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Multiplication operators on weighted Bloch spaces of the first Cartan domains
Journal of Inequalities and Applications volume 2020, Article number: 197 (2020)
Abstract
Let \(\Re_{I}(m,n)\) be the first Cartan domain. Motivated by some results of the multiplication operators on the holomorphic function spaces on the unit ball of \({\mathbb {C}}^{n}\), we study multiplication operators on weighted Bloch spaces of the first Cartan domain and obtain some necessary or sufficient conditions for the boundedness and compactness in this paper.
1 Introduction
The well-known first Cartan domain (see [22]) is defined as
where ZÌ… denotes the conjugate of the matrix Z, \(Z^{T}\) denotes the transpose of Z, and m, n are positive integers. For the sake of convenience, it is denoted by \(\Re_{I}\).
Let \(H(\Re_{I})\) be the space of all holomorphic functions on \(\Re_{I}\). For \(\alpha\geq0\), the weighted-type space \(H_{\alpha}^{\infty}(\Re _{I})\) on \(\Re_{I}\) consists of all \(f\in H(\Re_{I})\) such that
The little weighted-type space \(H_{\alpha,0}^{\infty}(\Re_{I})\) on \(\Re_{I}\) consists of all \(f\in H(\Re_{I})\) such that
If \(\alpha=0\), then \(H_{\alpha}^{\infty}(\Re_{I})\) is denoted as \(H^{\infty}(\Re_{I})\) and \(H_{\alpha,0}^{\infty}(\Re_{I})\) is denoted as \(H_{0}^{\infty}(\Re_{I})\). The weighted-type spaces on the unit disk or the unit ball frequently appear; see, for example, [6, 11, 12, 16, 17, 19].
Let \(\mathbb{B}=\{z\in {\mathbb {C}}^{n}:|z|<1\}\) be the open unit ball of \({\mathbb {C}}^{n}\). It is obvious that \(\mathbb{B}=\Re_{I}(1, n)\), which shows that \(\Re_{I}(m,n)\) is a generalization of \(\mathbb{B}\). The weighted Bloch space on \(\mathbb{B}\), usually denoted by \(\mathcal{B}^{\alpha}(\mathbb{B})\), consists of all \(f\in H(\mathbb{B})\) such that
where
It is well known that the quantity \(b(f)\) is a seminorm of \(\mathcal{B}^{\alpha }(\mathbb{B})\) and under the norm
\(\mathcal{B}^{\alpha}(\mathbb{B})\) is a Banach space. The weighted Bloch spaces also frequently appear in the literature; see, for example, [2, 13–15, 27].
Similarly, the weighted Bloch space \(\mathcal{B}^{\alpha}(\Re_{I})\) on \(\Re _{I}\) consists of all \(f\in H(\Re_{I})\) such that
Under the norm
\(\mathcal{B}^{\alpha}(\Re_{I})\) is a Banach space. The little Bloch space \(\mathcal{B}_{0}^{\alpha}(\Re_{I})\) on \(\Re_{I}\) consists of all \(f\in H(\Re_{I})\) such that
Let X and Y be two function spaces on \(\Re_{I}\). If it follows that \(\psi f\in Y\) for all \(f\in X\), then ψ is called a multiplier from X to Y, and usually \(M_{\psi}:f\mapsto\psi f\) is called a multiplication operator from X to Y. In general, ψf cannot necessarily belong to Y for some \(f\in X\). In order to explain this fact, we need to introduce the Bloch space \(\mathcal{B}(U^{2})\) (see [24]), where \(U^{2}=\{z=(z_{1},z_{2}):|z_{1}|<1,|z_{2}|<1\}\) is the unit polydisk in \({\mathbb {C}}^{2}\). We say a holomorphic function f belongs to \(\mathcal{B}(U^{2})\) if f satisfies the condition
If we consider
in \(\mathcal{B}(U^{2})\), it is easy to see that ψf does not belong to \(\mathcal{B}(U^{2})\) for \(\psi(z)=z_{1}\). Hence, a natural problem is to find some conditions when ψ satisfies \(M_{\psi}f\in Y\) for all \(f\in X\). It is well known that the theory of the multipliers or multiplication operators on function spaces has been studied for a long time. In 1966, Talyor started an investigation of the multipliers on \(D_{\alpha }\) in [21]. Later on, for example, Stegenga considered the multipliers of the Dirichlet space in [10]. Now multipliers or multiplication operators between or on various function spaces of the classical domains have been studied by many authors (see, for example, [5, 7, 23, 26]). But beyond that, there is a great interest in some generalizations of the multiplication operators on classical domains (for example, see [1, 4, 8, 15, 18] for the weighted composition operators). However, we do not find any result of multipliers or multiplication operators on the holomorphic function spaces of the first Cartan domain. In this paper, we study the multiplication operators on weighted Bloch spaces of the first Cartan domain and obtain some necessary or sufficient conditions for the boundedness and compactness.
