Abstract

Let be nonhomogeneous Cartan-Hartogs domain of the first type, a holomorphic self-map, and a fixed holomorphic function on . We study the weighted composition operator for a function holomorphic on . Our main results generalize both cases of the unit ploydisc and the unit ball obtained by Li and Stević (Li 2007 and Li 2008). Firstly, we obtain two crucial inequalities on ; furthermore, the boundedness and compactness of operator from the space of all bounded holomorphic functions to the -Bloch space on are investigated.

1. Introduction

Let be a bounded domain in and be the set of all holomorphic functions on . Let be complex Banach spaces on , let be a fixed holomorphic function on , and let be a holomorphic self-map of . The weighted composition operator with the multiplication symbol and the composition symbol is defined by for a function holomorphic on . It should be mentioned that this operator can be regarded as a generalization of a multiplication operator and a composition operator on various Banach spaces; one can see [1] and reference within for more information on composition operators.

Our primary objects of study in this article are bounded and compact weighted composition operators from the space of all bounded holomorphic functions to the -Bloch space on the Cartan-Hartogs domain of the first type, which is defined by Yin [2]. In the work of [3], Cartan first split the irreducible bounded symmetric domains into four types of Cartan domains and two exceptional domains whose complex dimensions are 16 and 27, respectively. Based on this pioneering work, Yin [2] constructed the Hua domains in the theory of several complex variables, which mainly contain the Cartan-Hartogs domains, Cartan-Egg domains, Hua domains, generalized Hua domains, and Hua construction. The Cartan-Hartogs domain of the first type is defined as follows: where is the Cartan domain of the first type, denotes the conjugate transpose of , denotes the determinant of a square matrix, are some positive integers, and is a positive real number. In particular, when , , and , the Cartan-Hartogs domain of the first type turns to be the case of the unit ball; it is obvious that the unit ball is a specific case of the Hua domain. In [4], they verified that the Hua domain is not a homogeneous domain or a Reinhardt domain unless a ball. For simplicity, the Cartan-Hartogs domain of the first type is characterized as . Moreover, throughout this paper, we only consider the case of for convenience. However, we would like to mention that all of results obtained in this work can be extended to the case of naturally.

In [5], Ohno investigated the boundedness and compactness of weighted composition operators between and the Bloch space in the open unit disc. In the setting of the unit ball, Du and Li [6] study the boundedness and compactness of weighted composition operators from to the Bloch space , whose norm is defined by the radial derivative . Li and Stević [7] gave another characterization for the boundedness and compactness of weighted composition operators from to the -Bloch space , whose norm is defined by the gradient . Actually, these two norms are equivalent (see [8] for details). In the setting of the unit polydisc, Li and Stević [9, 10] presented some necessary and sufficient conditions for the composition operators and weighted composition operators between and -Bloch space to be bounded and compact. Besides, there are various interesting works in the literature concerning the operators from the Bloch-type space with the normal weight or the logarithmic weight to in the unit disc, unit ball, or polydisc (cf. [6, 11–15]).

Allen and Colonna [16] investigated the boundedness and compactness of the weighted composition operators from to the Bloch space in the bounded homogeneous domain. In the case of the infinite dimensional bounded symmetric domains, Hamada in [17] studied the bounded weighted composition operators from to the Bloch space on the infinite dimensional bounded symmetric domain, which is realized as the open unit ball of a -triple in [18].

However, in the setting of the Hua domain, the related works only focus on the composition operators between the classic Bloch spaces, the Bloch-type space equipped with the special weight or the normal weight (see, e.g., [19–23]).

In the present paper, motivated by [7, 9], we characterize the boundedness and compactness of weighted composition operators from to -Bloch space on the Cartan-Hartogs domain of the first type. The remainder of this article is organized as follows. In Section 2, we collect background materials necessary for the understanding of the statements of our main results. In Section 3, two important inequalities on the Cartan-Hartogs domain of the first type are derived. The first one, let and , for a holomorphic function in the Cartan-Hartogs domain of the first type, there exists a constant such that

The second inequality is that, for , we have which is in a position to derive the Hua inequality (see [24]). Using these two inequalities and constructing some test functions on , Section 4 is devoted to studying the boundedness of the weighted composition operator , and in Section 5, the compactness of the weighted composition operator is also derived.

