# Matrix Quasinorms Induced by Maximal and Minimal Vector Norms

Authors: Jong-Do Park
Journal: Journal of Function Spaces
 (2016)

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## Abstract

In the set of all vector norms inCn, there exist maximal and minimal complex norms which coincide with the real Euclidean norm in Rn. The purpose of this paper is to introduce new quasinorms defined on complex matrices. These two matrix quasinorms are induced by maximal and minimal complex vector norms. We also prove the dual relation between these two quasinorms.


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## Body

## 1. Introduction

The standard Euclidean norm inRn (or Cn) is (1)x≔∑j=1nxj2,where x=(x1,…,xn)∈Rn (or Cn). We could easily extend this vector norm to matrices, just by taking a matrix Z=(zij)1≤i≤m,1≤j≤n as a vector in Rmn (or Cmn). This natural extension is called the Frobenius norm ZF (also called Hilbert-Schmidt norm or Schur norm) defined by(2)ZF≔∑j=1m∑k=1nzjk2=trZZ¯T.InCn, there are another two well-known norms L(·) and N∗(·),called the maximal norm and the minimal norm introduced by Siciak [1] and Hahn-Pflug [2], respectively. For z∈Cn, the explicit forms of L(z) and N∗(z) are given by(3)Lz=z2+z4-z•z2,(4)N∗z=12z2+z•z,where z•w≔∑j=1nzjwj for z,w∈Cn.It is known thatL(z) and N∗(z) have the following properties [1–3]:(i)
L(·) and N∗(·) are norms.(ii)
N∗(z)=L(z)=z for all z∈Rn.(iii)
N∗(z)≤z≤L(z) for all z∈Cn.(iv)
They are dual in the sense thatz•w≤L(z)N∗(w) for all z,w∈Cn.In fact, L(z) and N∗(z) are maximal and minimal norms satisfying (ii) and (iii). The Bergman kernel for the ball B∗≔{z∈Cn:N∗(z)<1} was computed explicitly in [4]. Moreover, recently many papers deal with function theoretic problems on B∗ in [5–9].In this paper, similarly to the Frobenius norm, we extendL(·) and N∗(·) to complex matrices satisfying (ii), (iii), and (iv). At first, we show that these two extensions are quasinorms (see Theorem 4). The second result is the duality of L(·) and N∗(·) like Hölder inequality (see Theorem 5). Also we construct Np(Z) for p≥1. If p=1, then N1(Z)=N∗(Z) and limp→∞⁡Np(Z)=L(Z). Finally, we proved the dual relation between Np and Nq when 1/p+1/q=1 (see Corollary 6 and Theorem 7).