For \(Z=(z_{ij})_{m\times n}\in {\mathbb {C}}^{m\times n}\), let \(|Z|^{2}=\sum_{\substack{1\leq i\leq m\\1\leq j\leq n}}|z_{ij}|^{2}\). Constants are denoted by C, they are positive and may differ from one occurrence to the next. The notation \(a\lesssim b\) means that there exists a positive constant C independent of the essential variables in the quantities a and b such that \(a\leq Cb\).
2 Some lemmas
First, we have the following obvious result.
Lemma 1
Let\(Z=(z_{ij})_{m\times n}\in\Re_{I}\), then\(|z_{ij}|<1\)for alliandj.
Proof
Let \(Z= (z_{ij} )_{m\times n}\in\Re_{I}\). Then \(I-Z\overline {Z}^{T}>0\). So, we have \(1-\sum_{j=1}^{n}|z_{ij}|^{2}>0\) for \(i=1,2,\ldots,n\), from which the desired result follows. □
By a direct calculation, we obtain the following formula.
Lemma 2
Let\(f\in H(\Re_{I})\). Then, for all\(Z\in \Re_{I}\), it follows that
We need the following result (see [20]) to obtain the point evaluation estimate for the Bloch functions.
Lemma 3
Let\(Z\in\Re_{I}\). Then there exist two unitary matricesUandVsuch that
where\(1>\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{m}\geq 0\)and\(\lambda_{1}^{2},\ldots,\lambda_{m}^{2}\)are eigenvalues of\(Z\overline{Z}^{T}\).
Lemma 4
Let\(\alpha>0\). Then there exists a positive constantCindependent of\(f\in \mathcal{B}^{\alpha}(\Re_{I})\)and\(Z\in\Re_{I}\)such that
Proof
If \(Z=0\), then the result holds obviously. Now, assume that \(Z=(z_{ij})_{m\times n}\neq0\). It follows from Lemma 3 that there exist two unitary matrices U and V such that
where \(1>\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{m}\geq 0\) and \(\lambda_{1}^{2},\ldots,\lambda_{m}^{2}\) are eigenvalues of \(Z\overline{Z}^{T}\). By (2), we have
It follows from (3) that
Assume \(t\in[0,1]\). Since \(\lambda_{1}\geq\lambda_{2}\geq\cdots \geq\lambda_{m}\geq0\), for each \(i\in\{1,2,\ldots,m\}\), we have
By this, we have
From the facts
and \(|Z|^{2}=\operatorname{tr}(Z\overline{Z}^{T})\), we obtain
Then, from (4) and (5) we obtain
where (7) is obtained by (6) by using the elementary fact
This completes the proof. □
Remark 1
In Lemma 4, there exists a parameter m which maybe is the biggest difference from the weighted Bloch spaces on the unit ball. Unfortunately, we do not find an effective method to avoid it. However, this also shows that this result is a generalization of the corresponding result on \(\mathcal{B}^{\alpha }(\mathbb{B})\).
In order to study the compactness of the operator \(M_{\psi}\) on \(\mathcal{B}^{\alpha }(\Re_{I})\), we need the following result which is similar to Proposition 3.11 in [3].
