Weighted Composition Operators between Bers-Type Spaces on Generalized Hua–Cartan–Hartogs Domains

Abstract

We address weighted composition operators between Bers-type spaces on generalized Hua–Cartan–Hartogs domains and provide the necessary and sufficient conditions for their boundedness and compactness. We then apply our results to study the boundedness and the compactness of weighted composition operators between Bers-type spaces on four different domains: generalized Hua domains, generalized Cartan–Hartogs domains, generalized Cartan–Hartogs domains over different Cartan domains and generalized ellipsoidal-type domains, by proving that the above four different domains are special cases of the generalized Hua–Cartan–Hartogs domains.

1. Introduction

Throughout this paper, we denote by $\mathrm{\Omega }$ a bounded domain of ${\mathbb{C}}^n$ and by $H(\mathrm{\Omega })$ the set of all holomorphic functions on $\mathrm{\Omega }$. We also denote by $\varphi$ a holomorphic map of $\mathrm{\Omega }$ on itself (self-map) and by $\psi$ a generic element of $H(\mathrm{\Omega })$. A weighted composition operator $\psi C_{\varphi }$ is defined as

$$\psi C_{\varphi }f(z)=\psi (z)(f\circ \varphi (z)),z\in \mathrm{\Omega }.$$

If $\psi \equiv 1$, $\psi C_{\varphi }$ reduces to the composition operator, usually denoted by $C_{\varphi }$. If $\varphi (z)=z$, we have the multiplication operator, usually denoted by $M_{\psi }$.

In recent years, there has been great interest in the study of (weighted) composition operators between the spaces of various domains. For example, on the unit disk, some properties of weighted composition operators between $H^{\infty }$ and Bloch space and between Bloch space and weighted Banach space have been extensively discussed by Allen in [1] and [2], respectively. Pu characterized the sufficient and necessary conditions of bounded and compact composition operators from Bloch space to Bers-type space and small Bers-type space [3]. Zhong, Wang and Liu [4,5] also discussed the sufficient and necessary conditions for the boundedness and compactness of a weighted composition operator between Bers-type spaces, obtaining the same results.

For the unit ball, Jin and Tang [6] investigated the sufficient and necessary conditions for the boundedness and compactness of a weighted composition operator between Bers-type spaces, which generalize the results obtained for the unit disk in [4,5]. Du and Li [7] studied the properties of weighted composition operators from $H^{\infty }$ space to Bloch space. In [8], Dai described the characteristics of Lipschitz space on the unit ball, and gave the necessary and sufficient conditions for the weighted composition operators on Lipschitz space to be bounded and compact. Zhou et al. studied the properties of weighted composition operators from Bers-type space to Bloch space [9].

Concerning the polydisc, Li and Zhang [10] discussed the equivalence of compactness conditions for composition operators between Bloch type spaces on the polydisc, and gave the simplest representation of the compactness conditions. The boundedness and compactness of composition operators of general weight Bloch spaces were studied by Hu in [11]. Stević investigated the boundedness and compactness of composition operators between special weight Bloch space and $H^{\infty }$ space in [12]. Later, in [13,14], together with Li, the conclusions were extended to the case of weighted composition operators.

In another realm of different framework spaces, mathematicians have studied some other properties of various operators. For example, ref. [15] studied the properties of the topological space of composition operators on the Banach algebra of bounded functions on an unbounded, locally finite metric space in the operator norm topology and essential norm topology. Guo [16] investigated the boundedness, essential norm and compactness of the generalized Stević–Sharma operator from the minimal Möbius invariant space into Bloch-type space. Mengestie studied certain modified or weighted composition–differentiation operators defined on Fock-type spaces and described conditions under which the operators admits closed-range, surjective, and order-bounded structures in [17]. Ref. [18] established several inequalities involving the Berezin number and the Berezin norm for various combinations of operators acting on a reproducing kernel Hilbert space.

In 1930, E. Cartan [19] fully characterized the irreducible bounded symmetric domains into six types: four types of Cartan domains and two exceptional domains of complex dimensions 16 and 27, respectively. The four types of Cartan domains are defined as follows:

$$\begin{array}{cc}\hfill {\Re }_{\mathrm{I}}(m,n)& :=\left\{Z\in {\mathbb{C}}^{m\times n}:I-Z{\overline{Z}}^{\prime }>0\right\},\hfill \\ \hfill {\Re }_{\mathrm{II}}(p)& :=\left\{Z\in {\mathbb{C}}^{p\times p}:I-Z\overline{Z}>0,Z=Z^{\prime }\right\},\hfill \\ \hfill {\Re }_{\mathrm{III}}(q)& :=\left\{Z\in {\mathbb{C}}^{q\times q}:I+Z\overline{Z}>0,Z=-Z^{\prime }\right\},\hfill \\ \hfill {\Re }_{\mathrm{IV}}(N)& :=\left\{z\in {\mathbb{C}}^N:1+|z{z}^{\prime }{|}^2-2z{\overline{z}}^{\prime }>0,1-{|z{z}^{\prime }|}^2>0\right\},\hfill \end{array}$$

where $m,n,p,q$ and N are positive integers, $Z\in {\mathbb{C}}^{m\times n}$ means that Z is a $m\times n$ complex matrix, $\overline{Z}$ denotes the conjugate of Z and $Z^{\prime }$ denotes the transpose of Z.

For the sake of convenience, the four Cartan domains will be denoted by the shorthands ${\Re }_{\mathrm{I}},{\Re }_{\mathrm{II}},{\Re }_{\mathrm{III}}$ and ${\Re }_{\mathrm{IV}}$, respectively, throughout the paper. Su revisited the classical extremal problem on Cartan domains in [20]. Wang and Liu studied the Bloch constant on a Cartan domain of the first kind in [21].

In 1998, building on the notion of Cartan domains, Yin was inspired by Roos to construct a new type of domain called the Cartan–Hartogs domain:

$$\begin{array}{cc}\hfill {\mathrm{Y}}_{\mathrm{I}}(N;m,n;k):& =\left\{\xi \in {\mathbb{C}}^N,Z\in {\Re }_{\mathrm{I}}(m,n):{|\xi |}^{2k}<det(I-Z{\overline{Z}}^{\prime })\right\},\hfill \\ \hfill {\mathrm{Y}}_{\mathrm{II}}(N;p;k):& =\left\{\xi \in {\mathbb{C}}^N,Z\in {\Re }_{\mathrm{II}}(p):{|\xi |}^{2k}<det(I-Z{\overline{Z}}^{\prime })\right\},\hfill \\ \hfill {\mathrm{Y}}_{\mathrm{III}}(N;q;k):& =\left\{\xi \in {\mathbb{C}}^N,Z\in {\Re }_{\mathrm{III}}(q):{|\xi |}^{2k}<det(I-Z{\overline{Z}}^{\prime })\right\},\hfill \\ \hfill {\mathrm{Y}}_{\mathrm{IV}}(N;n;k):& =\left\{\xi \in {\mathbb{C}}^N,z\in {\Re }_{\mathrm{IV}}{(n):|\xi |}^{2k}<(1+|z{z}^{\prime }{|}^2-{2|z|}^2)\right\},\hfill \end{array}$$

where ${\Re }_{\mathrm{I}}(m,n),{\Re }_{\mathrm{II}}(p),{\Re }_{\mathrm{III}}(q)$ and ${\Re }_{\mathrm{IV}}(n)$ denote, respectively, the Cartan domains of the first type, second type, third type and fourth type; $\overline{Z}$ denotes the conjugate of Z and $Z^{\prime }$ denotes the transpose of Z; $N,m,n,p$ and q are positive integers; and k is a positive real number.

In this framework, Bai [22] discussed the boundedness and compactness of weighted composition operators between Bers-type spaces on a Cartan–Hartogs domain of the first type, and obtained necessary and sufficient conditions. In [23], Su and Zhang studied the boundedness and compactness of weighted composition operators from $H^{\infty }$ to the special weight Bloch space on a Cartan–Hartogs domain of the first type. The boundedness and compactness of the composition operators between special weight Bloch spaces on a Cartan–Hartogs domain of the fourth type were studied by Su and Zhang in [24].

For $m=k=1$, the Cartan–Hartogs domain of the first type ${\mathrm{Y}}_{\mathrm{I}}$ reduces to the unit ball.

Yin then constructed four types of Cartan–Egg domains [25], later extending the Cartan–Egg domains to Hua domains [26]:

$$\begin{array}{cc}\hfill & {\mathrm{HE}}_{\mathrm{I}}(n_1,n_2,\cdots ,n_r;m,n;p_1,p_2,\cdots ,p_r)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{I}}(m,n):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det(I-Z{\overline{Z}}^{\prime }),j=1,2,\cdots ,r\right\},\hfill \\ \hfill & {\mathrm{HE}}_{\mathrm{II}}(n_1,n_2,\cdots ,n_r;p;p_1,p_2,\cdots ,p_r)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{II}}(p):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det(I-Z\overline{Z}),j=1,2,\cdots ,r\right\},\hfill \\ \hfill & {\mathrm{HE}}_{\mathrm{III}}(n_1,n_2,\cdots ,n_r;q;p_1,p_2,\cdots ,p_r)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{III}}(q):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det(I+Z\overline{Z}),j=1,2,\cdots ,r\right\},\hfill \\ \hfill & {\mathrm{HE}}_{\mathrm{IV}}(n_1,n_2,\cdots ,n_r;n;p_1,p_2,\cdots ,p_r)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},z\in {\Re }_{\mathrm{IV}}(n):\sum _{j=1}^r|{\xi }_j{|}^{2p_j}<(1+|{zz}^{\prime }{|}^2-2z{\overline{z}}^{\prime }),j=1,2,\cdots ,r\right\},\hfill \end{array}$$

where ${\xi }_j=({\xi }_{j1},\cdots ,{\xi }_{j{n}_j})$, $|{\xi }_j{|}^2={\sum }_{i=1}^{n_j}{|{\xi }_{ji}|}^2(j=1,2,\cdots ,r)$. $n_1,\cdots ,n_r,m,n,p$ and q are positive integers, and $p_1,\cdots ,p_r$ are positive real numbers.

The explicit formula of the Bergman kernel function on the four types of Hua domains have been obtained in [26]. Li, Su and Wang [27] discussed an extremal problem on Hua domains of the second type. In [28,29,30], Liu et al. studied the convexity of Hua domains of the first, second and third type, respectively, and computed the Carathéodory metric and Kobayashi metric on these three domains. Su et al. investigated the boundedness and compactness of composition operators between u-Bloch space and v-Bloch space on Hua domains of the first type [31]. The necessary and sufficient conditions for the boundedness and compactness of weighted composition operators between Bers-type spaces on four types of Hua domains are characterized by Jiang and Li in [32].

In 2003, Yin once again extended the Hua domains to the generalized Hua domains [33]:

$$\begin{array}{cc}\hfill & {\mathrm{GHE}}_{\mathrm{I}}(n_1,n_2,\cdots ,n_r;m,n;p_1,p_2,\cdots ,p_r;k)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{I}}(m,n):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det{(I-Z{\overline{Z}}^{\prime })}^k,j=1,2,\cdots ,r\right\},\hfill \\ \hfill & {\mathrm{GHE}}_{\mathrm{II}}(n_1,n_2,\cdots ,n_r;p;p_1,p_2,\cdots ,p_r;k)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{II}}(p):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det{(I-Z\overline{Z})}^k,j=1,2,\cdots ,r\right\},\hfill \\ \hfill & {\mathrm{GHE}}_{\mathrm{III}}(n_1,n_2,\cdots ,n_r;q;p_1,p_2,\cdots ,p_r;k)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{III}}(q):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det{(I+Z\overline{Z})}^k,j=1,2,\cdots ,r\right\},\hfill \\ \hfill & {\mathrm{GHE}}_{\mathrm{IV}}(n_1,n_2,\cdots ,n_r;n;p_1,p_2,\cdots ,p_r;k)\hfill \\ \hfill & =\left\{{\xi }_j\in {\mathbb{C}}^{n_j},z\in {\Re }_{\mathrm{IV}}(n):\sum _{j=1}^r|{\xi }_j{|}^{2p_j}<(1+|{zz}^{\prime }{{|}^2-2z{\overline{z}}^{\prime })}^k,j=1,2,\cdots ,r\right\},\hfill \end{array}$$

where ${\xi }_j=({\xi }_{j1},\cdots ,{\xi }_{j{n}_j})$; $|{\xi }_j{|}^2={\sum }_{i=1}^{n_j}{|{\xi }_{ji}|}^2(j=1,2,\cdots ,r)$. $n_1,\cdots ,n_r,m,n,p$ and q are positive integers, and $p_1,\cdots ,p_r,k$ are positive real numbers. When $k=1$, the generalized Hua domains are the Hua domains.

The explicit formula of the Bergman kernel function on the four types of generalized Hua domains has been obtained in [33]. Su and Wang studied the boundedness and compactness of operators $\psi C_{\varphi }:{\mathcal{B}}^{\alpha }\to {\mathcal{A}}_{\beta }$ between Bloch space and Bers space on generalized Hua domains of the first type, and obtained some sufficient conditions and necessary conditions [34].

In 2005, Yin extended the generalized Hua domains and proposed a type of domains named Hua Constructions [35]. Both Hua domains and generalized Hua domains are special cases of Hua Construction. Cartan–Hartogs domains, Cartan–Egg domains, Hua domains, generalized Hua domains and Hua Constructions are collectively referred to as Hua domains, see [36]. The Hua domains are generally not transitive except for the unit ball. It is thus very meaningful to study the problem of Hua domains.