Throughout the rest of the paper, denotes some constants which may change from line to line.

2. Preliminaries

In this section, before we state the main results, we would like to collect some notations and crucial lemmas in order to prove the main results.

Definition 1. We use to denote the space of all bounded holomorphic functions on . The space is a Banach algebra under the following supremum norm :

For a holomorphic function , the complex gradient of at will be denoted by , that is

Definition 2. Let . The -Bloch space consists of all holimorphic functions on satisfying

If we equip the norm it is clear that the -Bloch space becomes a Banach space under the norm which can be proved in a standard way.

For more information on and the Bloch-type space, we refer to [25, 26] and references therein.

Lemma 3 (see [27], Theorem 3.3.1) (Hadamard). Let be an Hermitian matrix. Then, and holds if and only if is a diagonal matrix.

Lemma 4 (see [28]). If , keep the same sign and , then,

Remark 5. When or , we get

Lemma 6 (see [26], Proposition 5.1). Let be the unit disc on . . Moreover, for all and .

This lemma shows that any bounded analytic function on is in the Bloch space. We will generalize this lemma to the Cartan-Hartogs domain of the first type in Section 3.

Lemma 7 (see [29], Appendix: Theorem 1.1). Let be an matrix ; then, there exist an unitary matrix and an unitary matrix such that where are characteristic values of .

Lemma 8 (see [29], Theorem 3.1.1). Let satisfying Then, there is an arranged square matrix such that and the minimum value is obtained when and .

Lemma 9 (see [28]). Let ; then,

Lemma 10. Let be a compact subset of and be a holomorphic function on . Then for every , there exists a constant such that

Proof. This lemma is a special case of Lemma 2.4 in [23] by taking ; here, we omit the details.

Lemma 11. Let be a holomorphic self-map of and a holomorphic function on . The weighted composition operator is compact if and only if is bounded and for any bounded sequence in converging to 0 uniformly on compact subsets of , .

Proof. Suppose that is compact. Let be a bounded sequence in with , and uniformly on compact subsets of as . By the definition of compactness of , the sequence has a subsequence converging to . By (19), there exists a constant such that for every . It follows that uniformly on compact subsets of as . Therefore, uniformly on compact subsets of as . Owing to the definition of , we obtain ; thus, .
Conversely, suppose that is a sequence in the ball ; then . It is obvious that is uniformly bounded on compact subsets of . By Montel’s theorem, we know that has a subsequence converges to uniformly on . Moveover, and . Hence the sequence is such that and uniformly on compact subsets of . We Following from the hypothesis implies that which yields that the set is relatively compact.

3. Two Important Inequalities

In this section, we obtain two important inequalities on , which are essential in proving our main results. We remark that two inequalities below seem to be known in the unit ball, but we need to prove them correct on the Cartan-Hartogs domain of the first type.

Theorem 12. Let and . There exists a positive constant independent of such that for all and .

Proof. Since and the definition of , we have . For each , let In view of (10) and (11), we have Due to , it leads to Since , it is easy to see that . Moreover, let , ; we have . Making use of the following inequality , it suffices to obtain In fact, to prove , we can consider . Let ; it follows that we should prove , which obviously holds. Moreover, according to Lemma 6, it leads to which gives the desired estimate.

Remark 13. When the target is the unit ball in , let , , and ; we have the inequality , which arrives at the same conclusion in ([7], Lemma 3).

Theorem 14. Let , and . If , , and . Then, the following inequality holds and holds if and only if .