Lemma 5
Let\(\alpha>0\)andψbe a holomorphic function on\(\Re_{I}\). Then the bounded operator\(M_{\psi}\)is compact on\(\mathcal{B}^{\alpha}(\Re_{I})\)if and only if for every bounded sequence\(\{f_{n}\}\)in\(\mathcal{B}^{\alpha }(\Re_{I})\)such that\(f_{n}\rightarrow0\)uniformly on any compact subset of\(\Re_{I}\)as\(n\rightarrow\infty\), it follows that
Proof
Suppose that the bounded operator \(M_{\psi}\) is compact on \(\mathcal{B}^{\alpha}(\Re_{I})\). Let \(\{{f_{n}}\}\) be a bounded sequence in \(\mathcal{B}^{\alpha}(\Re_{I})\) such that \(f_{n}\rightarrow0\) uniformly on any compact subset of \(\Re_{I}\) as \(n\rightarrow\infty \). If \(\|M_{\psi}f_{n}\|_{\mathcal{B}^{\alpha}(\Re_{I})}\nrightarrow0\) as \(n\rightarrow\infty\), then there exists a subsequence \(\{f_{n_{j}}\}\) of \(\{f_{n}\}\) such that
Since \(M_{\psi}\) is compact on \(\mathcal{B}^{\alpha}(\Re_{I})\), there exist a function \(g\in \mathcal{B}^{\alpha}(\Re_{I})\) and a subsequence of \(\{f_{n_{j}}\}\) (without loss of generality, still written by \(\{f_{n_{j}}\}\)) such that
Let K be a compact subset of \(\Re_{I}\). From Lemma 4, it follows that \(M_{\psi}f_{n_{j}}-g\rightarrow0\) uniformly on K as \(j\rightarrow \infty\). From this, for \(\varepsilon>0\), there exists a positive integer \(N_{1}\) such that
for all \(Z\in K\), whenever \(j>N_{1}\). Since \(f_{n_{j}}\rightarrow0\) uniformly on K as \(j\rightarrow\infty\), also there exists a positive integer \(N_{2}\) such that \(|f_{n_{j}}(Z)|<\varepsilon\) for all \(Z\in K\), whenever \(j>N_{2}\). Let \(N=\max\{N_{1},N_{2}\}\) and \(M=\max_{Z\in K}|\psi(Z)|\). From (9), we have
for all \(Z\in K\), whenever \(j>N\). From (10) and the arbitrariness of ε, we obtain \(g(Z)=0\) for all \(Z\in K\), which leads to \(g\equiv0\) on \(\Re _{I}\). This shows that \(\lim_{j\rightarrow\infty}\|M_{\psi}f_{n_{j}}\|_{\mathcal{B}^{\alpha}(\Re_{I})}=0\) which contradicts (8).
Now suppose that \(\{f_{n}\}\) is a bounded sequence in \(\mathcal{B}^{\alpha }(\Re_{I})\). Then it is locally uniformly bounded on \(\Re_{I}\), which shows that there exists a subsequence \(\{f_{n_{j}}\}\) of \(\{f_{n}\}\) such that \(f_{n_{j}}\to f\) uniformly on every compact subset of \(\Re_{I}\) as \(j\rightarrow\infty\). From this, we have \(f_{n_{j}}-f\to0\) uniformly on every compact subset of \(\Re_{I}\) as \(j\rightarrow\infty\). Consequently, we obtain
which shows that \(M_{\psi}\) is compact on \(\mathcal{B}^{\alpha}(\Re_{I})\). □
In the studies of the several complex variables, the mathematician Loo-Keng Hua found the following matrix inequality in 1955.
Lemma 6
Let\(I-A\overline{A}^{T}\)and\(I-B\overline{B}^{T}\)be two Hermitian and positive definite matrices. Then
By the way, as an easy application of Lemma 6, we see that, for each \(Z, S\in\Re_{I}\), the matrix \(I-Z\overline{S}^{T}\) is reversible. We also have the following result (see [20]).
Lemma 7
There exists a positive constantCindependent of all\(Z,S\in\Re_{I}\)such that
where\(I_{ij}\)is an\(m\times n\)matrix whose element of theith row and thejth column is 1, and the other elements are 0.
Let S be a fixed point in \(\Re_{I}\). If \(\alpha=\frac{1}{2}\), on \(\Re_{I}\) we define the function
and if \(\alpha\neq\frac{1}{2}\), on \(\Re_{I}\) we define the function
To prove that \(f_{S}\) belongs to \(\mathcal{B}^{\frac{1}{2}}(\Re_{I})\) (or \(g_{S}\) belongs to \(\mathcal{B}^{a}(\Re_{I})\) for \(\alpha \neq\frac{1}{2}\)), let us recall the definition of the function matrix derivative.