Ahn-Park [37] introduced the generalized Cartan–Hartogs domain:

$$\begin{array}{c}\hfill {\widehat{\mathrm{\Omega }}}_m=\left\{\xi \in {\mathbb{C}}^m,Z_k\in {\mathrm{\Omega }}_k:{|\xi |}^2<{N_{{\mathrm{\Omega }}_1}(Z_1,Z_1)}^{{\mu }_1}\cdots {N_{{\mathrm{\Omega }}_t}(Z_t,Z_t)}^{{\mu }_t},{\mu }_k>0\right\}.\end{array}$$

where ${\mathrm{\Omega }}_k$ is one of the six bounded symmetric domains, and $N_{{\mathrm{\Omega }}_k}(Z_k,Z_k)$ is the corresponding generic norm of ${\mathrm{\Omega }}_k$. $k=1,\cdots ,t.$m is a positive integer. This type of domain generalizes the Cartan–Hartogs domain introduced by Yin and Roos.

Wang et al. proved the vanishing theorem on generalized Cartan–Hartogs domains of the second type in [38,39], and ref. [40] discussed the boundedness and compactness of composition operators between weighted Bloch spaces on generalized Cartan–Hartogs domain of the first type. Considerable attention has been devoted to Rawnsley’s $\epsilon$-functions and to the comparison theorem for the Einstein–Kahler and Kobayashi metrics on generalized Cartan–Hartogs domains; see, e.g., [41,42]. Other conclusions on the generalized Cartan–Hartogs domains can be found in [43,44,45]. On the other hand, the properties of weighted composition operators between Bers-type spaces on generalized Cartan–Hartogs domains have not been studied.

Starting from these results, we introduce a novel type of domain, which we term the generalized Hua–Cartan–Hartogs domain:

$$\begin{array}{cc}\hfill & \mathcal{H}(n_1,\cdots ,n_r;p_1,\cdots ,p_r;A_1,\cdots ,A_t)\hfill \\ & =\{{\xi }_j\in {\mathbb{C}}^{n_j},Z_k\in {\Re }_{{\mathrm{A}}_k}:\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t}),j=1,\cdots ,r;k=1,\cdots ,t.\},\hfill \end{array}$$

where $n_1,\cdots ,n_r,t$ and r are positive integers, $p_1,\cdots ,p_r$ are positive real numbers and ${\xi }_j=({\xi }_{j_1},{\xi }_{j_2},\cdots ,{\xi }_{j_{n_j}})$, ${\Re }_{{\mathrm{A}}_1},{\Re }_{{\mathrm{A}}_2},\cdots ,{\Re }_{{\mathrm{A}}_t}\in \{{\Re }_{\mathrm{I}},{\Re }_{\mathrm{II}},{\Re }_{\mathrm{III}},{\Re }_{\mathrm{IV}}\}$. The generic norm $N_k(Z_k,\overline{W_k})$ is holomorphic for $Z_k$ and anti-holomorphic for $W_k$, where $Z_k,W_k\in {\Re }_{{\mathrm{A}}_k},k=1,\cdots ,t$. $N_k(Z_k,\overline{W_k})$ should meet the following two conditions:

$$\begin{array}{cc}\hfill (1)& 0<N_k(Z_k,\overline{Z_k})\le 1,k=1,\cdots ,t.\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill (2)& 2|N_k(Z_k,\overline{W_k})|\ge N_k(Z_k,\overline{Z_k})+N_k(W_k,\overline{W_k}),Z_k,W_k\in {\Re }_{{\mathrm{A}}_k},k=1,\cdots ,t.\hfill \end{array}$$

(2)

We use the shorthand $\mathcal{H}$ for the generalized Hua–Cartan–Hartogs domain and denote the points of $\mathcal{H}$ by $(Z_1,Z_2,\cdots ,Z_t,{\xi }_1,{\xi }_2,\cdots ,{\xi }_r)=(Z,\xi )$, where $(Z_1,Z_2,\cdots ,Z_t)=Z$, $({\xi }_1,{\xi }_2,\cdots ,{\xi }_r)=\xi$.

In the fourth section of this paper, we will further prove that generalized Hua domains, generalized Cartan–Hartogs domains, generalized Cartan–Hartogs domains with different types of Cartan domains as bases and generalized ellipsoidal-type domains are special generalized Hua–Cartan–Hartogs domains.

A Bers-type space on $\mathcal{H}$ is defined as follows:

Definition 1.

Let $\alpha >0$. A Bers-type space on $\mathcal{H}$, denoted by ${\mathcal{A}}_{\alpha }(\mathcal{H})$, consists of all holomorphic functions on $\mathcal{H}$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}=\underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-\sum _{j=1}^r{|{\xi }_j|}^{2p_j}]\alpha |f(Z,\xi )|<+\infty .$$

It is easy to see that ${\mathcal{A}}_{\alpha }(\mathcal{H})$ is a Banach space.

Since the unit ball is a special case of the generalized Hua domain, it is also a special case of the generalized Hua–Cartan–Hartogs domain. In this case, the Bers-type space defined on the generalized Hua–Cartan–Hartogs domain is consistent with the definition on the unit ball.

In this paper, we study the boundedness and compactness of a weighted composition operator between Bers-type spaces on the generalized Hua–Cartan–Hartogs domain, and obtain necessary and sufficient conditions, which are relevant generalizations of some previous conclusions.

In the fourth section of this paper, by proving that the following four domains are special cases of generalized Hua–Cartan–Hartogs domains, we will prove sufficient and necessary conditions for the boundedness and compactness of weighted composition operators between Bers-type spaces on four domains: generalized Hua domains, generalized Cartan–Hartogs domains, generalized Cartan–Hartogs domains with different types of Cartan domains as bases and generalized ellipsoidal-type domains. Here, we briefly anticipate them:

$(1)$ For $t=1$, if $Z\in {\Re }_{\mathrm{A}},\mathrm{A}=\mathrm{I},\mathrm{II}$, then let $N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=det{(I-Z{\overline{Z}}^{\prime })}^k$; if $z\in {\Re }_{\mathrm{IV}}$, then let $N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k,$ where k is positive real number. In this case, the generalized Hua–Cartan–Hartogs domains are equivalent to generalized Hua domains.

$(2)$ For $t=2$, if $Z_k\in {\Re }_{{\mathrm{A}}_{\mathrm{k}}},{\mathrm{A}}_{\mathrm{k}}=\mathrm{I},\mathrm{II}$, then let $N_k(Z_k,\overline{Z_k})=det{(I-Z_k{\overline{Z_k}}^{\prime })}^{s_k}$; if $z_k\in {\Re }_{\mathrm{IV}}$, then let $N_k(Z_k,\overline{Z_k})=(1+|z_k{z}_k^{\prime }{|}^2-2|z_k{{|}^2)}^{s_k}$, where $s_k$ are positive real numbers, $k=1,2.$ In this case, the generalized Hua–Cartan–Hartogs domains are generalized Cartan–Hartogs domains. The boundedness and compactness of a weighted composition operator between Bers-type spaces on generalized Cartan–Hartogs domains will be discussed by examples from generalized Cartan–Hartogs domains and generalized Cartan–Hartogs domains over different types of Cartan domains, respectively.

$(3)$ For $t=1$, if $Z\in {\Re }_{\mathrm{I}}(m,n)$, then let $N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=1-{\displaystyle \frac{{|Z|}^2}{m}}$; if $Z\in {\Re }_{\mathrm{II}}(p)$, then let $N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=1-{\displaystyle \frac{{|Z|}^2}{p}}$; if $Z\in {\Re }_{\mathrm{III}}(q)$, then let $N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=1-{\displaystyle \frac{{|Z|}^2}{[\frac{q}{2}]}}$, where $[{\displaystyle \frac{q}{2}}]$ denotes the integer part of ${\displaystyle \frac{q}{2}}$ and if $z\in {\Re }_{\mathrm{IV}}(N)$, let $N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=1-{|z|}^2$. In this case, we refer to this generalized Hua–Cartan–Hartogs domains as generalized ellipsoidal-type domains.

Constants will be denoted by $C,C_1,C_2,\cdots$; they are positive and may differ in the different cases. Without loss of generality, we assume that $n_j=1$, that is, ${\xi }_j\in \mathbb{C},j=1,2,\cdots ,r$, $\xi =({\xi }_1,{\xi }_2,\cdots ,{\xi }_r)$ and ${\parallel \xi \parallel }^2={\sum }_{j=1}^r{|{\xi }_j|}^{2p_j}$.

2. Preliminaries

This section is devoted to presenting and proving a few lemmas that will be used in the following theorems about the boundedness and compactness of weighted composition operators $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$.

Lemma 1

(see [46]). Given the sequence $a_k,\in \mathbb{C}$, $k=1,2,\cdots ,n$, then when $p\ge 1$

$$\sum _{k=1}^n|a_k{|}^p\le {\left[\sum _{k=1}^n|a_k|\right]}^p\le n^{p-1}\sum _{k=1}^n{|a_k|}^p.$$

(3)

and when $0<p<1$

$$\sum _{k=1}^n|a_k{|}^p\ge {\left[\sum _{k=1}^n|a_k|\right]}^p\ge n^{p-1}\sum _{k=1}^n{|a_k|}^p.$$

(4)

Lemma 2

(see [46]). (The product-type Minkowski inequality.) Let $a_k,b_k\ge 0$, $k=1,2,\cdots ,n$; then,

$${\left[\prod _{k=1}^n(a_k+b_k)\right]}^{{\displaystyle \frac{1}{n}}}\ge {\left(\prod _{k=1}^n{a}_k\right)}^{{\displaystyle \frac{1}{n}}}+{\left(\prod _{k=1}^n{b}_k\right)}^{{\displaystyle \frac{1}{n}}}.$$

(5)

with an equality that holds iff $a_k=C{b}_k,k=1,2,\cdots ,n.$

Lemma 3

(see [47]). Given the $m\times n$ matrix $(m\le n)$

$$Z=\left(\begin{array}{cccc}{z}_{11}& z_{12}& \dots & z_{1n}\\ z_{21}& z_{22}& \dots & z_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ z_{m1}& z_{m2}& \dots & z_{mn}\end{array}\right),$$

then there exists an $m\times m$ unitary matrix U and an $n\times n$ unitary matrix V such that

$$Z=U\left(\begin{array}{ccccccc}{\lambda }_1& 0& \dots & 0& 0& \dots & 0\\ 0& {\lambda }_2& \dots & 0& 0& \dots & 0\\ ⋮& ⋮& & ⋮& ⋮& & ⋮\\ 0& 0& \dots & {\lambda }_m& 0& \dots & 0\end{array}\right)V({\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_m\ge 0).$$

Lemma 4

(see [47]). Given two diagonal $m\times m$ matrices ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2$,

$${\mathrm{\Lambda }}_1=\left(\begin{array}{cccc}{\lambda }_1& 0& \dots & 0\\ 0& {\lambda }_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\lambda }_m\end{array}\right)({\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_m\ge 0)$$

and

$${\mathrm{\Lambda }}_2=\left(\begin{array}{cccc}{\mu }_1& 0& \dots & 0\\ 0& {\mu }_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\mu }_m\end{array}\right)({\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_m\ge 0)$$

satisfying

$${\lambda }_j{\mu }_k<1(j,k=1,\cdots ,m),$$

then there exists a permutation matrix P such that

$$\underset{U{\overline{U}}^{\prime }=I,V{\overline{V}}^{\prime }=I}{inf}|det(I-{\mathrm{\Lambda }}_{1}U{\mathrm{\Lambda }}_2{\overline{U}}^{\prime }V)|=|det(I-{\mathrm{\Lambda }}_{1}P{\mathrm{\Lambda }}_2{P}^{\prime })|.$$

The infimum is achieved for $U=\mathrm{\Theta }P,V=I$, where

$$\mathrm{\Theta }=\left(\begin{array}{cccc}{e}^{i{\theta }_1}& 0& \dots & 0\\ 0& e^{i{\theta }_2}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & e^{i{\theta }_m}\end{array}\right).$$

Lemma 5.

For any $\alpha >0$, $(Z,\xi )\in \mathcal{H}$ and $f\in {\mathcal{A}}_{\alpha }(\mathcal{H})$, one has

$$|f(Z,\xi )|\le {\displaystyle \frac{{\parallel f\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}}{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\alpha }}}.$$

Proof. 

This is easily obtained by the definition of Bers-type space. □

Lemma 6.

Given $\alpha ,\beta >0$, the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$ is compact if and only if $\psi C_{\varphi }$ is bounded in ${\mathcal{A}}_{\beta }(\mathcal{H})$ and for any bounded sequence ${\{f_k\}}_{k\ge 1}$ in ${\mathcal{A}}_{\alpha }(\mathcal{H})$ converging to 0 uniformly on every compact subset of $\mathcal{H}$ one has that

$$\underset{k\to \infty }{lim}{\parallel \psi C_{\varphi }{f}_k\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}=0.$$

Proof. 

This is easily proved by the usual methods, such as the proof of Lemma 12 in [34]. □

Lemma 7.

(Hua-type inequality). If $Z,S\in {\Re }_{{\mathrm{A}}_1}\times {\Re }_{{\mathrm{A}}_2}\times \cdots \times {\Re }_{{\mathrm{A}}_t}$, then

$$N_k(Z_k,\overline{Z_k})·N_k(S_k,\overline{S_k})\le {|N_k(Z_k,\overline{S_k})|}^2,k=1,2,\cdots ,t.$$

(6)

Proof. 

If $Z,S\in {\Re }_{{\mathrm{A}}_1}\times {\Re }_{{\mathrm{A}}_2}\times \cdots \times {\Re }_{{\mathrm{A}}_t}$, combine (2) and $a^2+b^2\ge 2ab$; we have

$$\begin{array}{cc}\hfill 2|N_k(Z_k,\overline{S_k})|& \ge N_k(Z_k,\overline{Z_k})+N_k(S_k,\overline{S_k})\hfill \\ \hfill & \ge 2{[N_k(Z_k,\overline{Z_k})]}^{{\displaystyle \frac{1}{2}}}·{[N_k(S_k,\overline{S_k})]}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Then, we have

$$N_k(Z_k,\overline{Z_k})·N_k(S_k,\overline{S_k})\le {|N_k(Z_k,\overline{S_k})|}^2,k=1,2,\cdots ,t.$$

and the lemma is proved. □

Lemma 8.