Proof. When , since , applying Lemma 7, there exist unitary matrixes and unitary matrixes such that Then, it turns out to and according to Lemma 8, there exists an arrange square matrix such that Hence, using (18), we have where is the rearrangement of . Moreover, referring to the condition of equalities for (31) and (32), we obtain the inequality which becomes an equality if and only if .
When , there exists an unitary matrix such that By (32), we obtain Thus, the inequality holds when , and holds if and only if . By the inequality of arithmetic and geometric means, we have and the equality holds if and only if . Therefore, combining (36) with (37) gives that The first inequality becomes an equality if and only if , and the second inequality becomes an equality if and only if , , or , which implies holds only when in (38). Hence, in this case, there is equality in (38) if and only if .

Corollary 15. Let , , and . If , , , and . Then, the following inequality holds

Proof. This proof only follows the elementary inequality ; here, we omit the details.

Corollary 16. Let . If and , then,

Proof. Substituting and into (28) leads to this inequality.

Remark 17. Since, we get which yields the Hua inequality discovered by Hua Loo-Keng in [24].

4. Boundedness of

In this section, we characterize the bounded weighted composition operator in the case . The following theorem describes such properties.

We will begin by introducing some notations. Let be a holomorphic self-map of , denoting

Theorem 18. For and , let be a holomorphic self-map of , a holomorphic function on , and . If then the weighted composition operator is bounded.
Conversely, if the weighted composition operator is bounded, then, where

Proof. Assume that (43) holds. There exists a positive constant such that for all and . Firstly, we know that Therefore, it leads to Namely, For a function , we obtain the following estimate Since and (22), it leads to which implies that is bounded.
Conversely, assume that is bounded. It follows that there exists a positive constant such that Let ; we have , which implies . For , define a test function by From (28), it follows that which implies and .
For the test function , we have where . It leads to Then, it follows that Let Since , (52) and (57), we obtain Since , we obtain The proof is completed.

Remark 19. Let , and , we obtain the following results in the case of the unit ball . Let . If then the weighted composition operator is bounded. Conversely, the weighted composition operator is bounded, then where

Li and Stević investigated the boundedness of this weighted composition operator in [7], which is as the same as the above results; therefore, our main results cover and substantially improve the work of [7].

5. Compactness of

In this section, we characterize the compact weighted composition operator .

Theorem 20. For and , let be a holomorphic self-map of , a holomorphic function on and . If then the weighted composition operator is compact.
Conversely, if the weighted composition operator is compact, then where is the same as (45).

Proof. Suppose that (64) holds. We have Following from Theorem 18, we obtain that is bounded. Let be a bounded sequence, and converges to 0 uniformly on compact subsets of . Let . By the assumptions, for any , there is a constant such that whenever . Taking (49), (67), (68), and Theorem 12 into account, it turns out that In addition, we set Note that is a compact subset of . For defined in (67), it leads to uniformly on as . Cauchy’s estimate gives that as on compact subsets, in particular on . Hence, as , by (49) we obtain According to the two inequalities (69) and (71), as , we have Consequently, making use of Lemma 11, we get is compact.
Conversely, suppose that is compact. Let be a sequence on such that , as . If the sequence is nonexistent, conditions (c) and (d) obviously hold. Moreover, let us introduce a test function sequence : The proof of Theorem 12 gives and . Due to (28), it gives that Taking , we have . This implies , as . Let be a compact subset of . For , it is easy to see that has a positive lower bound. Hence, we obtain uniformly on all compact subsets of , as .
Since is compact, according to Lemma 11, we have For the test function , we have where . Thus, and we have Let Since and (78), we obtain that where So we get if one of these two limits exists.
Next, let for a sequence in such that , as . Then, It is easy to obtain is a bounded sequence in and uniformly on every compact subset of . Moreover, we notice that and By the similar method as above, And by (82), All of the proofs are complete.

Remark 21. Let , , and ; we get the following results in the case of the unit ball . Let . If then, the weighted composition operator is compact. Conversely, the weighted composition operator is compact; then, where It turns out to be the same as the results obtained by Li and Stević in [7].

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11771184) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX20_2210).