Let
and each \(y_{ij}(x)\) be differentiable on the interval I. Then the well-known derivative of \(Y(x)\) is defined as
If we regard \(\operatorname{det}(I-Z\overline{S}^{T})\) as a function of Z, we have the following result.
Lemma 8
LetSbe a fixed point in\(\Re_{I}\). Then on\(\Re_{I}\)it follows that
Proof
Let \(A(x)\) be a differentiable function matrix and \(\operatorname {det}A(x)\neq0\) for each \(x\in I\). Then by the formula (see [9])
we have
By the definition of the function matrix derivative, it is easy to see that
From (13) and (14), the desired result follows. □
Lemma 9
Let\(\alpha>0\). For the fixed point\(S\in \Re_{I}\), the following statements hold.
-
(i)
If\(\alpha=\frac{1}{2}\), then the function\(f_{S}\)belongs to\(\mathcal{B}^{\frac{1}{2}}(\Re_{I})\). Moreover,
$$\begin{aligned} \sup_{S\in\Re_{I}} \Vert f_{S} \Vert _{\mathcal{B}^{\frac{1}{2}}(\Re_{I})}\lesssim 1. \end{aligned}$$(15) -
(ii)
If\(\alpha\neq\frac{1}{2}\), then the function\(g_{S}\)belongs to\(\mathcal{B}^{\alpha }(\Re_{I})\). Moreover,
$$\begin{aligned} \sup_{S\in\Re_{I}} \Vert g_{S} \Vert _{\mathcal{B}^{\alpha}(\Re_{I})}\lesssim 1. \end{aligned}$$(16)
Proof
We first prove (i). By Lemma 8, we have
Then we obtain
By Lemmas 6–8 and (17), we obtain
From (18), it follows that \(f_{S}\in \mathcal{B}^{\frac{1}{2}}(\Re_{I})\) and (15) holds.
Next, we prove (ii). Obviously, we have
Then
Also by Lemmas 6–8 and (19), we have
From (20), it follows that \(g_{S}\in \mathcal{B}^{\alpha}(\Re_{I})\) and (16) holds. □
Remark 2
It is easy to see that \({f_{S}}\) and \({g_{S}}\) uniformly converge to zero on any compact subset of \(\Re_{I}\) as \(S\rightarrow \partial\Re_{I}\).
3 Boundedness and compactness of \(M_{\psi}\) on \(\mathcal{B}^{\alpha }(\Re_{I})\)
First, we have the following result of the operator \(M_{\psi}\) on \(\mathcal{B}^{\alpha}(\Re_{I})\).
Theorem 1
Let\(\alpha>0\)and\(\psi\in H(\Re _{I})\). Then the following statements hold.
-
(i)
For\(0< m\alpha<1\), if\(\psi\in H^{\infty}(\Re_{I})\cap \mathcal{B}^{\alpha}(\Re_{I})\), then the operator\(M_{\psi}\)is bounded on\(\mathcal{B}^{\alpha}(\Re_{I})\).
-
(ii)
For\(m\alpha=1\), if\(\psi\in H^{\infty}(\Re_{I})\)and
$$\begin{aligned} M_{1}:=\sup_{Z\in\Re_{I}} \bigl[\operatorname{det}\bigl(I-Z \overline{Z}^{T}\bigr) \bigr]^{\alpha} \bigl\vert \nabla \psi(Z) \bigr\vert \log\frac{2}{\operatorname{det}(I-Z\overline{Z}^{T})}< +\infty, \end{aligned}$$(21)then the operator\(M_{\psi}\)is bounded on\(\mathcal{B}^{\alpha}(\Re _{I})\).
-
(iii)
For\(m\alpha>1\), if\(\psi\in H^{\infty}(\Re_{I})\)and
$$\begin{aligned} M_{2}:=\sup_{Z\in\Re_{I}}\frac{ \vert \nabla\psi(Z) \vert }{ [\operatorname{det}(I-Z\overline{Z}^{T})]^{m\alpha-\alpha-1}}< + \infty, \end{aligned}$$(22)then the operator\(M_{\psi}\)is bounded on\(\mathcal{B}^{\alpha}(\Re _{I})\).