If $Z,S\in {\Re }_{{\mathrm{A}}_1}\times {\Re }_{{\mathrm{A}}_2}\times \cdots \times {\Re }_{{\mathrm{A}}_t}$, then

$$N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})·N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})\le {|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t})|}^2.$$

(7)

Proof. 

We can calculate (7) directly from (6). □

Lemma 9.

If $(Z,\xi ),(S,\varsigma )\in \mathcal{H}$, then

$$|\sum _{j=1}^r{\xi }_j^{p_j}{\overline{\varsigma }}_j^{p_j}|\le \parallel \xi \parallel \parallel \varsigma \parallel$$

(8)

and

$$\parallel \xi \parallel \parallel \varsigma \parallel <|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t})|.$$

Proof. 

$$\begin{array}{cc}\hfill |\sum _{j=1}^r{\xi }_j^{p_j}{\overline{\varsigma }}_j^{p_j}|\le \sum _{j=1}^r|{\xi }_j{|}^{p_j}·{|{\overline{\varsigma }}_j|}^{p_j}& \le {(\sum _{j=1}^r|{\xi }_j{|}^{2p_j}·\sum _{j=1}^r{|{\overline{\varsigma }}_j|}^{2p_j})}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & ={(\parallel \xi \parallel }^2{\parallel \varsigma \parallel }^2{)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & =\parallel \xi \parallel \parallel \varsigma \parallel .\hfill \end{array}$$

For $(Z,\xi ),(S,\varsigma )\in \mathcal{H}$, we have

$${\parallel \xi \parallel }^2<N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t}),$$

$${\parallel \varsigma \parallel }^2<N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t}).$$

From (7), it follows that

$$\begin{array}{cc}\hfill \parallel \xi \parallel \parallel \varsigma \parallel & <{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})·N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le |N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t})|,\hfill \end{array}$$

and the lemma is proved. □

Lemma 10.

If $(Z,\xi ),(S,\varsigma )\in \mathcal{H}$, then

$$\begin{array}{cc}\hfill & [N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2][N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2]\hfill \\ \hfill & \le [|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t}){|-\parallel \xi \parallel \parallel \varsigma \parallel ]}^2.\hfill \end{array}$$

(9)

Proof. 

Assume that $a,b,c$ and d are non-negative real numbers with $b\le a,d\le c$; then, we have $(a^2-b^2)(c^2-d^2)\le {(ac-bd)}^2$. From this inequality and (7), we obtain

$$\begin{array}{cc}\hfill & [N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2][N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2]\hfill \\ \hfill & \le {\{{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})]}^{{\displaystyle \frac{1}{2}}}{[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})]}^{{\displaystyle \frac{1}{2}}}-\parallel \xi \parallel \parallel \varsigma \parallel \}}^2\hfill \\ \hfill & \le [|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t}){|-\parallel \xi \parallel \parallel \varsigma \parallel ]}^2,\hfill \end{array}$$

and the lemma is proved. □

Lemma 11.

Let $\alpha >0$; if $(S,\varsigma )\in \mathcal{H}$, then $f_{(S,\varsigma )}(Z,\xi )\in {\mathcal{A}}_{\alpha }(\mathcal{H})$, where the function $f_{(S,\varsigma )}(Z,\xi )$ is defined as

$$f_{(S,\varsigma )}(Z,\xi )={\displaystyle \frac{[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\alpha }}{{[N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t})-{\sum }_{j=1}^r{\xi }_j^{p_j}{\overline{\varsigma }}_j^{p_j}]}^{2\alpha }}}$$

with $\parallel f_{(S,\varsigma )}{\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}\le 1$.

Proof. 

From (8) and (9), we have

$$\begin{array}{cc}\hfill & [N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\alpha }|f_{(S,\varsigma )}(Z,\xi )|\hfill \\ \hfill & =[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\alpha }{\displaystyle \frac{[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\alpha }}{|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t})-{\sum }_{j=1}^r{\xi }_j^{p_j}{\overline{\varsigma }}_j^{p_j}{|}^{2\alpha }}}\hfill \\ \hfill & \le [N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\alpha }{\displaystyle \frac{[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\alpha }}{[|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t})|-|{\sum }_{j=1}^r{\xi }_j^{p_j}{\overline{\varsigma }}_j^{p_j}{|]}^{2\alpha }}}\hfill \\ \hfill & \le [N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\alpha }{\displaystyle \frac{[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\alpha }}{[|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t}){|-\parallel \xi \parallel \parallel \varsigma \parallel ]}^{2\alpha }}}\hfill \\ \hfill & \le 1,\hfill \end{array}$$

which shows that

$$\parallel f_{(S,\varsigma )}{\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}=\underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\alpha }|f_{(S,\varsigma )}(Z,\xi )|\le 1,$$

and $f_{(S,\varsigma )}\in {\mathcal{A}}_{\alpha }(\mathcal{H})$. The lemma is thus proved. □

Lemma 12.

If $Z,S\in {\Re }_{{\mathrm{A}}_1}\times {\Re }_{{\mathrm{A}}_2}\times \cdots \times {\Re }_{{\mathrm{A}}_t}$, then

$$2^t|N_1(Z_1,\overline{S_1})\cdots N_t(Z_t,\overline{S_t})|\ge N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})+N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t}).$$

(10)

Proof. 

We can calculate (10) directly from (2). □

Lemma 13.

If $(Z,\xi ),(S,\varsigma )\in \mathcal{H}$, then

$$2^t|\prod _{k=1}^t{N}_k(Z_k,\overline{S_k})-\parallel \xi \parallel \parallel \varsigma \parallel |\ge \prod _{k=1}^t{N}_k(Z_k,\overline{Z_k})-{\parallel \xi \parallel }^2+\prod _{k=1}^t{N}_k(S_k,\overline{S_k})-{\parallel \varsigma \parallel }^2.$$

(11)

Proof. 

Since $(Z,\xi ),(S,\varsigma )\in \mathcal{H}$, we have

$${\parallel \xi \parallel }^2<\prod _{k=1}^t{N}_k(Z_k,\overline{Z_k}),$$

$${\parallel \varsigma \parallel }^2<\prod _{k=1}^t{N}_k(S_k,\overline{S_k}).$$

Given a permutation $i_1,i_2,\cdots ,i_t$ of $1,2,\cdots ,t$, we write

$$M_{i_1,\cdots ,i_k;i_{k+1},\cdots ,i_t}=\prod _{l=1}^k{N}_{i_l}(Z_{i_l},\overline{Z_{i_l}})\prod _{l=k+1}^t{N}_{i_l}(S_{i_l},\overline{S_{i_l}}),k=1,2,\cdots ,t-1.$$

Then

$$\begin{array}{cc}\hfill & M_{i_1,\cdots ,i_k;i_{k+1},\cdots ,i_t}+M_{i_{k+1},\cdots ,i_t;i_1,\cdots ,i_k}-2\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & =\prod _{l=1}^k{N}_{i_l}(Z_{i_l},\overline{Z_{i_l}})\prod _{l=k+1}^t{N}_{i_l}(S_{i_l},\overline{S_{i_l}})+\prod _{l=1}^k{N}_{i_l}(S_{i_l},\overline{S_{i_l}})\prod _{l=k+1}^t{N}_{i_l}(Z_{i_l},\overline{Z_{i_l}})-2\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & \ge 2\prod _{l=1}^k{N}_{i_l}{(Z_{i_l},\overline{Z_{i_l}})}^{{\displaystyle \frac{1}{2}}}\prod _{l=k+1}^t{N}_{i_l}{(S_{i_l},\overline{S_{i_l}})}^{{\displaystyle \frac{1}{2}}}\prod _{l=1}^k{N}_{i_l}{(S_{i_l},\overline{S_{i_l}})}^{{\displaystyle \frac{1}{2}}}\prod _{l=k+1}^t{N}_{i_l}{(Z_{i_l},\overline{Z_{i_l}})}^{{\displaystyle \frac{1}{2}}}-2\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & =2\prod _{l=1}^t{N}_{i_l}{(Z_{i_l},\overline{Z_{i_l}})}^{{\displaystyle \frac{1}{2}}}\prod _{l=1}^t{N}_{i_l}{(S_{i_l},\overline{S_{i_l}})}^{{\displaystyle \frac{1}{2}}}-2\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & >2\parallel \xi \parallel \parallel \varsigma \parallel -2\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & =0.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill & 2^t|\prod _{k=1}^t{N}_k(Z_k,\overline{S_k})-\parallel \xi \parallel \parallel \varsigma \parallel |\hfill \\ \hfill & \ge 2^t|\prod _{k=1}^t{N}_k(Z_k,\overline{S_k})|-2^t\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & \ge \prod _{k=1}^t[N_k(Z_k,\overline{Z_k})+N_k(S_k,\overline{S_k})]-2\parallel \xi \parallel \parallel \varsigma \parallel -(2^t-2)\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & =\prod _{k=1}^t{N}_k(Z_k,\overline{Z_k})+\prod _{k=1}^t{N}_k(S_k,\overline{S_k})-2\parallel \xi \parallel \parallel \varsigma \parallel +\mathcal{M}-(2^t-2)\parallel \xi \parallel \parallel \varsigma \parallel \hfill \\ \hfill & \ge \prod _{k=1}^t{N}_k(Z_k,\overline{Z_k})+\prod _{k=1}^t{N}_k(S_k,\overline{S_k})-{\parallel \xi \parallel }^2-{\parallel \varsigma \parallel }^2\hfill \\ \hfill & =\prod _{k=1}^t{N}_k(Z_k,\overline{Z_k})-{\parallel \xi \parallel }^2+\prod _{k=1}^t{N}_k(S_k,\overline{S_k})-{\parallel \varsigma \parallel }^2,\hfill \end{array}$$

where $\mathcal{M}$ denotes the sum of all terms in ${\prod }_{k=1}^t[N_k(Z_k,\overline{Z_k})+N_k(S_k,\overline{S_k})]$ except ${\prod }_{k=1}^t{N}_k(Z_k,\overline{Z_k})$ and ${\prod }_{k=1}^t{N}_k(S_k,\overline{S_k})$, and $\mathcal{M}-(2^t-2)\parallel \xi \parallel \parallel \varsigma \parallel \ge 0$. The lemma is thus proved. □

3. Main Results

In this Section, we present few characterization theorems about the boundedness and compactness of weighted composition operators $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$.

Theorem 1.

Consider two positive numbers $\alpha ,\beta >0$, a holomorphic self-map ϕ of $\mathcal{H}$ and a function $\psi \in H(\mathcal{H})$. Then, the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$ is bounded if and only if

$$\tilde{M}=\underset{(Z,\xi )\in \mathcal{H}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty ,$$

where $(W,\eta )=\varphi (Z,\xi )$.

Proof. 

Assume that $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$ is bounded. Then, for each $f\in {\mathcal{A}}_{\alpha }(\mathcal{H})$, there exists a positive constant C such that $\parallel \psi C_{\varphi }{f\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}\le C{\parallel f\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}$. For the fixed point $(S,\varsigma )\in \mathcal{H}$, consider the function

$$f_{(S,\varsigma )}(Z,\xi )={\displaystyle \frac{[N_1(A_1,\overline{A_1})\cdots N_t(A_t,\overline{A_t})-{\parallel \zeta \parallel }^2{]}^{\alpha }}{{[N_1(Z_1,\overline{A_1})\cdots N_t(Z_t,\overline{A_t})-{\sum }_{j=1}^r{\xi }_j^{p_j}{\overline{\zeta }}_j^{p_j}]}^{2\alpha }}},(Z,\xi )\in \mathcal{H}$$

where $(A,\zeta )=\varphi (S,\varsigma )$. From Lemma 11, it follows that $f_{(S,\varsigma )}\in {\mathcal{A}}_{\alpha }(\mathcal{H})$ and $\parallel f_{(S,\varsigma )}{\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}\le 1$. By direct computation, we have

$$\begin{array}{cc}\hfill & \parallel \psi C_{\varphi }{f}_{(S,\varsigma )}{\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}\hfill \\ \hfill & =\underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_{(S,\varsigma )}(Z,\xi )|\hfill \\ \hfill & \ge [N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_{(S,\varsigma )}(S,\varsigma )|\hfill \\ \hfill & =[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\beta }|\psi (S,\varsigma )f_{(S,\varsigma )}(A,\zeta )|\hfill \\ \hfill & =[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\beta }|\psi (S,\varsigma )|{\displaystyle \frac{[N_1(A_1,\overline{A_1})\cdots N_t(A_t,\overline{A_t})-{\parallel \zeta \parallel }^2{]}^{\alpha }}{{[N_1(A_1,\overline{A_1})\cdots N_t(A_t,\overline{A_t})-{\sum }_{j=1}^r{\zeta }_j^{p_j}{\overline{\zeta }}_j^{p_j}]}^{2\alpha }}}\hfill \\ \hfill & =|\psi (S,\varsigma )|{\displaystyle \frac{[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\beta }}{[N_1(A_1,\overline{A_1})\cdots N_t(A_t,\overline{A_t})-{\parallel \zeta \parallel }^2{]}^{\alpha }}}.\hfill \end{array}$$

We also know that $\parallel \psi C_{\varphi }{f}_{(S,\varsigma )}{\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}\le C{\parallel f_{(S,\varsigma )}\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}\le C<+\infty$, which shows that

$$|\psi (S,\varsigma )|{\displaystyle \frac{[N_1(S_1,\overline{S_1})\cdots N_t(S_t,\overline{S_t})-{\parallel \varsigma \parallel }^2{]}^{\beta }}{[N_1(A_1,\overline{A_1})\cdots N_t(A_t,\overline{A_t})-{\parallel \zeta \parallel }^2{]}^{\alpha }}}\le C,(A,\zeta )=\varphi (S,\varsigma ),$$

that is,

$$\tilde{M}=\underset{(Z,\xi )\in \mathcal{H}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty ,(W,\eta )=\varphi (Z,\xi ).$$

Conversely, if

$$\tilde{M}=\underset{(Z,\xi )\in \mathcal{H}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty ,$$

then, for all $f\in {\mathcal{A}}_{\alpha }(\mathcal{H})$, we obtain from Lemma 5

$$\begin{array}{cc}\hfill & \parallel \psi C_{\varphi }{f\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}\hfill \\ \hfill & =\underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }f(Z,\xi )|\hfill \\ \hfill & =\underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi (Z,\xi )f(W,\eta )|\hfill \\ \hfill & \le \underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi (Z,\xi )|{\displaystyle \frac{{\parallel f\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}\hfill \\ \hfill & =\tilde{M}{\parallel f\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})},\hfill \end{array}$$

which implies that $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$ is bounded. This proves the desired result. □

Theorem 2.

