↓, . . . , yn↓) denotes the vectors obtained by the rearranging the coordinates of x, y in the decreasing orders, respectively, and x↓≺ y↓ means∑k
j=1x↓j ≤∑k
j=1yj↓ for all 1≤ k ≤ n.
It is easy to check that the functional (λ1(x), λ2(x)) 7→ |||x|||p is a symmetric gauge function on R2. For any x, y ∈ Rn, we say x is weakly majorized by y, denoted by x≺wy if λ(x)↓ ≺ λ(y)↓, that is,
λ2(x)≤ λ2(y) and
∑2 i=1
λi(x)≤
∑2 i=1
λi(y).
In light of these concepts, we achieve the following norm inequality.
Theorem 3.5. Let ||| · |||p denote the second type of Schatten p-norm onRn given as in (3.3). For any x, y∈ Kn, there holds
x≺w y =⇒ |||x|||p ≤ |||y|||p.
Proof. For any x, y ∈ Kn with x ≺w y, we note that both (λ1(x), λ2(x)) and (λ1(y), λ2(y)) lie in R2+, and λ(x)↓ ≺ λ(y)↓. Then, using Problem II.5.12(iv) [2, page 53] we obtain the desired inequality since (λ1(x), λ2(x)) 7→ |||x|||p is a sym-
metric gauge function. □
In linear algebra, functional analysis and some other related areas of mathematics, a quasinorm is often used in the analysis, which is similar to a norm except that the triangle inequality is replaced by
∥x + y∥ ≤ K(∥x∥ + ∥y∥)
for some K > 0. It is already known that for 0 < p < 1, the functional x 7→
(∑n
i=1|xi|p)1p defines a quasinorm∥ · ∥p onRn and satisfies
∥x + y∥p≤ 21p(∥x∥p+∥y∥p).
Meanwhile, a question arises from the above discussion. Is the second type of Schatten p-norm ||| · |||p a quasinorm for 0 < p < 1 ? To answer this question, we need the following technical lemma.
Lemma 3.6. Let a1, a2, . . . , an be nonnegative real numbers and 0 < p < 1. Then,
we have ( n
∑
i=1
ai
)p
≤
∑n i=1
api.
Proof. This is a fundamental result, please refer to [11] for a proof. □
Theorem 3.7. Let ||| · |||p denote the second type of Schatten p-norm on Rn given as in (3.3). For any x, y∈ Rn and 0 < p < 1, there holds
|||x + y|||p ≤ 21p−1(|||x|||p+|||y|||p) . Proof. We note that
|||x + y|||p = (|λ1(x + y)|p+|λ2(x + y)|p)1p
≤ 21p−1(|λ1(x + y)| + |λ2(x + y)|)
≤ 21p−1(|λ1(x)| + |λ2(x)| + |λ1(y)| + |λ2(y)|)
≤ 21p−1(
(|λ1(x)|p+|λ2(x)|p)1p + (|λ1(y)|p+|λ2(y)|p)1p )
= 21p−1(|||x|||p+|||y|||p) ,
where the three inequalities hold by the convexity of the function t 7→ t1/p, the inequality (3.4) for p = 1, and Lemma 3.6, respectively. □
4. Concluding remark
In this paper, we have successfully extended the concept of Schatten p-norm on matrices space to the setting ofRn space via Euclidean Jordan algebra. Two types of Schatten p-norm onRnspace are defined and their connection is discussed. As a matter of fact, Tao et al. [13, Theorem 4.1] establish that for p≥ 1, the functional
x7→
[ r
∑
i=1
|λi(x)|p ]1/p
forms a trace p-norm in any Euclidean Jordan algebra with rank r. In view of Theorem 3.7, we suspect that there is possibility to improve it although we cannot get the proof done yet. Hence, we make a conjecture as below, which is for our future study.
Conjecture 1. Let (V, ◦, ⟨·, ·⟩) be a Euclidean Jordan algebra of rank r. Then for 0 < p < 1, the functional
x7→
[ r
∑
i=1
|λi(x)|p ]1/p
is a quasinorm on V.
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Manuscript received September 10, 2018 revised December 8, 2018
C. H. Hunag
General Education Center Wenzao Ursuline University of Languages Kaohsiung 80793, Taiwan E-mail address: q100300@mail.wzu.edu.tw
J. S. Chen
Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan E-mail address: jschen@math.ntnu.edu.tw
C. C. Hu
Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan E-mail address: cchu@ntnu.edu.tw