-
(iv)
If the operator\(M_{\psi}\)is bounded on\(\mathcal{B}^{\alpha}(\Re _{I})\), then\(\psi\in H_{\alpha}^{\infty}(\Re_{I})\cap \mathcal{B}^{\alpha}(\Re _{I})\).
Proof
We first prove (i). For \(f\in \mathcal{B}^{\alpha}(\Re_{I})\), by Lemma 4, we have
Then from (23) we obtain
From the assumption and (24), it follows that the operator \(M_{\psi}\) is bounded on \(\mathcal{B}^{\alpha}(\Re_{I})\).
We prove statement (ii). For each \(f\in \mathcal{B}^{\alpha}(\Re_{I})\), by Lemmas 2 and 4, we have
By (25), we obtain
which shows that the operator \(M_{\psi}\) is bounded on \(\mathcal{B}^{\alpha }(\Re_{I})\).
Statement (iii) can be obtained similarly. Here we omit.
Now, we begin to prove (iv). Choose \(f(Z)\equiv1\) on \(\Re_{I}\). Then by the boundedness of the operator \(M_{\psi}\) on \(\mathcal{B}^{\alpha }(\Re_{I})\), we have
which shows
that is, \(\psi\in \mathcal{B}^{\alpha}(\Re_{I})\). Again applying the boundedness of the operator \(M_{\psi}\) on \(\mathcal{B}^{\alpha}(\Re_{I})\) to the function \(g(Z)=z_{11}\), by Lemma 1 and (27) we obtain
By (28), we have
that is, \(\psi\in H_{\alpha}^{\infty}(\Re_{I})\). Combining (27) and (29), we obtain \(\psi\in H_{\alpha}^{\infty}(\Re_{I})\cap \mathcal{B}^{\alpha}(\Re_{I})\). □
Next, we discuss the compactness of the operator \(M_{\psi}\) on \(\mathcal{B}^{\alpha}(\Re_{I})\).
Theorem 2
Let\(\alpha>0\)andψbe the holomorphic function on\(\Re_{I}\). Then the following statements hold.
-
(i)
For\(0< m\alpha<1\), if\(\psi\in H_{0}^{\infty}(\Re_{I})\cap \mathcal{B}_{0}^{\alpha}(\Re_{I})\), then the operator\(M_{\psi}\)is compact on\(\mathcal{B}^{\alpha}(\Re _{I})\).
-
(ii)
For\(m\alpha=1\), if\(\psi\in H_{0}^{\infty}(\Re_{I})\cap \mathcal{B}^{\alpha}(\Re_{I})\)and
$$\begin{aligned} \lim_{Z\rightarrow\partial\Re_{I}} \bigl[\operatorname{det}\bigl(I-Z\overline {Z}^{T} \bigr) \bigr]^{\alpha} \bigl\vert \nabla\psi(Z) \bigr\vert \log \frac{2}{\operatorname{det}(I-Z\overline{Z}^{T})}=0, \end{aligned}$$(30)then the operator\(M_{\psi}\)is compact on\(\mathcal{B}^{\alpha}(\Re _{I})\).
-
(iii)
For\(m\alpha>1\), if\(\psi\in H_{0}^{\infty}(\Re_{I})\cap \mathcal{B}^{\alpha}(\Re_{I})\)and
$$\begin{aligned} \lim_{Z\rightarrow\partial\Re_{I}}\frac{ \vert \nabla\psi(Z) \vert }{ [\operatorname{det}(I-Z\overline{Z}^{T}) ]^{m\alpha-\alpha-1}}=0, \end{aligned}$$(31)then the operator\(M_{\psi}\)is compact on\(\mathcal{B}^{\alpha}(\Re _{I})\).