Consider two positive numbers $\alpha ,\beta >0$, a holomorphic self-map ϕ of $\mathcal{H}$, and a function $\psi \in H(\mathcal{H})$. Then, the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$ is compact if and only if $\psi \in {\mathcal{A}}_{\beta }(\mathcal{H})$ and

$$\underset{\varphi (Z,\xi )\to \partial \mathcal{H}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}=0,$$

(12)

where $(W,\eta )=\varphi (Z,\xi )$.

Proof. 

Assume that the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$ is compact; then, $\psi C_{\varphi }$ is bounded. By taking $f\equiv 1$, we have

$$\begin{array}{cc}\hfill & [N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi (Z,\xi )|\hfill \\ & =[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }f(Z,\xi )|<+\infty .\hfill \end{array}$$

This shows that $\psi \in {\mathcal{A}}_{\beta }(\mathcal{H})$. Let us now consider a sequence $(S^i,{\varsigma }^i)(i=1,2,\cdots .)$ in $\mathcal{H}$ such that $\varphi (S^i,{\varsigma }^i)\to \partial \mathcal{H}$ as $i\to \infty$. If such a sequence does not exist, then (12) holds. If such a sequence exists, let $(A^i,{\zeta }^i)=\varphi (S^i,{\varsigma }^i),i=1,2,\cdots$, and define the sequence of functions $f_i(Z,\xi )=f_{(S^i,{\varsigma }^i)}(Z,\xi ),i=1,2,\cdots .$

$$f_i(Z,\xi )=f_{(S^i,{\varsigma }^i)}(Z,\xi )={\displaystyle \frac{[N_1(A_1^i,\overline{A_1^i})\cdots N_t(A_t^i,\overline{A_t^i})-\parallel {\zeta }^i{{\parallel }^2]}^{\alpha }}{{[N_1(Z_1,\overline{A_1^i})\cdots N_t(Z_t,\overline{A_t^i})-{\sum }_{j=1}^r{\xi }_j^{p_j}{\overline{{\zeta }_j^i}}^{p_j}]}^{2\alpha }}}.$$

From Lemma 11, it follows that $f_{(S^i,{\varsigma }^i)}\in {\mathcal{A}}_{\alpha }(\mathcal{H})$ and $\parallel f_{(S^i,{\varsigma }^i)}{\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}\le 1$, that is, $\{f_i\}$ is bounded in ${\mathcal{A}}_{\alpha }(\mathcal{H})$. Taking $i\to \infty$, it follows that $(A^i,{\zeta }^i)\to \partial \mathcal{H}$, and therefore $[N_1(A_1^i,\overline{A_1^i})\cdots N_t(A_t^i,\overline{A_t^i})-\parallel {\zeta }^i{\parallel }^2]\to 0$, as $i\to \infty$.

According to the two inequalities (8) and (11),

$$\begin{array}{cc}\hfill |N_1(Z_1,\overline{A_1^i})\cdots N_t(Z_t,\overline{A_t^i})-\sum _{j=1}^r{\xi }_j^{p_j}{\overline{{\zeta }_j^i}}^{p_j}|& \ge |N_1(Z_1,\overline{A_1^i})\cdots N_t(Z_t,\overline{A_t^i})|-|\sum _{j=1}^r{\xi }_j^{p_j}{\overline{{\zeta }_j^i}}^{p_j}|\hfill \\ \hfill & \ge |N_1(Z_1,\overline{A_1^i})\cdots N_t(Z_t,\overline{A_t^i})|-\parallel \xi \parallel \parallel {\zeta }^i\parallel \hfill \\ \hfill & \ge 2^{-t}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2]\hfill \\ \hfill & +2^{-t}[N_1(A_1^i,\overline{A_1^i})\cdots N_t(A_t^i,\overline{A_t^i})-\parallel {\zeta }^i{\parallel }^2]\hfill \\ \hfill & \ge 2^{-t}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2].\hfill \end{array}$$

$[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-\parallel \xi {\parallel }^2]$ has thus a positive lower bound in the compact subset. Hence, $f_i\rightrightarrows 0$ on every compact subset of $\mathcal{H}$ as $i\to \infty$. Then, by making use of Lemma 6,

$$\underset{i\to \infty }{lim}{\parallel \psi C_{\varphi }{f}_i\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}=0.$$

$$\begin{array}{cc}\hfill & \parallel \psi C_{\varphi }{f}_i{\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}\hfill \\ \hfill & =\underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_i(Z,\xi )|\hfill \\ \hfill & \ge [N_1(S_1^i,\overline{S_1^i})\cdots N_t(S_t^i,\overline{S_t^i})-\parallel {\varsigma }^i{\parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_i(S^i,{\varsigma }^i)|\hfill \\ \hfill & =[N_1(S_1^i,\overline{S_1^i})\cdots N_t(S_t^i,\overline{S_t^i})-\parallel {\varsigma }^i{\parallel }^2{]}^{\beta }|\psi (S^i,{\varsigma }^i)f_i(A^i,{\zeta }^i)|\hfill \\ \hfill & =[N_1(S_1^i,\overline{S_1^i})\cdots N_t(S_t^i,\overline{S_t^i})-\parallel {\varsigma }^i{\parallel }^2{]}^{\beta }|\psi (S^i,{\varsigma }^i)|{\displaystyle \frac{[N_1(A_1^i,\overline{A_1^i})\cdots N_t(A_t^i,\overline{A_t^i})-\parallel {\zeta }^i{{\parallel }^2]}^{\alpha }}{{[N_1(A_1^i,\overline{A_1^i})\cdots N_t(A_t^i,\overline{A_t^i})-{\sum }_{j=1}^r{{\zeta }_j^i}^{p_j}{\overline{{\zeta }_j^i}}^{p_j}]}^{2\alpha }}}\hfill \\ \hfill & =|\psi (S^i,{\varsigma }^i)|{\displaystyle \frac{[N_1(S_1^i,\overline{S_1^i})\cdots N_t(S_t^i,\overline{S_t^i})-\parallel {\varsigma }^i{{\parallel }^2]}^{\beta }}{[N_1(A_1^i,\overline{A_1^i})\cdots N_t(A_t^i,\overline{A_t^i})-\parallel {\zeta }^i{{\parallel }^2]}^{\alpha }}}.\hfill \end{array}$$

This leads to

$$\underset{i\to \infty }{lim}|\psi (S^i,{\varsigma }^i)|{\displaystyle \frac{[N_1(S_1^i,\overline{S_1^i})\cdots N_t(S_t^i,\overline{S_t^i})-\parallel {\varsigma }^i{{\parallel }^2]}^{\beta }}{[N_1(A_1^i,\overline{A_1^i})\cdots N_t(A_t^i,\overline{A_t^i})-\parallel {\zeta }^i{{\parallel }^2]}^{\alpha }}}=0,$$

where $(A^i,{\zeta }^i)=\varphi (S^i,{\varsigma }^i)$, and then

$$\underset{\varphi (Z,\xi )\to \partial \mathcal{H}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}=0,$$

where $(W,\eta )=\varphi (Z,\xi )$.

Conversely, suppose that (12) holds, then

$$\underset{(Z,\xi )\in \mathcal{H}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty ,$$

that is, $\psi C_{\varphi }$ is bounded. Let ${\{f_i\}}_{i\ge 1}$ be a bounded sequence of functions in ${\mathcal{A}}_{\alpha }(\mathcal{H})$ which converges to 0 uniformly on every compact subset of $\mathcal{H}$. Upon denoting by $\parallel f_i{\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}\le C_1,i=1,2,\cdots$, we have, by (12), that $\forall \epsilon >0$, $\exists \sigma >0$, such that $\forall (Z,\xi )\in E=\{(Z,\xi )\in \mathcal{H}:\mathrm{dist}(\varphi (Z,\xi ),\partial \mathcal{H})<\sigma \}$; we have

$$|\psi (Z,\xi )|{\displaystyle \frac{[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}<\epsilon .$$

(13)

By (13) and Lemma 5, we obtain

$$\begin{array}{cc}\hfill & \underset{(Z,\xi )\in E}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_i(Z,\xi )|\hfill \\ \hfill & =\underset{(Z,\xi )\in E}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi (Z,\xi )||f_i(W,\eta )|\hfill \\ \hfill & \le \underset{(Z,\xi )\in E}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi (Z,\xi )|{\displaystyle \frac{\parallel f_i{\parallel }_{{\mathcal{A}}_{\alpha }(\mathcal{H})}}{[N_1(W_1,\overline{W_1})\cdots N_t(W_t,\overline{W_t})-{\parallel \eta \parallel }^2{]}^{\alpha }}}\hfill \\ \hfill & \le C_1\epsilon .\hfill \end{array}$$

On the other hand, if we set

$$E_{\sigma }=\{(Z,\xi )\in \mathcal{H}:\mathrm{dist}(\varphi (Z,\xi ),\partial \mathcal{H})\ge \sigma \},$$

it is clear that $E_{\sigma }$ is a compact subset of $\mathcal{H}$. By the hypothesis, we know that $\{f_i\}$ converges to 0 uniformly on every compact subset of $\mathcal{H}$. Since $\psi \in {\mathcal{A}}_{\beta }(\mathcal{H})$, we can assume that ${\parallel \psi \parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}\le C_2$. Then, for such $\epsilon >0$,

$$\begin{array}{cc}\hfill & \underset{(Z,\xi )\in E_{\sigma }}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_i(Z,\xi )|\hfill \\ \hfill & =\underset{(Z,\xi )\in E_{\sigma }}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi (Z,\xi )||f_i(\varphi (Z,\xi ))|\hfill \\ \hfill & \le \underset{(Z,\xi )\in E_{\sigma }}{sup}|\psi (Z,\xi )|[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }·\underset{(Z,\xi )\in E_{\sigma }}{sup}|f_i(\varphi (Z,\xi ))|\hfill \\ \hfill & \le C_2\epsilon .\hfill \end{array}$$

By combining the above two cases, for $i\to \infty$, we have

$$\begin{array}{cc}\hfill \parallel \psi C_{\varphi }{f}_i{\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}& =\underset{(Z,\xi )\in \mathcal{H}}{sup}[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_i(Z,\xi )|\hfill \\ \hfill & \le (\underset{(Z,\xi )\in E}{sup}+\underset{(Z,\xi )\in E_{\sigma }}{sup})[N_1(Z_1,\overline{Z_1})\cdots N_t(Z_t,\overline{Z_t})-{\parallel \xi \parallel }^2{]}^{\beta }|\psi C_{\varphi }{f}_i(Z,\xi )|\hfill \\ \hfill & \le (C_1+C_2)\epsilon .\hfill \end{array}$$

That is, ${lim}_{i\to \infty }{\parallel \psi C_{\varphi }{f}_i\parallel }_{{\mathcal{A}}_{\beta }(\mathcal{H})}=0$. From Lemma 6, we obtain that the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathcal{H})\to {\mathcal{A}}_{\beta }(\mathcal{H})$ is compact. This completes the proof of the theorem. □

4. Some Special Examples

In this Section, we discuss several special cases of generalized Hua–Cartan–Hartogs domains, and obtain four specific domains in which the necessary and sufficient condition for the boundedness and compactness of weighted composition operators between Bers-type spaces may be specifically characterized.

4.1. The Generalized Hua Domains

For $t=1$, consider the quantity

$$\begin{array}{c}\hfill N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=\left\{\begin{array}{l}det{(I-Z{\overline{Z}}^{\prime })}^k,Z\in {\Re }_{\mathrm{I}}(m,n),\\ det{(I-Z{\overline{Z}}^{\prime })}^k,Z\in {\Re }_{\mathrm{II}}(p),\\ (1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k,z\in {\Re }_{\mathrm{IV}}(N),\end{array}\right.\end{array}$$

(14)

where k is a positive real number. In this case, the generalized Hua–Cartan–Hartogs domains are the generalized Hua domains:

$$\begin{array}{cc}\hfill & {\mathrm{GHE}}_{\mathrm{I}}:=\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{I}}(m,n):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det{(I-Z{\overline{Z}}^{\prime })}^k,j=1,\cdots ,r\},\hfill \\ \hfill & {\mathrm{GHE}}_{\mathrm{II}}:=\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{II}}(p):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det{(I-Z{\overline{Z}}^{\prime })}^k,j=1,\cdots ,r\},\hfill \\ \hfill & {\mathrm{GHE}}_{\mathrm{IV}}:=\{{\xi }_j\in {\mathbb{C}}^{n_j},z\in {\Re }_{\mathrm{IV}}(N):\sum _{j=1}^r|{\xi }_j{|}^{2p_j}<(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k,j=1,\cdots ,r\}.\hfill \end{array}$$

${\mathrm{GHE}}_{\mathrm{I}},{\mathrm{GHE}}_{\mathrm{II}}$ and ${\mathrm{GHE}}_{\mathrm{IV}}$ denote, respectively, the generalized Hua domains of the first, second and fourth kind. The Bers-type spaces on ${\mathrm{GHE}}_{\mathrm{I}},{\mathrm{GHE}}_{\mathrm{II}},{\mathrm{GHE}}_{\mathrm{IV}}$ may be defined as follows.

