Abstract
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the open unit ball of a complex Banach space \({\mathbb {X}}\), which may beinfinite dimensional. The weighted composition operator and weighted space defined on \({\mathbf {B}}_{{\mathbb {X}}}\) are introduced. We obtain the boundedness and compactness of the weightedcomposition operator from the Bloch-type spaces to the weighted spaces, and some properties with the Bloch-type spaces are given. Our main results generalize theprevious works on the Euclidean unit ball \({\mathbb {B}}^n\) to the case of \({\mathbf {B}}_{{\mathbb {X}}}\).
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1 Introduction
Let \({\mathbb {C}}^n\) be the space of n-dimensional complex variables \(z=(z_1, z_2, \ldots , z_n)\). The unit ball
and \({\mathbb {B}}^1\equiv {\mathbb {U}}\) denotes the unit disk in \({\mathbb {C}}\). Let \(H({\mathbb {B}}^n)\) be the family of holomorphic function from \({\mathbb {B}}^n\) to \({\mathbb {C}}\).
A positive continuous function \(\mu \) on [0, 1) is called normal if there is \(\delta \in [0, 1)\) and \(0<a<b<\infty \) such that
Then a normal function \(\mu \) is strictly decreasing on \([\delta ,1)\) and \(\mu (r)\rightarrow 0\) as \(r\rightarrow 1.\) Denote by Aut\(({\mathbb {B}}^n)\) the holomorphic automorphism group of \({\mathbb {B}}^n\). If \(u\in H({\mathbb {B}}^n)\) and \(\varphi \in \) Aut\(({\mathbb {B}}^n)\), then the corresponding weighted composition operator is defined by
The boundedness and the compactness of the operators \(uC_{\varphi }\) on Bloch-type spaces, Zygmund spaces, Hardy spaces and the weighted Bergman space attract a lot ofattentions (see, e.g., [6, 8, 12, 14, 16, 18, 19, 22, 23]).
The classical Bloch functions on the open unit disk \({\mathbb {U}}\) have been widelystudied (see, e.g., [1, 17]), and the corresponding notion in higher dimension was firstintroduced by Hahn [11]. Timoney [20] studied in depth the Bloch functions on bounded homogeneous domain in \({\mathbb {C}}^n\) with using Bergman metric (also see, Allen and Colonna [2]). Furthermore, Blasco et al. [3, 4] extended the Bloch space on \({\mathbb {B}}^n\) to the case of unit ball \({\mathbb {B}}_{{\mathbb {H}}}\) of an infinite dimensional complex Hilbert space \({\mathbb {H}}\). The bounded symmetric domains in complex Banach spaces are exactly the open unit balls of JB*-triples which are complex Banach spaces equipped with aJordan triple structure. Moreover, a complex Banach space is a JB*-triple if and only if its open unit ball is homogeneous (see, e.g., Deng and Ouyang [7], Kaup [13]). Recently, Chu et al. [5] generalized the Bloch space on \({\mathbb {B}}^n\) to the case of an infinitedimensional bounded symmetric domain realized as the open unit ball of a JB*-triple \({\mathbb {X}}\) by taking the place of the Bergman metric with the Kobayashi metric (compare with definition in Timoney [20]). In addition, they obtain the criteria for boundedness andcompactness on composition operator between the Bloch spaces on infinitedimensional bounded symmetric domain. By [5], Hamada [10] continued to study the weighted composition operators from the Hardy space \(H^{\infty }\) to the Bloch space on bounded symmetric domains. Hamada [9] obtained the boundedness and compactness of the extended Cesàro operators between the Bloch-type spaces, which extended the results in Tang [21] to the case of unit ball of a infinite dimensional complex Banach space.
In this paper, we conform to the definitions of Bloch-type spaces and little Bloch-type spaces as [9], which generalize the corresponding spaces on \({\mathbb {B}}^n\) to the case of the open unit ball \({\mathbf {B}}_{{\mathbb {X}}}\) of an infinite dimensional complex Banach space \({\mathbb {X}}\) with arbitrary norm \(\Vert \cdot \Vert \). All the weighted composition operator and weighted space are extended to \({\mathbf {B}}_{{\mathbb {X}}}\) (see, Sect. 2). We study the boundedness and compactness of the weighted composition operator from \(\omega \)-Bloch space \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) (resp. little \(\omega \)-Bloch space \({\mathcal {B}}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\)) to the weighted space on \(H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) (resp. little weighted space \(H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}}))\) (see, Sect. 4). In Sect. 3, we give the relations between the Bloch-type spaces \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) and the little Bloch-type spaces \({\mathcal {B}}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\). Using the work of Hamada [9], we successfully construct two test functions (see, Lemma 2.4), which play a key role in the proof of our main results. Since the Bloch-type spaces \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) coincide with the Bloch-type spaces \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {H}}})\) when \({\mathbb {X}}\) is a complex Hilbert space \({\mathbb {H}}\) [see Remark 2.1 in (iii)], our main results extend the corresponding works on \({\mathbb {B}}^n\) (see, e.g., Krantz and Stević [12]) to the case of \({\mathbf {B}}_{{\mathbb {X}}}\) of an infinite dimensional complex Banach space.
2 Preliminaries and Auxiliary Results
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space \({\mathbb {X}}\) with arbitrary norm \(\Vert \cdot \Vert \). Let \(H({\mathbf {B}}_{{\mathbb {X}}})\) denote the set of holomorphic mappings from \({\mathbf {B}}_{{\mathbb {X}}}\) into \({\mathbb {C}}.\) For \(x\in {\mathbb {X}}{\setminus }\{0\}\), we define
Then \(T(x)\ne {{\emptyset }}\) in view of the Hahn–Banach theorem.