## 2. Statements of Main Results

In 1981, Siciak [1] found the existence of complexmaximal extension of · called the Lie norm L(z) for z=(z1,…,zn)∈Cn. The maximal norm L(z) satisfies(i)
L(x)=x for x∈Rn,(ii)
N(z)≤L(z) for any complex norm N(z) with N(x)=x for x∈Rn. In addition, the explicit form of L(z) is known as in (3). In fact, the ball {z∈Cn:L(z)<1} is called the Lie ball which is a classical bounded symmetric domain of type IV.For theminimal complex extension, it is known that such a norm does not exist, since there is a sequence of complex extensions of ∥·∥ which converges to 0 at certain points. One can see a counterexample in [2]. In 1988, Hahn and Pflug [2] constructed the complexminimal extension of · called the minimal norm N∗(z) in a slightly different sense as follows:(i)
N∗(x)=x for x∈Rn.(ii)
N∗(z)≤N(z) for any complex norm N(z) with N(x)=x for x∈Rn and N(z)≤z for z∈Cn. Moreover, the explicit form of N∗(z) is known as in (4). The Bergman kernel for the ball {z∈Cn:N∗(z)<1} was computed explicitly in [4].In [3], Morimoto and Fujita proved the following relation between L(z) and N∗(z) when z∈Cn.Proposition 1 (see [3]).
LetL(z) and N∗(z) be norms defined as in (3) and (4) for z∈Cn. Then, for z,w∈Cn, (i)
z•w≤L(z)N∗(w),(ii)
L(z)=sup{z•w:N∗(w)=1}.Now we deal with the generalization ofN∗(·) and L(·) when z is a complex matrix. In 2002, Youssfi [9] introduced the natural generalization of N∗(·) from complex vectors to complex matrices. Let M be the set of all m×n complex matrices. For Z=(zjk)1≤j≤m,1≤k≤n∈M, we define the row vectors Z(j) by (5)Zj=zj1,zj2,…,zjn,where j=1,2,…,m. The Bergman kernel and the Szegö kernel for the ball {Z∈M:N∗(Z)<1} have been computed by using proper holomorphic liftings (see [9]).Definition 2.
ForZ=(zjk)1≤j≤m,1≤k≤n∈M, we define N∗(Z) by (6)N∗Z≔12∑j=1mZj2+Zj•Zj=12∑j=1m∑k=1nzjk2+∑j=1m∑k=1nzjk2.Similarly, we defineL(Z) and M(Z) for Z∈M as follows.Definition 3.
ForZ=(zjk)1≤j≤m,1≤k≤n∈M, we define L(Z) and M(Z) by (7)LZ≔∑j=1mZj2+∑j=1mZj22-∑j=1mZj•Zj2=∑j=1m∑k=1nzjk2+∑j=1m∑k=1nzjk22-∑j=1m∑k=1nzjk22,MZ≔∑j=1mZj2-∑j=1mZj22-∑j=1mZj•Zj2=∑j=1m∑k=1nzjk2-∑j=1m∑k=1nzjk22-∑j=1m∑k=1nzjk22.Ifm=1, then L(Z) and N∗(Z) are the Lie norm and the minimal norm in Cn, respectively, and if n=1, then L(Z)=N∗(Z) is the Euclidean norm in Cm. Moreover, it is easily proved that the inequality(8)12ZF≤N∗Z≤ZF≤LZ≤2ZFholds, where the Frobenius norm ZF is defined as in (2).At first we prove that extensionsL(Z) and N∗(Z) of these norms to complex matrices are quasinorms of M.Theorem 4.
ForZ∈M, L(Z) and N∗(Z) are quasinorms. Precisely, for Z,W∈M we have (i)
L(Z+W)≤m{L(Z)+L(W)},(ii)
N∗(Z+W)≤m{N∗(Z)+N∗(W)}.The generalization of Proposition1(i) will be proved as follows.Theorem 5.
ForZ,W∈M, we have (9)∑j=1m∑k=1nzjkwjk≤LZN∗W.We also generalize Proposition1(ii) for complex matrices as follows.Corollary 6.
ForZ∈M, we have (10)imin1≤j≤m⁡LZj≤sup⁡∑j=1mZj•Wj:1≤∑j=1mN∗Wj≤m≤mLZ,iimin1≤j≤m⁡LZj≤sup⁡∑j=1mZj•Wj:∑j=1mN∗Wj=1≤LZ.Forp≥1 and Z∈M, we define(11)NpZ≔12∑j=1mLZjp+MZjp1/p,where (12)MZj≔Zj2-Zj22-Zj•Zj2.One can easily see that N1(Z)=N∗(Z) and limp→∞Np(Z)=L(Z). Then, we finally proved the following.Theorem 7.
ForZ∈M, we have(13)imin1≤j≤m⁡NpZj≤sup⁡∑j=1mZj•Wj:1≤∑j=1mNqWj≤m1/p≤m1/pNpZ,(14)iimin1≤j≤m⁡NpZj≤sup⁡∑j=1mZj•Wj:∑j=1mNqWj=1≤NpZ,where 1/p+1/q=1.Remark 8.
Ifm=1, then Corollary 6 and Theorem 7 are identical to previous results proved by Morimoto and Fujita [3].