Definition 2.

Let $\alpha >0$.

$(\mathrm{i})$ The Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{I}})$ consists of all $f\in H({\mathrm{GHE}}_{\mathrm{I}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{I}})}=\underset{(Z,\xi )\in {\mathrm{GHE}}_{\mathrm{I}}}{sup}[det{(I-Z{\overline{Z}}^{\prime })}^k-\sum _{j=1}^r|{\xi }_j{|}^{2p_j}{]}^{\alpha }|f(Z,\xi )|<+\infty .$$

$(\mathrm{ii})$ The Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{II}})$ consists of all $f\in H({\mathrm{GHE}}_{\mathrm{II}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{II}})}=\underset{(Z,\xi )\in {\mathrm{GHE}}_{\mathrm{II}}}{sup}[det{(I-Z{\overline{Z}}^{\prime })}^k-\sum _{j=1}^r|{\xi }_j{|}^{2p_j}{]}^{\alpha }|f(Z,\xi )|<+\infty .$$

$(\mathrm{iii})$ The Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{IV}})$ consists of all $f\in H({\mathrm{GHE}}_{\mathrm{IV}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{IV}})}=\underset{(z,\xi )\in {\mathrm{GHE}}_{\mathrm{IV}}}{sup}[(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k-\sum _{j=1}^r|{\xi }_j{|}^{2p_j}{]}^{\alpha }|f(z,\xi )|<+\infty .$$

Definition 1 coincides with Definition 2 when the generalized Hua–Cartan–Hartogs domains are the generalized Hua domains.

To prove that the above generalized Hua domains are special cases of the generalized Hua–Cartan–Hartogs domains, we need to verify that $det{(I-Z{\overline{W}}^{\prime })}^k,Z,W\in {\Re }_{\mathrm{I}},{\Re }_{\mathrm{II}}$ and ${(1+z{z}^{\prime }{\overline{ww}}^{\prime }-2z{\overline{w}}^{\prime })}^k,z,w\in {\Re }_{\mathrm{IV}}$ satisfy conditions (1) and (2). Indeed, we have

Lemma 14.

The following statements hold.

$(\mathrm{i})$ If $Z,W\in {\Re }_{\mathrm{I}}(m,n)$ and $0<km\le 1$, then

$$2|det{(I-Z{\overline{W}}^{\prime })}^k|\ge det{(I-Z{\overline{Z}}^{\prime })}^k+det{(I-W{\overline{W}}^{\prime })}^k.$$

(15)

$(\mathrm{ii})$ If $Z,W\in {\Re }_{\mathrm{II}}(p)$ and $0<kp\le 1$, then

$$2|det{(I-Z{\overline{W}}^{\prime })}^k|\ge det{(I-Z{\overline{Z}}^{\prime })}^k+det{(I-W{\overline{W}}^{\prime })}^k.$$

(16)

Proof. 

$(1)$ Assume that $Z,W\in {\Re }_{\mathrm{I}}(m,n)$ and that $m\le n$.

For $m=n$, applying Lemma 3, there exist $m\times m$ unitary matrices $U_1,U_2,V_1$ and $V_2$ such that

$$Z=U_1\left(\begin{array}{cccc}{\lambda }_1& 0& \dots & 0\\ 0& {\lambda }_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\lambda }_m\end{array}\right)V_1=U_1{\mathrm{\Lambda }}_1{V}_1(1>{\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_m\ge 0),$$

(17)

and

$$W=U_2\left(\begin{array}{cccc}{\mu }_1& 0& \dots & 0\\ 0& {\mu }_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\mu }_m\end{array}\right)V_2=U_2{\mathrm{\Lambda }}_2{V}_2(1>{\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_m\ge 0).$$

(18)

Then, it turns out that

$$\begin{array}{cc}\hfill det(I-Z{\overline{W}}^{\prime })& =det(I-U_1{\mathrm{\Lambda }}_1{V}_1{\overline{V_2}}^{\prime }{\overline{{\mathrm{\Lambda }}_2}}^{\prime }{\overline{U_2}}^{\prime })\hfill \\ \hfill & =det(U_1{\overline{U_1}}^{\prime }-U_1{\mathrm{\Lambda }}_1{V}_1{\overline{V_2}}^{\prime }{\overline{{\mathrm{\Lambda }}_2}}^{\prime }{\overline{U_2}}^{\prime })\hfill \\ \hfill & =det{U}_{1}det({\overline{U_1}}^{\prime }-{\mathrm{\Lambda }}_1{V}_1{\overline{V_2}}^{\prime }{\overline{{\mathrm{\Lambda }}_2}}^{\prime }{\overline{U_2}}^{\prime })\hfill \\ \hfill & =det(I-{\mathrm{\Lambda }}_1{V}_1{\overline{V_2}}^{\prime }{\overline{{\mathrm{\Lambda }}_2}}^{\prime }{V}_2{\overline{V_1}}^{\prime }{V}_1{\overline{V_2}}^{\prime }{\overline{U_2}}^{\prime }{U}_1),\hfill \end{array}$$

and, according to Lemma 4, there exists a square matrix P, such that

$$|det(I-Z{\overline{W}}^{\prime })|\ge |det(I-{\mathrm{\Lambda }}_{1}P{\mathrm{\Lambda }}_2{P}^{\prime })|=\prod _{i=1}^m(1-{\lambda }_i{\mu }_{k_i}),$$

where $k_1,k_2,\cdots ,k_m$ is a permutation of $1,2,\cdots ,m$. Using (4) and (5), we have

$$\begin{array}{cc}\hfill 2|det{(I-Z{\overline{W}}^{\prime })}^k|& =2^{1-mk}{2}^{mk}|det{(I-Z{\overline{W}}^{\prime })}^k|\hfill \\ \hfill & =2^{1-mk}{\left[2^m|det(I-Z{\overline{W}}^{\prime })|\right]}^k\hfill \\ \hfill & \ge 2^{1-mk}{\left[2^m\prod _{i=1}^m(1-{\lambda }_i{\mu }_{k_i})\right]}^k\hfill \\ \hfill & =2^{1-mk}{\left[\prod _{i=1}^m(2-2{\lambda }_i{\mu }_{k_i})\right]}^k\hfill \\ \hfill & \ge 2^{1-mk}{\left[\prod _{i=1}^m(2-{\lambda }_i^2-{\mu }_{k_i}^2)\right]}^k\hfill \\ \hfill & =2^{1-mk}{\left\{{\left\{\prod _{i=1}^m[(1-{\lambda }_i^2)+(1-{\mu }_{k_i}^2)]\right\}}^{{\displaystyle \frac{1}{m}}}\right\}}^{mk}\hfill \\ \hfill & \ge 2^{1-mk}{\left\{{\left[\prod _{i=1}^m(1-{\lambda }_i^2)\right]}^{{\displaystyle \frac{1}{m}}}+{\left[\prod _{i=1}^m(1-{\mu }_{k_i}^2)\right]}^{{\displaystyle \frac{1}{m}}}\right\}}^{mk}\hfill \\ \hfill & \ge 2^{1-mk}{2}^{mk-1}\left\{{\left[\prod _{i=1}^m(1-{\lambda }_i^2)\right]}^{{\displaystyle \frac{1}{m}}\times mk}+{\left[\prod _{i=1}^m(1-{\mu }_{k_i}^2)\right]}^{{\displaystyle \frac{1}{m}}\times mk}\right\}\hfill \\ \hfill & ={\left[\prod _{i=1}^m(1-{\lambda }_i^2)\right]}^k+{\left[\prod _{i=1}^m(1-{\mu }_{k_i}^2)\right]}^k\hfill \\ \hfill & =det{(I-Z{\overline{Z}}^{\prime })}^k+det{(I-W{\overline{W}}^{\prime })}^k.\hfill \end{array}$$

(19)

For $m<n$, there exists a unitary matrix $U^{(n)}$ such that

$$Z=(Z_1^{(m)},0)U,W=(W_1^{(m)},W_2)U.$$

By (19), we obtain

$$\begin{array}{cc}\hfill 2|det{(I-Z{\overline{W}}^{\prime })}^k|& =2|det{(I-Z_1{\overline{W_1}}^{\prime })}^k|\hfill \\ \hfill & \ge det{(I-Z_1{\overline{Z_1}}^{\prime })}^k+det{(I-W_1{\overline{W_1}}^{\prime })}^k\hfill \\ \hfill & \ge det{(I-Z_1{\overline{Z_1}}^{\prime })}^k+det{(I-W_1{\overline{W_1}}^{\prime }-W_2{\overline{W_2}}^{\prime })}^k\hfill \\ \hfill & =det{(I-Z{\overline{Z}}^{\prime })}^k+det{(I-W{\overline{W}}^{\prime })}^k.\hfill \end{array}$$

This completes the proof of (15).

$(2)$ Assume $Z,W\in {\Re }_{\mathrm{II}}(p)$; then, the polar decompositions of Z and W are similar to (17) and (18); thus, (16) can be proved in the same way. □

Lemma 15

(see [47]). The linear transformation

$$w_1=z_1+i{z}_2,w_2=z_1-i{z}_2,$$

$$w_3=i{z}_3-z_4,w_4=i{z}_3+z_4,$$

maps the domain ${\Re }_{\mathrm{IV}}(4)$

$$1+|z_1^2+z_2^2+z_3^2+z_4^2{|}^2-2(|z_1{|}^2+|z_2{|}^2+|z_3{|}^2+|z_4{|}^2)>0,$$

$$1-|z_1^2+z_2^2+z_3^2+z_4^2{|}^2>0,$$

onto the domain ${\Re }_{\mathrm{I}}(2,2):$

$$I-W{\overline{W}}^{\prime }>0,W=\left(\begin{array}{cc}{w}_1& w_3\\ w_4& w_2\end{array}\right).$$

Lemma 16

(see [47]). If Z and W are $m\times n$ matrices which satisfy $I-Z{\overline{Z}}^{\prime }\ge 0,I-W{\overline{W}}^{\prime }>0$, then $I-{\overline{W}}^{\prime }Z{\overline{({\overline{W}}^{\prime }Z)}}^{\prime }>0$ and $I-{\overline{W}}^{\prime }Z$ is nonsingular.

Lemma 17.

Let $z,w\in {\Re }_{\mathrm{IV}}(N)$; then, there exist two second-order square matrices Z and W such that

$$1+z{z}^{\prime }{\overline{ww}}^{\prime }-2z{\overline{w}}^{\prime }=det(I-{\overline{W}}^{\prime }Z)\ne 0,$$

$$1+|z{z}^{\prime }{|}^2-2{|z|}^2=det(I-Z{\overline{Z}}^{\prime }),$$

$$1+|w{w}^{\prime }{|}^2-2{|w|}^2=det(I-W{\overline{W}}^{\prime }).$$

Proof. 

If $z,w\in {\Re }_{\mathrm{IV}}(N)$, then there exists an $N\times N$ real orthogonal square matrix $\mathrm{\Gamma }$ such that the two N-dimensional vectors z and w can be written as

$$z=(z_1^{*},z_2^{*},z_3^{*},z_4^{*},0,\cdots ,0)\mathrm{\Gamma },$$

$$w=(w_1^{*},w_2^{*},w_3^{*},w_4^{*},0,\cdots ,0)\mathrm{\Gamma }.$$

According to Lemma 15, we have

$$I-Z{\overline{Z}}^{\prime }>0,I-W{\overline{W}}^{\prime }>0,$$

where

$$Z=\left(\begin{array}{cc}{z}_1^{*}+i{z}_2^{*}& i{z}_3^{*}-z_4^{*}\\ i{z}_3^{*}+z_4^{*}& z_1^{*}-i{z}_2^{*}\end{array}\right)\in {\Re }_{\mathrm{I}}(2,2),$$

(20)

$$W=\left(\begin{array}{cc}{w}_1^{*}+i{w}_2^{*}& i{w}_3^{*}-w_4^{*}\\ i{w}_3^{*}+w_4^{*}& w_1^{*}-i{w}_2^{*}\end{array}\right)\in {\Re }_{\mathrm{I}}(2,2).$$

(21)

By Lemma 16, we obtain $det(I-{\overline{W}}^{\prime }Z)\ne 0$. Hence,

$$\begin{array}{cc}\hfill 1+z{z}^{\prime }{\overline{ww}}^{\prime }-2z{\overline{w}}^{\prime }& =1+[{(z_1^{*})}^2+{(z_2^{*})}^2+{(z_3^{*})}^2+{(z_4^{*})}^2]\overline{[{(w_1^{*})}^2+{(w_2^{*})}^2+{(w_3^{*})}^2+{(w_4^{*})}^2]}\hfill \\ & -2(z_1^{*}\overline{w_1^{*}}+z_2^{*}\overline{w_2^{*}}+z_3^{*}\overline{w_3^{*}}+z_4^{*}\overline{w_4^{*}})\hfill \\ \hfill & =det(I-{\overline{W}}^{\prime }Z)\ne 0.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill 1+|z{z}^{\prime }{|}^2-2{|z|}^2& =1+|{(z_1^{*})}^2+{(z_2^{*})}^2+{(z_3^{*})}^2+{(z_4^{*})}^2{|}^2-2(|z_1^{*}{|}^2+|z_2^{*}{|}^2+|z_3^{*}{|}^2+|z_4^{*}{|}^2)\hfill \\ \hfill & =det(I-Z{\overline{Z}}^{\prime }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill 1+|w{w}^{\prime }{|}^2-2{|w|}^2& =1+|{(w_1^{*})}^2+{(w_2^{*})}^2+{(w_3^{*})}^2+{(w_4^{*})}^2{|}^2-2(|w_1^{*}{|}^2+|w_2^{*}{|}^2+|w_3^{*}{|}^2+|w_4^{*}{|}^2)\hfill \\ \hfill & =det(I-W{\overline{W}}^{\prime }).\hfill \end{array}$$

This completes the proof of the Lemma. □

Lemma 18.