Proof
Here we only prove statement (ii). Statements (i) and (iii) can be similarly proved. By Lemma 5 we only need to prove that, if \(\{f_{i}\}\) is a sequence in \(\mathcal{B}^{\alpha}(\Re_{I})\) such that \(\sup_{i\in {\mathbb {N}}}\|f_{i}\|_{\mathcal{B}^{\alpha}(\Re_{I})}\leq M\) and \(f_{i}\rightarrow0\) uniformly on any compact subset of \(\Re_{I}\) as \(i\rightarrow\infty\), then \(\lim_{i\rightarrow\infty}\|M_{\psi}f_{i}\|_{\mathcal{B}^{\alpha}(\Re _{I})}=0\). We observe that \(\psi\in H_{0}^{\infty}(\Re_{I})\) and condition (30) imply that, for every \(\varepsilon>0\), there exists \(\sigma>0\) such that on \(K=\{Z\in\Re_{I}:\operatorname{dist}(Z,\partial\Re_{I})<\sigma\}\) it follows that
For such ε and σ, by using (32) and Lemma 4, we have
It is obvious to see that if \(f_{i}\rightarrow0\) uniformly on a compact subset of \(\Re_{I}\) as \(i\rightarrow\infty\), then \(|\nabla f_{i}(Z)|\) does as \(i\rightarrow\infty\). Since \(\Re_{I}\setminus K\) is a compact subset of \(\Re_{I}\), \(f_{i}\rightarrow0\) uniformly on \(\Re_{I}\setminus K\) as \(i\rightarrow\infty\). From this and (33), we get
which shows that \(M_{\psi}\) is compact on \(\mathcal{B}^{\alpha}(\Re_{I})\). □
Theorem 3
Let\(\alpha>0\)and\(\psi\in H(\Re _{I})\). Then the following statements hold.
-
(i)
For\(\alpha=\frac{1}{2}\), if the operator\(M_{\psi}\)is compact on\(\mathcal{B}^{\alpha}(\Re_{I})\), then
$$\begin{aligned}& \lim_{Z\to\partial\Re_{I}}\operatorname{det}\bigl(I-Z\overline{Z}^{T}\bigr) \biggl\{ \sum_{\substack{1\leq i\leq m\\1\leq j\leq n}} \biggl\vert \frac {\partial\psi(Z)}{\partial z_{ij}}\log\frac{2}{\operatorname{det}(I-Z\overline{Z}^{T})}\\& \quad{} +\psi(Z)\operatorname{tr} \bigl[\bigl(I-Z \overline{Z}^{T}\bigr)^{-1}I_{ij} \overline{Z}^{T} \bigr] \biggr\vert ^{2} \biggr\} ^{\frac{1}{2}}=0. \end{aligned}$$ -
(ii)
For\(\alpha\neq\frac{1}{2}\), if the operator\(M_{\psi}\)is compact on\(\mathcal{B}^{\alpha}(\Re_{I})\), then
$$\begin{aligned} \lim_{Z\rightarrow\partial\Re_{I}}\operatorname{det}\bigl(I-Z\overline{Z}^{T}\bigr) \biggl\{ \sum_{\substack{1\leq i\leq m\\1\leq j\leq n}} \biggl\vert \frac {1}{2\alpha-1}\frac{\partial\psi(Z)}{\partial z_{ij}}+\psi(Z)\operatorname{tr} \bigl[\bigl(I-Z \overline{Z}^{T}\bigr)^{-1}I_{ij}\overline {Z}^{T} \bigr] \biggr\vert ^{2} \biggr\} ^{\frac{1}{2}}=0. \end{aligned}$$
Proof
We first prove statement (i). Suppose that the operator \(M_{\psi}\) is compact on \(\mathcal{B}^{\alpha}(\Re_{I})\). Consider a sequence \(\{S_{n}\}\) in \(\Re _{I}\) such that \(S_{n}\to\partial\Re_{I}\) as \(n\to\infty\). Using this sequence, we define the functions
Then, by Lemma 9, we know that the sequence \(\{f_{n}\}\) is uniformly bounded in \(\mathcal{B}^{\alpha}(\Re_{I})\) and uniformly converges to zero on any compact subset of \(\Re_{I}\) as \(n\to\infty\). Hence, by Lemma 5,
Using (35), it follows from a direct computation that (i) holds.
Consider
We can similarly prove statement (ii) and the details are omitted. □
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Acknowledgements
The author would like to thank the anonymous referee for providing valuable comments for the improvement of this paper. This work was supported by the Sichuan Science and Technology Program (2018JY0200) and the Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (2016QZJ01).
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Jiang, Zj. Multiplication operators on weighted Bloch spaces of the first Cartan domains. J Inequal Appl 2020, 197 (2020). https://doi.org/10.1186/s13660-020-02460-3
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DOI: https://doi.org/10.1186/s13660-020-02460-3