## 3. Proofs

Throughout this section, it is convenient to define(15)aj≔∑k=1nzjk2,bj≔∑k=1nzjk2,cj≔∑k=1nwjk2,dj≔∑k=1nwjk2,ej≔∑k=1nzjk+wjk2,fj≔∑k=1nzjk+wjk2.Note thataj≥bj≥0, cj≥dj≥0, and ej≥fj≥0.Lemma 9.
Assume thatx1,…,xm,y1,…,ym satisfy xj≥yj≥0 for all j=1,…,m. Then, we have (16)i∑j=1mxj2-yj2≤∑j=1mxj2-∑j=1myj2,ii∑j=1mxj+∑j=1mxj2-∑j=1myj2≤∑j=1mxj+xj2-yj2.Proof.
(i) If we apply Cauchy-Schwarz inequality, then we have(17)∑j=1mxj2-yj2=∑j=1mxj+yjxj-yj≤∑j=1mxj+yj∑j=1mxj-yj=∑j=1mxj2-∑j=1myj2.
(ii) Note that(18)∑j=1mxj+xj2-yj22-∑j=1mxj+∑j=1mxj2-∑j=1myj2=2∑j≠lxj+xj2-yj2xl+xl2-yl2+∑j=1mxj2-yj2-∑j=1mxj2-∑j=1myj2.
The last term is estimated as follows:(19)∑j=1mxj2-∑j=1myj2=∑j=1mxj2-yj2+2∑j≠lxjxl-yjyl≤∑j=1mxj2-yj2+2∑j≠lxjxl-yjyl.
It follows that(20)∑j=1mxj+xj2-yj22-∑j=1mxj+∑j=1mxj2-∑j=1myj2≥2∑j≠lxj+xj2-yj2xl+xl2-yl2-2∑j≠lxjxl-yjyl≥2∑j≠lxjxl-2∑j≠lxjxl>2-2∑j≠lxjxl>0,since yjyl≥0 for all j,l. The proof of (ii) is finished.


### 3.1. Proof of Theorem4(i)

Forj=1,2,…,m, we define the row vectors Z(j) and W(j) by (21)Zj=zj1,zj2,…,zjn,(22)Wj=wj1,wj2,…,wjn.Since L(z) is a norm with z∈Cn, for each j=1,2,…,m, we have (23)LZj+Wj≤LZj+LWj,so that(24)∑j=1mLZ+Wj≤∑j=1mLZj+∑j=1mLWj.Now we will obtain the upper bound of(25)∑j=1mLZj+∑j=1mLWj.By Cauchy-Schwarz inequality and Lemma 9(i), we have (26)∑j=1mLZj=∑j=1m∑k=1nzjk2+∑k=1nzjk22-∑k=1nzjk22=∑j=1maj+aj2-bj2≤m∑j=1maj+∑j=1maj2-bj2≤m∑j=1maj+∑j=1maj2-∑j=1mbj2=mLZ.Similarly, we have (27)∑j=1mLWj≤mLW.Thus, we see that(28)∑j=1mLZj+∑j=1mLWj≤mLZ+LW.By Lemma 9(ii), we have(29)LZ+W=∑j=1mej+∑j=1mej2-∑j=1mfj2≤∑j=1mej+ej2-fj2=∑j=1mLZ+Wj.Combining (24), (28), and (29), we obtain that L(·) is a quasinorm.


### 3.2. Proof of Theorem4(ii)

Now we show thatN∗(·) is a quasinorm. Note that (30)N∗Z+W=12∑j=1mej+fj<∑j=1m12ej+fj=∑j=1mN∗Z+Wj.Since N∗(Z) is a norm when m=1 in [3], we have (31)∑j=1mN∗Z+Wj≤∑j=1mN∗Zj+∑j=1mN∗Wj.By Cauchy-Schwarz inequality, (32)∑j=1mN∗Zj=∑j=1m12aj+bj≤m12∑j=1maj+bj=mN∗Zand similarly (33)∑j=1mN∗Wj≤mN∗W.Combining the above inequalities, we obtain that N∗(·) is also a quasinorm.


### 3.3. Proof of Theorem5

We will prove that Proposition1(i) holds also for the matrices. By Proposition 1(i), we have (34)∑k=1nzjkwjk≤LZjN∗Zj=aj+aj2-bj2×12cj+djfor each j=1,…,m. By Cauchy-Schwarz inequality and Lemma 9(i), we have (35)∑j=1m∑k=1nzjkwjk≤∑j=1maj+aj2-bj2×12cj+dj≤∑j=1maj+aj2-bj2∑j=1m12cj+dj≤∑j=1maj+∑j=1maj2-∑j=1mbj2×12∑j=1mcj+∑j=1mdj=LZN∗W.