If $z,w\in {\Re }_{\mathrm{IV}}(N)$ and $0<k\le {\displaystyle \frac{1}{2}}$, then

$$(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k+(1+|w{w}^{\prime }{|}^2-{2|w|}^2{)}^k\le 2|{(1+z{z}^{\prime }{\overline{ww}}^{\prime }-2z{\overline{w}}^{\prime })}^k|.$$

(22)

Proof. 

For $z,w\in {\Re }_{\mathrm{IV}}(N)$, Lemma 17 ensures that there exist two second-order square matrices $Z,W\in {\Re }_{\mathrm{I}}(2,2)$ such that

$$(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k=det{(I-Z{\overline{Z}}^{\prime })}^k,$$

$$(1+|w{w}^{\prime }{|}^2-{2|w|}^2{)}^k=det{(I-W{\overline{W}}^{\prime })}^k,$$

$${(1+z{z}^{\prime }{\overline{ww}}^{\prime }-2z{\overline{w}}^{\prime })}^k=det{(I-Z{\overline{W}}^{\prime })}^k,$$

where Z and W are identical to (20) and (21), respectively. By (15), we have

$$(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k+(1+|w{w}^{\prime }{|}^2-{2|w|}^2{)}^k\le 2|{(1+z{z}^{\prime }{\overline{ww}}^{\prime }-2z{\overline{w}}^{\prime })}^k|.$$

This completes the proof of the Lemma. □

Lemma 19.

If $z\in {\Re }_{\mathrm{IV}}(N)$, then $0<1+|z{z}^{\prime }{|}^2-2{|z|}^2\le 1$.

Proof. 

Since $z\in {\Re }_{\mathrm{IV}}(N)$, there exists a real orthogonal square matrix $\mathrm{\Gamma }$ such that

$$z=e^{i\theta }({\lambda }_1,i{\lambda }_2,0,\cdots ,0)\mathrm{\Gamma }({\lambda }_1\ge {\lambda }_2>0,1>{\lambda }_1+{\lambda }_2),$$

where $e^{i\theta }=cos\theta +isin\theta$. Therefore,

$$z{z}^{\prime }=e^{i\theta }({\lambda }_1,i{\lambda }_2,0,\cdots ,0)\mathrm{\Gamma }{\mathrm{\Gamma }}^{\prime }{({\lambda }_1,i{\lambda }_2,0,\cdots ,0)}^{\prime }{e}^{i\theta }=e^{2i\theta }({\lambda }_1^2-{\lambda }_2^2),$$

which implies that

$$0<|z{z}^{\prime }|=|e^{2i\theta }({\lambda }_1^2-{\lambda }_2^2)|={\lambda }_1^2-{\lambda }_2^2<1.$$

At the same time,

$$\begin{array}{cc}\hfill {2|z|}^2& =2z{\overline{z}}^{\prime }=2e^{i\theta }({\lambda }_1,i{\lambda }_2,0,\cdots ,0)\mathrm{\Gamma }{\overline{\mathrm{\Gamma }}}^{\prime }{({\lambda }_1,-i{\lambda }_2,0,\cdots ,0)}^{\prime }{\overline{e^{i\theta }}}^{\prime }\hfill \\ & =2|e^{i\theta }{|}^2({\lambda }_1^2+{\lambda }_2^2)=2({\lambda }_1^2+{\lambda }_2^2)\ge {\lambda }_1^2-{\lambda }_2^2\ge {({\lambda }_1^2-{\lambda }_2^2)}^2={|z{z}^{\prime }|}^2,\hfill \end{array}$$

that is, $|z{z}^{\prime }{|}^2-2{|z|}^2\le 0$; then, $1+|z{z}^{\prime }{|}^2-2{|z|}^2\le 1$. Furthermore, when $z\in {\Re }_{\mathrm{IV}}(N)$, we have $1+|z{z}^{\prime }{|}^2-2{|z|}^2>0$. Hence, $0<1+|z{z}^{\prime }{|}^2-2{|z|}^2\le 1$. This completes the proof of the Lemma. □

By the definition of the Cartan domains of the first kind and second kind, $det{(I-Z{\overline{Z}}^{\prime })}^k,Z\in {\Re }_{\mathrm{I}},{\Re }_{\mathrm{II}}$ clearly meet (1). From Lemma 19, $(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k,z\in {\Re }_{\mathrm{IV}}$ meets (1). Meanwhile, upon using Lemmas 14 and 18, we have that when k meets certain conditions, $det{(I-Z{\overline{W}}^{\prime })}^k,Z,W\in {\Re }_{\mathrm{I}},{\Re }_{\mathrm{II}}$ and ${(1+z{z}^{\prime }{\overline{ww}}^{\prime }-2z{\overline{w}}^{\prime })}^k,z,w\in {\Re }_{\mathrm{IV}}$ meet conditions (2). Therefore, those generalized Hua domains are special cases of generalized Hua–Cartan–Hartogs domains. Using Theorems 1 and 2, we obtain the conditions for the boundedness and compactness of weighted composition operators between Bers-type spaces on generalized Hua domains.

Let us now consider a holomorphic self-map $\varphi$ of $\mathrm{GHE}\in \{{\mathrm{GHE}}_{\mathrm{I}},{\mathrm{GHE}}_{\mathrm{II}},{\mathrm{GHE}}_{\mathrm{IV}}\}$. Let us write $(W,\eta )=\varphi (Z,\xi )$ for $(Z,\xi )\in {\mathrm{GHE}}_{\mathrm{I}},{\mathrm{GHE}}_{\mathrm{II}}$ and $(w,\eta )=\varphi (z,\xi )$ for $(z,\xi )\in {\mathrm{GHE}}_{\mathrm{IV}}$.

Corollary 1.

If $\alpha ,\beta >0$ are positive numbers, ϕ is a holomorphic self-map of $\mathrm{GHE}$ and $\psi \in H(\mathrm{GHE})$, then the following statements hold.

$(\mathrm{i})$ If $0<km\le 1$, then the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{I}})\to {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{I}})$ is bounded iff

$${\tilde{M}}_I=\underset{(Z,\xi )\in {\mathrm{GHE}}_{\mathrm{I}}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z{\overline{Z}}^{\prime })}^k-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W{\overline{W}}^{\prime })}^k-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty .$$

$(\mathrm{ii})$ If $0<kp\le 1$, then the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{II}})\to {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{II}})$ is bounded iff

$${\tilde{M}}_{II}=\underset{(Z,\xi )\in {\mathrm{GHE}}_{\mathrm{II}}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z{\overline{Z}}^{\prime })}^k-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W{\overline{W}}^{\prime })}^k-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty .$$

$(\mathrm{iii})$ If $0<k\le {\displaystyle \frac{1}{2}}$, then the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{IV}})\to {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{IV}})$ is bounded iff

$${\tilde{M}}_{IV}=\underset{(z,\xi )\in {\mathrm{GHE}}_{\mathrm{IV}}}{sup}|\psi (z,\xi )|{\displaystyle \frac{[(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k-{\parallel \xi \parallel }^2{]}^{\beta }}{[(1+|w{w}^{\prime }{|}^2-{2|w|}^2{)}^k-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty .$$

Corollary 2.

If $\alpha ,\beta >0$ are positive numbers, ϕ is a holomorphic self-map of $\mathrm{GHE}$ and $\psi \in H(\mathrm{GHE})$, then the following statements hold.

$(\mathrm{i})$ If $0<km\le 1$, then the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{I}})\to {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{I}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{I}})$ and

$$\underset{\varphi (Z,\xi )\to \partial {\mathrm{GHE}}_{\mathrm{I}}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z{\overline{Z}}^{\prime })}^k-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W{\overline{W}}^{\prime })}^k-{\parallel \eta \parallel }^2{]}^{\alpha }}}=0.$$

$(\mathrm{ii})$ If $0<kp\le 1$, then the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{II}})\to {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{II}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{II}})$ and

$$\underset{\varphi (Z,\xi )\to \partial {\mathrm{GHE}}_{\mathrm{II}}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z{\overline{Z}}^{\prime })}^k-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W{\overline{W}}^{\prime })}^k-{\parallel \eta \parallel }^2{]}^{\alpha }}}=0.$$

$(\mathrm{iii})$ If $0<k\le {\displaystyle \frac{1}{2}}$, then the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GHE}}_{\mathrm{IV}})\to {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{IV}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{GHE}}_{\mathrm{IV}})$ and

$$\underset{\varphi (z,\xi )\to \partial {\mathrm{GHE}}_{\mathrm{IV}}}{lim}|\psi (z,\xi )|{\displaystyle \frac{[(1+|z{z}^{\prime }{|}^2-{2|z|}^2{)}^k-{\parallel \xi \parallel }^2{]}^{\beta }}{[(1+|w{w}^{\prime }{|}^2-{2|w|}^2{)}^k-{\parallel \eta \parallel }^2{]}^{\alpha }}}=0.$$

Remark 1.

Corollaries 1 and 2 express the sufficient and necessary conditions for the boundedness and compactness of weighted composition operators between Bers-type spaces on generalized Hua domains of the first, second and fourth kind. This is a new result, not obtained before. In order to prove analogue results for the generalized Hua domains of the third type, we should prove that “if $Z,W\in {\Re }_{\mathrm{III}}(q)$, then $2|det{(I-Z{\overline{W}}^{\prime })}^k|\ge det{(I-Z{\overline{Z}}^{\prime })}^k+det{(I-W{\overline{W}}^{\prime })}^k$". We have tried to prove this inequality by the polar decomposition of ${\Re }_{III}$, without success so far. This will be the subject of future analysis.

For $p_1=p_2=\cdots =p_r=1,m=1,k=1$, the generalized Hua domain of the first kind ${\mathrm{GHE}}_{\mathrm{I}}$ is the unit ball ${\mathrm{B}}_{\mathrm{n}}$. The sufficient and necessary conditions for the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{B}}_{\mathrm{n}})\to {\mathcal{A}}_{\beta }({\mathrm{B}}_{\mathrm{n}})$ to be bounded and compact are summarized as follows:

Corollary 3.

Given $\alpha ,\beta >0$, a holomorphic self-map ϕ of ${\mathrm{B}}_{\mathrm{n}}$ and $\psi \in H({\mathrm{B}}_{\mathrm{n}})$, then the weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{B}}_{\mathrm{n}})\to {\mathcal{A}}_{\beta }({\mathrm{B}}_{\mathrm{n}})$

$(\mathrm{i})$ is bounded if and only if

$$\underset{z\in {\mathrm{B}}_{\mathrm{n}}}{sup}|\psi (z)|{\displaystyle \frac{{(1-|z|}^2{)}^{\beta }}{{(1-|\varphi (z)|}^2{)}^{\alpha }}}<+\infty .$$

$(\mathrm{ii})$ is compact if and only if $\psi \in {\mathcal{A}}_{\beta }({\mathrm{B}}_{\mathrm{n}})$ and

$$\underset{\varphi (z)\to 1}{lim}|\psi (z)|{\displaystyle \frac{{(1-|z|}^2{)}^{\beta }}{{(1-|\varphi (z)|}^2{)}^{\alpha }}}=0.$$

This result is consistent with the conclusion of Jin and Tang in [11].

From Corollary 3, by setting $n=1$, we may obtain the boundedness and compactness conditions for weighted composition operators between Bers-type space on the unit disk, which are consistent with the results of [9,10].

Corollary 4.

Given $\alpha ,\beta >0$, a holomorphic self-map ϕ of $\mathrm{D}$ and $\psi \in H(\mathrm{D})$, then the weighted composition $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }(\mathrm{D})\to {\mathcal{A}}_{\beta }(\mathrm{D})$

$(\mathrm{i})$ is bounded if and only if

$$\underset{z\in \mathrm{D}}{sup}|\psi (z)|{\displaystyle \frac{{(1-|z|}^2{)}^{\beta }}{{(1-|\varphi (z)|}^2{)}^{\alpha }}}<+\infty .$$

$(\mathrm{ii})$ is compact if and only if $\psi \in {\mathcal{A}}_{\beta }(\mathrm{D})$ and

$$\underset{\varphi (z)\to 1}{lim}|\psi (z)|{\displaystyle \frac{{(1-|z|}^2{)}^{\beta }}{{(1-|\varphi (z)|}^2{)}^{\alpha }}}=0.$$

4.2. The Generalized Cartan–Hartogs Domains

For $t=2$, let us introduce

$$\begin{array}{c}\hfill N_k(Z_k,\overline{Z_k})=\left\{\begin{array}{l}det{(I-Z_k{\overline{Z_k}}^{\prime })}^{s_k},Z_k\in {\Re }_{\mathrm{I}}(m,n),\hfill \\ det{(I-Z_k{\overline{Z_k}}^{\prime })}^{s_k},Z_k\in {\Re }_{\mathrm{II}}(p),\hfill \\ (1+|z_k{z}_k^{\prime }{|}^2-2|z_k{{|}^2)}^{s_k},z_k\in {\Re }_{\mathrm{IV}}(N),\hfill \end{array}\right.\end{array}$$

(23)

where $s_k$ are positive real numbers, $k=1,2.$ In this case, those generalized Hua–Cartan–Hartogs domains are generalized Cartan–Hartogs domains. We consider the case in which $Z_1$ and $Z_2$ simultaneously belong to ${\Re }_{\mathrm{I}},{\Re }_{\mathrm{II}}$ and ${\Re }_{\mathrm{IV}}$, and assume $Z_1\in {\Re }_{\mathrm{I}}(m,n),Z_2\in {\Re }_{\mathrm{I}}(g,l)$; then, let $N_1(Z_1,\overline{Z_1})N_2(Z_2,\overline{Z_2})=det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}$. We refer to this domain as ${\mathrm{GW}}_{\mathrm{I},\mathrm{I}}$, with the formal definition

$$\begin{array}{cc}\hfill {\mathrm{GW}}_{\mathrm{I},\mathrm{I}}:=\{& {\xi }_j\in {\mathbb{C}}^{n_j},Z_1\in {\Re }_{\mathrm{I}}(m,n),Z_2\in {\Re }_{\mathrm{I}}(g,l):\hfill \\ & \sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2},j=1,\cdots ,r\}.\hfill \end{array}$$

Definition 3.

