Abstract
In the set of all vector norms in , there exist maximal and minimal complex norms which coincide with the real Euclidean norm in . 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.
1. Introduction
The standard Euclidean norm in (or ) is where (or ). We could easily extend this vector norm to matrices, just by taking a matrix as a vector in (or ). This natural extension is called the Frobenius norm (also called Hilbert-Schmidt norm or Schur norm) defined by
In , there are another two well-known norms and ,called the maximal norm and the minimal norm introduced by Siciak [1] and Hahn-Pflug [2], respectively. For , the explicit forms of and are given bywhere for .
It is known that and have the following properties [1–3]:(i) and are norms.(ii) for all .(iii) for all .(iv)They are dual in the sense that for all .In fact, and are maximal and minimal norms satisfying (ii) and (iii). The Bergman kernel for the ball was computed explicitly in [4]. Moreover, recently many papers deal with function theoretic problems on in [5–9].
In this paper, similarly to the Frobenius norm, we extend and 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 and like Hölder inequality (see Theorem 5). Also we construct for . If , then and . Finally, we proved the dual relation between and when (see Corollary 6 and Theorem 7).
2. Statements of Main Results
In 1981, Siciak [1] found the existence of complex maximal extension of called the Lie norm for . The maximal norm satisfies(i) for ,(ii) for any complex norm with for . In addition, the explicit form of is known as in (3). In fact, the ball is called the Lie ball which is a classical bounded symmetric domain of type IV.
For the minimal 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 complex minimal extension of called the minimal norm in a slightly different sense as follows:(i) for .(ii) for any complex norm with for and for . Moreover, the explicit form of is known as in (4). The Bergman kernel for the ball was computed explicitly in [4].
In [3], Morimoto and Fujita proved the following relation between and when .
Proposition 1 (see [3]). Let and be norms defined as in (3) and (4) for . Then, for , (i),(ii)
Now we deal with the generalization of and when is a complex matrix. In 2002, Youssfi [9] introduced the natural generalization of from complex vectors to complex matrices. Let be the set of all complex matrices. For , we define the row vectors by where . The Bergman kernel and the Szegö kernel for the ball have been computed by using proper holomorphic liftings (see [9]).
Definition 2. For , we define by
Similarly, we define and for as follows.
Definition 3. For , we define and by
If , then and are the Lie norm and the minimal norm in , respectively, and if , then is the Euclidean norm in . Moreover, it is easily proved that the inequalityholds, where the Frobenius norm is defined as in (2).
At first we prove that extensions and of these norms to complex matrices are quasinorms of .
Theorem 4. For , and are quasinorms. Precisely, for we have (i),(ii).
The generalization of Proposition 1(i) will be proved as follows.
Theorem 5. For , we have
We also generalize Proposition 1(ii) for complex matrices as follows.
Corollary 6. For , we have
For and , we definewhere One can easily see that and . Then, we finally proved the following.
Theorem 7. For , we havewhere .
Remark 8. If , 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
Note that , , and .
Lemma 9. Assume that satisfy for all . Then, we have
Proof. (i) If we apply Cauchy-Schwarz inequality, then we have(ii) Note that The last term is estimated as follows:It follows that since for all . The proof of (ii) is finished.
3.1. Proof of Theorem 4(i)
For , we define the row vectors and by Since is a norm with , for each , we have so thatNow we will obtain the upper bound ofBy Cauchy-Schwarz inequality and Lemma 9(i), we have Similarly, we have Thus, we see thatBy Lemma 9(ii), we haveCombining (24), (28), and (29), we obtain that is a quasinorm.
3.2. Proof of Theorem 4(ii)
Now we show that is a quasinorm. Note that Since is a norm when in [3], we have By Cauchy-Schwarz inequality, and similarly Combining the above inequalities, we obtain that is also a quasinorm.
3.3. Proof of Theorem 5
We will prove that Proposition 1(i) holds also for the matrices. By Proposition 1(i), we have for each . By Cauchy-Schwarz inequality and Lemma 9(i), we have
3.4. Proof of Corollary 6
Note thatIf , then by (36). So, by Theorem 5, so that On the other hand, we have where for . By Proposition 1(ii), the last term is greater than or equal to It follows that The proof of (i) of Corollary 6 is finished. The proof of (ii) is similar to that of (i).
3.5. Proof of Theorem 7
We use the following inequalities for vectors.
Proposition 10 (see [3]). For , we have (i)(ii), where .
By Proposition 10(i), we have where . From , we obtainThe Hölder inequality tells us that if for all , thenIf we substitute in (44), then we obtainNow we can prove Theorem 7 similarly to the proof of Corollary 6. From (43) and (45), if , then
Similarly to the proof of Corollary 6, we write for . Then, using (45) and Proposition 10(ii), we haveIt follows thatFrom (46) and (48), we complete the proof of (i) of Theorem 7. The proof of (ii) is similar to that of (i).
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF-2015R1D1A1A01060295).