### 3.4. Proof of Corollary6

Note that(36)N∗W≤∑j=1mN∗Wj≤mN∗W.If 1≤∑j=1m‍N∗(W(j))≤m, then N∗(W)≤m by (36). So, by Theorem 5, (37)∑j=1mZj•Wj≤mLZ,so that (38)sup⁡∑j=1mZj•Wj:1≤∑j=1mN∗Wj≤m≤mLZ.On the other hand, we have (39)sup⁡∑j=1mZj•Wj:1≤∑j=1mN∗Wj≤m=sup⁡∑j=1mkjZj•Wkjj:1≤∑j=1mkj≤m,N∗Wkjj=1,where kj≔N∗(W(j)) for 1≤j≤m. By Proposition 1(ii), the last term is greater than or equal to (40)inf⁡∑j=1mkjLZj:1≤∑j=1mkj≤m≥min1≤j≤m⁡LZj∑j=1mkj≥min1≤j≤m⁡LZj.It follows that (41)min1≤j≤m⁡LZj≤sup⁡∑j=1mZj•Wj:1≤∑j=1mN∗Wj≤m.The proof of (i) of Corollary 6 is finished. The proof of (ii) is similar to that of (i).


### 3.5. Proof of Theorem7

We use the following inequalities for vectors.Proposition 10 (see [3]).
Forz,w∈Cn, we have (i)
2z·w≤L(z)L(w)+M(z)M(w)(ii)
Np(z)=sup{z•w:Nq(w)=1}, where 1/p+1/q=1.By Proposition10(i), we have (42)2∑j=1mZj•Wj≤∑j=1mLZjLWj+MZjMWj≤∑j=1mLZjp+MZjpp∑j=1mLWjq+MWjqq,where 1/p+1/q=1. From 2=21/p+1/q, we obtain(43)∑j=1mZj•Wj≤NpZNqW.The Hölder inequality tells us that if xj≥0 for all 1≤j≤m, then(44)∑j=1mxj1/q≤∑j=1mxj1/q≤m1/p∑j=1mxj1/q.If we substitute xj=(1/2)(L(W(j))p+M(W(j))) in (44), then we obtain(45)NqW≤∑j=1mNqWj≤m1/pNqW.Now we can prove Theorem 7 similarly to the proof of Corollary 6. From (43) and (45), if 1≤∑j=1mNq(W(j))≤m1/p, then(46)∑j=1mZj•Wj≤m1/pNpZ.Similarly to the proof of Corollary6, we write kj≔Nq(W(j)) for 1≤j≤m. Then, using (45) and Proposition 10(ii), we have(47)sup⁡∑j=1mZj•Wj:1≤∑j=1mNqWj≤m1/p=sup⁡∑j=1mkjZj•Wkjj:1≤∑j=1mkj≤m1/p,NqWkjj=1≥inf⁡∑j=1mkjNpZj:1≤∑j=1mkj≤m≥min1≤j≤m⁡NpZj∑j=1mkj≥min1≤j≤m⁡NpZj.It follows that(48)min1≤j≤m⁡NpZj≤sup⁡∑j=1mZj•Wj:1≤∑j=1mNqWj≤m1/p.From (46) and (48), we complete the proof of (i) of Theorem 7. The proof of (ii) is similar to that of (i).


## 3.1. Proof of Theorem4(i)

Forj=1,2,…,m, we define the row vectors Z(j) and W(j) by (21)Zj=zj1,zj2,…,zjn,(22)Wj=wj1,wj2,…,wjn.Since L(z) is a norm with z∈Cn, for each j=1,2,…,m, we have (23)LZj+Wj≤LZj+LWj,so that(24)∑j=1mLZ+Wj≤∑j=1mLZj+∑j=1mLWj.Now we will obtain the upper bound of(25)∑j=1mLZj+∑j=1mLWj.By Cauchy-Schwarz inequality and Lemma 9(i), we have (26)∑j=1mLZj=∑j=1m∑k=1nzjk2+∑k=1nzjk22-∑k=1nzjk22=∑j=1maj+aj2-bj2≤m∑j=1maj+∑j=1maj2-bj2≤m∑j=1maj+∑j=1maj2-∑j=1mbj2=mLZ.Similarly, we have (27)∑j=1mLWj≤mLW.Thus, we see that(28)∑j=1mLZj+∑j=1mLWj≤mLZ+LW.By Lemma 9(ii), we have(29)LZ+W=∑j=1mej+∑j=1mej2-∑j=1mfj2≤∑j=1mej+ej2-fj2=∑j=1mLZ+Wj.Combining (24), (28), and (29), we obtain that L(·) is a quasinorm.