Let $\alpha >0$; the Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})$ consists of all $f\in H({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})}=\underset{(Z,\xi )\in {\mathrm{GW}}_{\mathrm{I},\mathrm{I}}}{sup}[det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}-\sum _{j=1}^r|{\xi }_j{|}^{2p_j}{]}^{\alpha }|f(Z,\xi )|<+\infty .$$

For $Z_1\in {\Re }_{\mathrm{I}}(m,n),Z_2\in {\Re }_{\mathrm{I}}(g,l)$, by the definition of the Cartan domain of the first kind, $det{(I-Z_k{\overline{Z_k}}^{\prime })}^{s_k},k=1,2$ clearly meet (1); if $s_k(k=1,2)$ satisfy $0<s_{1}m\le 1,0<s_{2}g\le 1$, then Lemma 14 implies that $det{(I-Z_k{\overline{W_k}}^{\prime })}^{s_k},k=1,2$ satisfy conditions (2). Therefore, the generalized Cartan–Hartogs domain ${\mathrm{GW}}_{\mathrm{I},\mathrm{I}}$ is a generalized Hua–Cartan–Hartogs domain. Definition 3 coincides with Definition 1 when the generalized Hua–Cartan–Hartogs domains are the generalized Cartan–Hartogs domains. We may easily obtain conditions for the boundedness and compactness of a weighted composition operator between Bers-type spaces on ${\mathrm{GW}}_{\mathrm{I},\mathrm{I}}$.

Assume that $\varphi$ is a holomorphic self-map of ${\mathrm{GW}}_{\mathrm{I},\mathrm{I}}$, and consider $(W,\eta )=\varphi (Z,\xi )$ for $(Z,\xi )\in {\mathrm{GW}}_{\mathrm{I},\mathrm{I}}$.

Corollary 5.

For $\alpha ,\beta >0$, $0<s_{1}m\le 1$, $0<s_{2}g\le 1$, a holomorphic self-map ϕ of ${\mathrm{GW}}_{\mathrm{I},\mathrm{I}}$ and $\psi \in H({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})$, the following statements hold.

$(\mathrm{i})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})\to {\mathcal{A}}_{\beta }({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})$ is bounded iff

$${\tilde{N}}_{I,I}=\underset{(Z,\xi )\in {\mathrm{GW}}_{\mathrm{I},\mathrm{I}}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W_1{\overline{W_1}}^{\prime })}^{s_1}det{(I-W_2{\overline{W_2}}^{\prime })}^{s_2}-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty .$$

$(\mathrm{ii})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})\to {\mathcal{A}}_{\beta }({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{GW}}_{\mathrm{I},\mathrm{I}})$ and

$$\underset{\varphi (Z,\xi )\to \partial {\mathrm{GW}}_{\mathrm{I},\mathrm{I}}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W_1{\overline{W_1}}^{\prime })}^{s_1}det{(I-W_2{\overline{W_2}}^{\prime })}^{s_2}-{\parallel \eta \parallel }^2{]}^{\alpha }}}=0.$$

In particular, for $r=1$, ${\mathrm{GW}}_{\mathrm{I},\mathrm{I}}$ is the ordinary generalized Cartan–Hartogs domain ${\mathrm{W}}_{\mathrm{I},\mathrm{I}}$.

$${\mathrm{W}}_{\mathrm{I},\mathrm{I}}:=\left\{(Z_1,Z_2,\xi )\in {\Re }_{\mathrm{I}}(m,n)\times {\Re }_{\mathrm{I}}(g,l)\times {\mathbb{C}}^n:{|\xi |}^2<det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}\right\}.$$

If we set $r=1$ in Corollary 7, we obtain sufficient and necessary conditions for the boundedness and compactness of weighted composition operators between Bers-type spaces on the generalized Cartan–Hartogs domains ${\mathrm{W}}_{\mathrm{I},\mathrm{I}}$.

4.3. The Generalized Cartan–Hartogs Domains over Different Cartan Domains

For $t=2$, we introduce $N_k(Z_k,\overline{Z_k})$ as in (23), $k=1,2.$ In this case, those generalized Hua–Cartan–Hartogs domains are generalized Cartan–Hartogs domains. Here, we consider the case in which $Z_1$ and $Z_2$ belong to different Cartan domains. Taking $Z_1\in {\Re }_{\mathrm{I}}(m,n),Z_2\in {\Re }_{\mathrm{II}}(p)$ as an example, let $N_1(Z_1,\overline{Z_1})N_2(Z_2,\overline{Z_2})=det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}$ and refer to this domain as ${\mathrm{GW}}_{\mathrm{I},\mathrm{II}}$ by setting

$$\begin{array}{cc}\hfill {\mathrm{GW}}_{\mathrm{I},\mathrm{II}}:=\{& {\xi }_j\in {\mathbb{C}}^{n_j},Z_1\in {\Re }_{\mathrm{I}}(m,n),Z_2\in {\Re }_{\mathrm{II}}(p):\hfill \\ & \sum _{j=1}^r{|{\xi }_j|}^{2p_j}<det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2},j=1,2,\cdots ,r\}.\hfill \end{array}$$

Definition 4.

For $\alpha >0$, the Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})$ consists of all $f\in H({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})}=\underset{(Z,\xi )\in {\mathrm{GW}}_{\mathrm{I},\mathrm{II}}}{sup}[det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}-\sum _{j=1}^r|{\xi }_j{|}^{2p_j}{]}^{\alpha }|f(Z,\xi )|<+\infty .$$

For $Z_1\in {\Re }_{\mathrm{I}}(m,n),Z_2\in {\Re }_{\mathrm{II}}(p)$, by the definition of the Cartan domains of the first kind and second kind, $det{(I-Z_k{\overline{Z_k}}^{\prime })}^{s_k},k=1,2$ clearly meet (1); if $s_k(k=1,2)$ satisfy $0<s_{1}m\le 1,0<s_{2}p\le 1$, then by Lemma 14 we know that $det{(I-Z_k{\overline{W_k}}^{\prime })}^{s_k},k=1,2$ satisfy conditions (2). Therefore, the generalized Cartan–Hartogs domain ${\mathrm{GW}}_{\mathrm{I},\mathrm{II}}$ is a special case of generalized Hua–Cartan–Hartogs domains. Definition 4 coincides with Definition 1 when the generalized Hua–Cartan–Hartogs domains are generalized Cartan–Hartogs domains. Taking Theorems 1 and 2 into account, we obtain the conditions for the boundedness and compactness of weighted composition operators between Bers-type spaces on ${\mathrm{GW}}_{\mathrm{I},\mathrm{II}}$.

Assume that $\varphi$ is a holomorphic self-map of ${\mathrm{GW}}_{\mathrm{I},\mathrm{II}}$ and write $(W,\eta )=\varphi (Z,\xi )$ for $(Z,\xi )\in {\mathrm{GW}}_{\mathrm{I},\mathrm{II}}$. We have the following:

Corollary 6.

Given $\alpha ,\beta >0$, $0<s_{1}m\le 1$, $0<s_{2}p\le 1$, a holomorphic self-map ϕ of ${\mathrm{GW}}_{\mathrm{I},\mathrm{II}}$ and $\psi \in H({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})$, then the following statements hold.

$(\mathrm{i})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})\to {\mathcal{A}}_{\beta }({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})$ is bounded iff

$${\tilde{N}}_{I,II}=\underset{(Z,\xi )\in {\mathrm{GW}}_{\mathrm{I},\mathrm{II}}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W_1{\overline{W_1}}^{\prime })}^{s_1}det{(I-W_2{\overline{W_2}}^{\prime })}^{s_2}-{\parallel \eta \parallel }^2{]}^{\alpha }}}<+\infty .$$

$(\mathrm{ii})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})\to {\mathcal{A}}_{\beta }({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{GW}}_{\mathrm{I},\mathrm{II}})$ and

$$\underset{\varphi (Z,\xi )\to \partial {\mathrm{GW}}_{\mathrm{I},\mathrm{II}}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{[det{(I-Z_1{\overline{Z_1}}^{\prime })}^{s_1}det{(I-Z_2{\overline{Z_2}}^{\prime })}^{s_2}-{\parallel \xi \parallel }^2{]}^{\beta }}{[det{(I-W_1{\overline{W_1}}^{\prime })}^{s_1}det{(I-W_2{\overline{W_2}}^{\prime })}^{s_2}-{\parallel \eta \parallel }^2{]}^{\alpha }}}=0.$$

4.4. The Generalized Ellipsoidal-Type Domains

For $t=1$, let us introduce

$$\begin{array}{c}\hfill N_1(Z_1,\overline{Z_1})=N(Z,\overline{Z})=\left\{\begin{array}{c}1-{\displaystyle \frac{{|Z|}^2}{m}},Z\in {\Re }_{\mathrm{I}}(m,n),\hfill \\ 1-{\displaystyle \frac{{|Z|}^2}{p}},Z\in {\Re }_{\mathrm{II}}(p),\hfill \\ 1-{\displaystyle \frac{{|Z|}^2}{[\frac{q}{2}]}},Z\in {\Re }_{\mathrm{III}}(q),\hfill \\ 1-{|z|}^2,z\in {\Re }_{\mathrm{IV}}(N),\hfill \end{array}\right.\end{array}$$

where $[{\displaystyle \frac{q}{2}}]$ denotes the integer part of ${\displaystyle \frac{q}{2}}$. In this case, we refer to those generalized Hua–Cartan–Hartogs domains as the generalized ellipsoidal-type domains.

$$\begin{array}{cc}\hfill & {\mathrm{L}}_{\mathrm{I}}:=\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{I}}(m,n):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<1-{\displaystyle \frac{{|Z|}^2}{m}},j=1,2,\cdots ,r\},\hfill \\ \hfill & {\mathrm{L}}_{\mathrm{II}}:=\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{II}}(p):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<1-{\displaystyle \frac{{|Z|}^2}{p}},j=1,2,\cdots ,r\},\hfill \\ \hfill & {\mathrm{L}}_{\mathrm{III}}:=\{{\xi }_j\in {\mathbb{C}}^{n_j},Z\in {\Re }_{\mathrm{III}}(q):\sum _{j=1}^r{|{\xi }_j|}^{2p_j}<1-{\displaystyle \frac{{|Z|}^2}{[\frac{q}{2}]}},j=1,2,\cdots ,r\},\hfill \\ \hfill & {\mathrm{L}}_{\mathrm{IV}}:=\{{\xi }_j\in {\mathbb{C}}^{n_j},z\in {\Re }_{\mathrm{IV}}(N):\sum _{j=1}^r|{\xi }_j{|}^{2p_j}<1-{|z|}^2,j=1,2,\cdots ,r\}.\hfill \end{array}$$

${\mathrm{L}}_{\mathrm{I}}$, ${\mathrm{L}}_{\mathrm{II}}$, ${\mathrm{L}}_{\mathrm{III}}$ and ${\mathrm{L}}_{\mathrm{IV}}$ denote generalized ellipsoidal-type domains of the first, second, third and fourth kind, respectively. The Bers-type spaces on ${\mathrm{L}}_{\mathrm{I}}$, ${\mathrm{L}}_{\mathrm{II}}$, ${\mathrm{L}}_{\mathrm{III}}$ and ${\mathrm{L}}_{\mathrm{IV}}$ may be defined as follows:

Definition 5.

Let $\alpha >0$.

$(\mathrm{i})$ The Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{I}})$ consists of all $f\in H({\mathrm{L}}_{\mathrm{I}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{I}})}=\underset{(Z,\xi )\in {\mathrm{L}}_{\mathrm{I}}}{sup}{\left(1-{\displaystyle \frac{{|Z|}^2}{m}}-\sum _{j=1}^r{|{\xi }_j|}^{2p_j}\right)}^{\alpha }|f(Z,\xi )|<+\infty .$$

$(\mathrm{ii})$ The Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{II}})$ consists of all $f\in H({\mathrm{L}}_{\mathrm{II}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{II}})}=\underset{(Z,\xi )\in {\mathrm{L}}_{\mathrm{II}}}{sup}{\left(1-{\displaystyle \frac{{|Z|}^2}{p}}-\sum _{j=1}^r{|{\xi }_j|}^{2p_j}\right)}^{\alpha }|f(Z,\xi )|<+\infty .$$

$(\mathrm{iii})$ The Bers-type space $f\in {\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{III}})$ consists of all $f\in H({\mathrm{L}}_{\mathrm{III}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{III}})}=\underset{(Z,\xi )\in {\mathrm{L}}_{\mathrm{III}}}{sup}{\left(1-{\displaystyle \frac{{|Z|}^2}{[\frac{q}{2}]}}-\sum _{j=1}^r{|{\xi }_j|}^{2p_j}\right)}^{\alpha }|f(Z,\xi )|<+\infty .$$

$(\mathrm{iv})$ The Bers-type space ${\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{IV}})$ consists of all $f\in H({\mathrm{L}}_{\mathrm{IV}})$ satisfying

$${\parallel f\parallel }_{{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{IV}})}=\underset{(z,\xi )\in {\mathrm{L}}_{\mathrm{IV}}}{sup}{\left(1-{|z|}^2-\sum _{j=1}^r{|{\xi }_j|}^{2p_j}\right)}^{\alpha }|f(z,\xi )|<+\infty .$$

Definition 5 coincides with Definition 1 when the generalized Hua–Cartan–Hartogs domains are the generalized ellipsoidal-type domains.