## 3.2. Proof of Theorem4(ii)

Now we show thatN∗(·) is a quasinorm. Note that (30)N∗Z+W=12∑j=1mej+fj<∑j=1m12ej+fj=∑j=1mN∗Z+Wj.Since N∗(Z) is a norm when m=1 in [3], we have (31)∑j=1mN∗Z+Wj≤∑j=1mN∗Zj+∑j=1mN∗Wj.By Cauchy-Schwarz inequality, (32)∑j=1mN∗Zj=∑j=1m12aj+bj≤m12∑j=1maj+bj=mN∗Zand similarly (33)∑j=1mN∗Wj≤mN∗W.Combining the above inequalities, we obtain that N∗(·) is also a quasinorm.


## 3.3. Proof of Theorem5

We will prove that Proposition1(i) holds also for the matrices. By Proposition 1(i), we have (34)∑k=1nzjkwjk≤LZjN∗Zj=aj+aj2-bj2×12cj+djfor each j=1,…,m. By Cauchy-Schwarz inequality and Lemma 9(i), we have (35)∑j=1m∑k=1nzjkwjk≤∑j=1maj+aj2-bj2×12cj+dj≤∑j=1maj+aj2-bj2∑j=1m12cj+dj≤∑j=1maj+∑j=1maj2-∑j=1mbj2×12∑j=1mcj+∑j=1mdj=LZN∗W.


## 3.4. Proof of Corollary6

Note that(36)N∗W≤∑j=1mN∗Wj≤mN∗W.If 1≤∑j=1m‍N∗(W(j))≤m, then N∗(W)≤m by (36). So, by Theorem 5, (37)∑j=1mZj•Wj≤mLZ,so that (38)sup⁡∑j=1mZj•Wj:1≤∑j=1mN∗Wj≤m≤mLZ.On the other hand, we have (39)sup⁡∑j=1mZj•Wj:1≤∑j=1mN∗Wj≤m=sup⁡∑j=1mkjZj•Wkjj:1≤∑j=1mkj≤m,N∗Wkjj=1,where kj≔N∗(W(j)) for 1≤j≤m. By Proposition 1(ii), the last term is greater than or equal to (40)inf⁡∑j=1mkjLZj:1≤∑j=1mkj≤m≥min1≤j≤m⁡LZj∑j=1mkj≥min1≤j≤m⁡LZj.It follows that (41)min1≤j≤m⁡LZj≤sup⁡∑j=1mZj•Wj:1≤∑j=1mN∗Wj≤m.The proof of (i) of Corollary 6 is finished. The proof of (ii) is similar to that of (i).


## 3.5. Proof of Theorem7

We use the following inequalities for vectors.Proposition 10 (see [3]).
Forz,w∈Cn, we have (i)
2z·w≤L(z)L(w)+M(z)M(w)(ii)
Np(z)=sup{z•w:Nq(w)=1}, where 1/p+1/q=1.By Proposition10(i), we have (42)2∑j=1mZj•Wj≤∑j=1mLZjLWj+MZjMWj≤∑j=1mLZjp+MZjpp∑j=1mLWjq+MWjqq,where 1/p+1/q=1. From 2=21/p+1/q, we obtain(43)∑j=1mZj•Wj≤NpZNqW.The Hölder inequality tells us that if xj≥0 for all 1≤j≤m, then(44)∑j=1mxj1/q≤∑j=1mxj1/q≤m1/p∑j=1mxj1/q.If we substitute xj=(1/2)(L(W(j))p+M(W(j))) in (44), then we obtain(45)NqW≤∑j=1mNqWj≤m1/pNqW.Now we can prove Theorem 7 similarly to the proof of Corollary 6. From (43) and (45), if 1≤∑j=1mNq(W(j))≤m1/p, then(46)∑j=1mZj•Wj≤m1/pNpZ.Similarly to the proof of Corollary6, we write kj≔Nq(W(j)) for 1≤j≤m. Then, using (45) and Proposition 10(ii), we have(47)sup⁡∑j=1mZj•Wj:1≤∑j=1mNqWj≤m1/p=sup⁡∑j=1mkjZj•Wkjj:1≤∑j=1mkj≤m1/p,NqWkjj=1≥inf⁡∑j=1mkjNpZj:1≤∑j=1mkj≤m≥min1≤j≤m⁡NpZj∑j=1mkj≥min1≤j≤m⁡NpZj.It follows that(48)min1≤j≤m⁡NpZj≤sup⁡∑j=1mZj•Wj:1≤∑j=1mNqWj≤m1/p.From (46) and (48), we complete the proof of (i) of Theorem 7. The proof of (ii) is similar to that of (i).



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