In order to prove that the above generalized ellipsoidal-type domains are special cases of generalized Hua–Cartan–Hartogs domains, we need to prove the following lemma first.

Lemma 20.

The following statements hold.

$(\mathrm{i})$ If $Z\in {\Re }_{\mathrm{I}}(m,n)$, then $0<1-{\displaystyle \frac{{|Z|}^2}{m}}\le 1$ and for all $Z,W\in {\Re }_{\mathrm{I}}(m,n)$ we have

$$\left(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{m}}\right)+\left(1-{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{m}}\right)\le 2|\left(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{m}}\right)|.$$

$(\mathrm{ii})$ If $Z\in {\Re }_{\mathrm{II}}(p)$, then $0<1-{\displaystyle \frac{{|Z|}^2}{p}}\le 1$, and for all $Z,W\in {\Re }_{\mathrm{II}}(p)$ we have

$$\left(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{p}}\right)+\left(1-{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{p}}\right)\le 2|\left(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{p}}\right)|.$$

$(\mathrm{iii})$ If $Z\in {\Re }_{\mathrm{III}}(q)$, then $0<1-{\displaystyle \frac{{|Z|}^2}{[\frac{q}{2}]}}\le 1$, and for all $Z,W\in {\Re }_{\mathrm{III}}(q)$ we have

$$\left(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{2[\frac{q}{2}]}}\right)+\left(1-{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{2[\frac{q}{2}]}}\right)\le 2|\left(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{2[\frac{q}{2}]}}\right)|.$$

$(\mathrm{iv})$ If $z\in {\Re }_{\mathrm{IV}}(N)$, then $0<1-{|z|}^2\le 1$, and for all $z,w\in {\Re }_{\mathrm{IV}}(N)$ we have

$${(1-|z|}^2{)+(1-|w|}^2)\le 2|1-z{\overline{w}}^{\prime }|.$$

Proof. 

$(\mathrm{i})$ For $Z\in {\Re }_{\mathrm{I}}(m,n)$, we have $0\le {|Z|}^2=\mathrm{tr}(Z{\overline{Z}}^{\prime })<m$, that is, $0<1-{\displaystyle \frac{{|Z|}^2}{m}}\le 1$. According to the two inequalities $|\mathrm{tr}(Z{\overline{W}}^{\prime }){|}^2\le \mathrm{tr}(Z{\overline{Z}}^{\prime })\mathrm{tr}(W{\overline{W}}^{\prime })$ and $a^2+b^2\ge 2ab$, we obtain

$$\begin{array}{cc}\hfill 2|(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{m}})|& \ge 2-2{\displaystyle \frac{|\mathrm{tr}(Z{\overline{W}}^{\prime })|}{m}}\ge 2-2{\displaystyle \frac{\sqrt{\mathrm{tr}(Z{\overline{Z}}^{\prime })}\sqrt{\mathrm{tr}(W{\overline{W}}^{\prime })}}{m}}\hfill \\ & =2-2\sqrt{{\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{m}}}\sqrt{{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{m}}}\ge 2-({\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{m}}+{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{m}})\hfill \\ & =1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{m}}+1-{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{m}}.\hfill \end{array}$$

$(\mathrm{ii})$ For $Z\in {\Re }_{\mathrm{II}}(p)$, we may write

$$Z=\left(\begin{array}{cccc}{z}_{11}& {\displaystyle \frac{1}{\sqrt{2}}}{z}_{12}& \dots & {\displaystyle \frac{1}{\sqrt{2}}}{z}_{1p}\\ {\displaystyle \frac{1}{\sqrt{2}}}{z}_{21}& z_{22}& \dots & {\displaystyle \frac{1}{\sqrt{2}}}{z}_{2p}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{\sqrt{2}}}{z}_{p1}& {\displaystyle \frac{1}{\sqrt{2}}}{z}_{p2}& \dots & z_{pp}\end{array}\right),$$

then $0\le {|Z|}^2=\mathrm{tr}(Z{\overline{Z}}^{\prime })<p$, and we easily obtain the proof.

$(\mathrm{iii})$ For $Z\in {\Re }_{\mathrm{III}}(q)$, we have $0\le {|Z|}^2={\displaystyle \frac{1}{2}}\mathrm{tr}(Z{\overline{Z}}^{\prime })<[{\displaystyle \frac{q}{2}}]$, that is, $0<1-{\displaystyle \frac{{|Z|}^2}{[\frac{q}{2}]}}\le 1$. According to the two inequalities $|\mathrm{tr}(Z{\overline{W}}^{\prime }){|}^2\le \mathrm{tr}(Z{\overline{Z}}^{\prime })\mathrm{tr}(W{\overline{W}}^{\prime })$ and $a^2+b^2\ge 2ab$, we obtain

$$\begin{array}{cc}\hfill 2|(1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{2[\frac{q}{2}]}})|& \ge 2-2{\displaystyle \frac{|\mathrm{tr}(Z{\overline{W}}^{\prime })|}{2[\frac{q}{2}]}}\ge 2-2{\displaystyle \frac{\sqrt{\mathrm{tr}(Z{\overline{Z}}^{\prime })}\sqrt{\mathrm{tr}(W{\overline{W}}^{\prime })}}{2[\frac{q}{2}]}}\hfill \\ & =2-2\sqrt{{\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{2[\frac{q}{2}]}}}\sqrt{{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{2[\frac{q}{2}]}}}\ge 2-({\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{2[\frac{q}{2}]}}+{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{2[\frac{q}{2}]}})\hfill \\ & =1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{Z}}^{\prime })}{2[\frac{q}{2}]}}+1-{\displaystyle \frac{\mathrm{tr}(W{\overline{W}}^{\prime })}{2[\frac{q}{2}]}}.\hfill \end{array}$$

$(\mathrm{iv})$ For $z\in {\Re }_{\mathrm{IV}}(N)$, we have $0\le {|z|}^2<1$, that is, $0<1-{|z|}^2\le 1$. According to the two inequalities $|z{\overline{w}}^{\prime }|\le |z||w|$ and $a^2+b^2\ge 2ab$, we obtain

$$\begin{array}{c}\hfill 2|1-z{\overline{w}}^{\prime }|\ge 2-2|z{\overline{w}}^{\prime }{|\ge 2-2|z||w|\ge 2-(|z|}^2+{|w|}^2{)=1-|z|}^2+1-{|w|}^2.\end{array}$$

This completes the proof of the lemma. □

The above Lemma shows that $1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{m}},Z,W\in {\Re }_{\mathrm{I}}(m,n);$$1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{p}},Z,W\in {\Re }_{\mathrm{II}}(p);$ $1-{\displaystyle \frac{\mathrm{tr}(Z{\overline{W}}^{\prime })}{2[\frac{q}{2}]}},Z,W\in {\Re }_{\mathrm{III}}(q)$ and $(1-z{\overline{w}}^{\prime }),z,w\in {\Re }_{\mathrm{IV}}(N)$ satisfy conditions (1) and (2). Therefore, the generalized ellipsoidal-type domains are special cases of the generalized Hua–Cartan–Hartogs domains, and we may obtain the boundedness and compactness conditions for weighted composition operators between Bers-type spaces on these domains.

Assume that $\varphi$ is a holomorphic self-map of $\mathrm{L}\in \{{\mathrm{L}}_{\mathrm{I}},{\mathrm{L}}_{\mathrm{II}},{\mathrm{L}}_{\mathrm{III}},{\mathrm{L}}_{\mathrm{IV}}\}$. Let us write $(W,\eta )=\varphi (Z,\xi )$ for $(Z,\xi )\in {\mathrm{L}}_{\mathrm{I}},{\mathrm{L}}_{\mathrm{II}},{\mathrm{L}}_{\mathrm{III}}$, and write $(w,\eta )=\varphi (z,\xi )$ for $(z,\xi )\in {\mathrm{L}}_{\mathrm{IV}}$.

Corollary 7.

Given $\alpha ,\beta >0$, a holomorphic self-map ϕ of $\mathrm{L}$, and $\psi \in H(\mathrm{L})$, then the following statements hold.

$(\mathrm{i})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{I}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{I}})$ is bounded iff

$${\tilde{Q}}_I=\underset{(Z,\xi )\in {\mathrm{L}}_{\mathrm{I}}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{(1-\frac{{|Z|}^2}{m}-{\parallel \xi \parallel }^2{)}^{\beta }}{(1-\frac{{|W|}^2}{m}-{\parallel \eta \parallel }^2{)}^{\alpha }}}<+\infty .$$

$(\mathrm{ii})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{II}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{II}})$ is bounded iff

$${\tilde{Q}}_{II}=\underset{(Z,\xi )\in {\mathrm{L}}_{\mathrm{II}}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{(1-\frac{{|Z|}^2}{p}-{\parallel \xi \parallel }^2{)}^{\beta }}{(1-\frac{{|W|}^2}{p}-{\parallel \eta \parallel }^2{)}^{\alpha }}}<+\infty .$$

$(\mathrm{iii})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{III}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{III}})$ is bounded iff

$${\tilde{Q}}_{III}=\underset{(Z,\xi )\in {\mathrm{L}}_{\mathrm{III}}}{sup}|\psi (Z,\xi )|{\displaystyle \frac{(1-\frac{{|Z|}^2}{[\frac{q}{2}]}-{\parallel \xi \parallel }^2{)}^{\beta }}{(1-\frac{{|W|}^2}{[\frac{q}{2}]}-{\parallel \eta \parallel }^2{)}^{\alpha }}}<+\infty .$$

$(\mathrm{iv})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{IV}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{IV}})$ is bounded iff

$${\tilde{Q}}_{IV}=\underset{(z,\xi )\in {\mathrm{L}}_{\mathrm{IV}}}{sup}|\psi (z,\xi )|{\displaystyle \frac{{(1-|z|}^2-{\parallel \xi \parallel }^2{)}^{\beta }}{{(1-|w|}^2-{\parallel \eta \parallel }^2{)}^{\alpha }}}<+\infty .$$

Corollary 8.

Given $\alpha ,\beta >0$, a holomorphic self-map ϕ of $\mathrm{L}$, and $\psi \in H(\mathrm{L})$, then the following statements hold.

$(\mathrm{i})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{I}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{I}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{I}})$ and

$$\underset{\varphi (Z,\xi )\to \partial {\mathrm{L}}_{\mathrm{I}}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{(1-\frac{{|Z|}^2}{m}-{\parallel \xi \parallel }^2{)}^{\beta }}{(1-\frac{{|W|}^2}{m}-{\parallel \eta \parallel }^2{)}^{\alpha }}}=0.$$

$(\mathrm{ii})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{II}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{II}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{II}})$ and

$$\underset{\varphi (Z,\xi )\to \partial {\mathrm{L}}_{\mathrm{II}}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{(1-\frac{{|Z|}^2}{p}-{\parallel \xi \parallel }^2{)}^{\beta }}{(1-\frac{{|W|}^2}{p}-{\parallel \eta \parallel }^2{)}^{\alpha }}}=0.$$

$(\mathrm{iii})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{III}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{III}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{III}})$ and

$$\underset{\varphi (Z,\xi )\to \partial {\mathrm{L}}_{\mathrm{III}}}{lim}|\psi (Z,\xi )|{\displaystyle \frac{(1-\frac{{|Z|}^2}{[\frac{q}{2}]}-{\parallel \xi \parallel }^2{)}^{\beta }}{(1-\frac{{|W|}^2}{[\frac{q}{2}]}-{\parallel \eta \parallel }^2{)}^{\alpha }}}=0.$$

$(\mathrm{iv})$ The weighted composition operator $\psi C_{\varphi }:{\mathcal{A}}_{\alpha }({\mathrm{L}}_{\mathrm{IV}})\to {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{IV}})$ is compact iff $\psi \in {\mathcal{A}}_{\beta }({\mathrm{L}}_{\mathrm{IV}})$ and

$$\underset{\varphi (z,\xi )\to \partial {\mathrm{L}}_{\mathrm{IV}}}{lim}|\psi (z,\xi )|{\displaystyle \frac{{(1-|z|}^2-{\parallel \xi \parallel }^2{)}^{\beta }}{{(1-|w|}^2-{\parallel \eta \parallel }^2{)}^{\alpha }}}=0.$$

Remark 2.

In this paper, we define a very general class of domains, called generalized Hua–Cartan–Hartogs domains, and further define Bers-type function spaces on generalized Hua–Cartan–Hartogs domains. Then, the boundedness and compactness of weighted composition operators between Bers-type function spaces on generalized Hua–Cartan–Hartogs domain are discussed. We further demonstrate the following: generalized Hua domains, generalized Cartan–Hartogs domains, generalized Cartan–Hartogs domains over different Cartan domains and generalized ellipsoidal-type domains are all special cases of the generalized Hua–Cartan–Hartogs domain. Thus, as corollaries, the necessary and sufficient conditions for the boundedness and compactness of weighted composition operators between Bers-type spaces on the above four domains can be obtained immediately. The following issues can be further considered:

(1) As for the generalized Hua domain of the third type, we have not been able to prove whether it satisfies condition (Equation (2)), so we cannot prove that the generalized Hua domain of the third type is also a special case of the generalized Hua–Cartan–Hartogs domain; thus, our results cannot be applied to the generalized Hua domains of the third type. However, we guess that our results are also valid when applied to the generalized Hua domains of the third type.

(2) We can further consider the essential norm estimation, properties of spectrum and topological properties of weighted composition operators between Bers-type spaces on the generalized Hua–Cartan–Hartogs domain.

(3) The properties of various operators between other function spaces on the generalized Hua–Cartan–Hartogs domain can also be considered.