Let \(\omega \) be a normal function on [0, 1), and \(\omega \) can be extended to a function on \({\mathbf {B}}_{{\mathbb {X}}}\) by \(\omega (z)=\omega (\Vert z\Vert ).\) A function \(f\in H({\mathbf {B}}_{{\mathbb {X}}})\) is called a Bloch-type function with respect to \(\omega \) if
where \({\mathcal {R}}f(z)=Df(z)z\) is the radial derivative of f and Df(z) is the Fréchetderivative of f at z.
The class of all Bloch-type functions with respect to \(\omega \) on \({\mathbf {B}}_{{\mathbb {X}}}\) is called a Bloch-type space on \({\mathbf {B}}_{{\mathbb {X}}}\) and is denoted by \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\). With the norm
the Bloch-type space \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) becomes a Banach space (see, Proposition 2.5 in Hamada [9]).
The little Bloch-type space \({\mathcal {B}}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\) is a subspace of \(\mathcal {B}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) consisting of all f such that
Remark 2.1
- (i):
-
The Bloch-type space \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) and little Bloch-type space \({\mathcal {B}}_{{\mathcal {R}},\omega _0} ({\mathbf {B}}_{{\mathbb {X}}})\) were first introduced by Hamada [9], which generalize the corresponding spaces defined on the Euclidean unit ball \({\mathbb {B}}^n\) or on the unit disk \({\mathbb {U}}.\)
- (ii):
-
By choosing different functions \(\omega \), we have the following special spaces:
-
If \(\omega (z)=1-\Vert z\Vert ^2\) in (2.1) and (2.2), respectively, then we obtain the Bloch space \({\mathcal {B}}_{{\mathcal {R}}}({\mathbf {B}}_{{\mathbb {X}}})\) and little Bloch space \(\mathcal {B}_{{\mathcal {R}},0}({\mathbf {B}}_{{\mathbb {X}}})\) in the unit ball of a complex Banach space (the case in \({\mathbb {B}}^n\), see, e.g., [15]).
-
If \(\omega (z)=(1-\Vert z\Vert ^2)^{\alpha }\) with \(\alpha \in (0,\infty )\) in (2.1) and (2.2), respectively, then we obtain the \(\alpha \)-Bloch space \({\mathcal {B}}^{\alpha }_{{\mathcal {R}}}({\mathbf {B}}_{{\mathbb {X}}})\) and little \(\alpha \)-Bloch space \({\mathcal {B}}^{\alpha }_{{\mathcal {R}},0}({\mathbf {B}}_{{\mathbb {X}}})\) in the unit ball of a complex Banach space (the case in \({\mathbb {B}}^n\), see, e.g., [16]).
-
If \(\omega (z)=(1-\Vert z\Vert ^2)\big (\prod \nolimits _{j=1}^k\ln ^{[j]}\frac{e^k}{1-\Vert z\Vert ^2}\big )\) in (2.1) and (2.2), respectively, then we obtain the iterated logarithmic Bloch space \({\mathcal {B}}_{\log ,{\mathcal {R}}}({\mathbf {B}}_{{\mathbb {X}}})\) and little Bloch space \(\mathcal {B}_{\log ,{\mathcal {R}},0}({\mathbf {B}}_{{\mathbb {X}}})\) in the unit ball of a complex Banach space (the case in \({\mathbb {B}}^n\), see, e.g., [12]).
- (iii):
-
In the case \({\mathbf {B}}_{{\mathbb {X}}}={\mathbf {B}}_{{\mathbb {H}}}\) is the unit ball of a complex Hilbert space \({\mathbb {H}}\), Hamada [9] proved that the condition (2.1) is equal to
$$\begin{aligned} \Vert f\Vert _{{\mathcal {B}}_{{\mathcal {R}},\omega }}=\sup \{\omega (z)\Vert Df(z)\Vert :z\in {\mathbf {B}}_{{\mathbb {X}}}\}<+\infty , \end{aligned}$$(2.3)
and the condition (2.2) is equal to
The same situation holds with (2.3) and (2.4) when \({\mathbf {B}}_{{\mathbb {X}}}={\mathbb {B}}^n\) (see, Tang [21]). In fact, if \(f\in H({\mathbf {B}}_{{\mathbb {X}}})\), then the relation \(|{\mathcal {R}}f(z)|\leqslant \Vert Df(z)\Vert \) make sure that
but the converse is not true.
The weighted space \(H_{\omega }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) consisting of all \(f\in H({\mathbf {B}}_{{\mathbb {X}}})\) such that
where \(\omega \) is normal.
The little weighted space \(H_{\omega _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is a subspace of \(H_{\omega }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) consisting of all f such that
If \(u\in H({\mathbf {B}}_{{\mathbb {X}}})\), and \(\varphi \in \) Aut\(({\mathbf {B}}_{{\mathbb {X}}})\), then the operator \(uC_{\varphi }\) is defined by
Remark 2.2
We note that (2.5) extends the corresponding weighted compositionoperator in (1.1) on \({\mathbb {B}}^n\) to the case of unit ball \({\mathbf {B}}_{{\mathbb {X}}}\) of a complex Banach space \({\mathbb {X}}\).
Next, we formulate and prove several auxiliary results which are used in the main theorems. Lemma 2.3 was proved by Hamada [9], (the corresponding results in \({\mathbb {B}}^n\), see Tang [21]). Lemma 2.5 is a generalization of the result on \({\mathbb {B}}^n\) (see, Krantz and Stević [12]) to the case of unit ball \({\mathbf {B}}_{{\mathbb {X}}}\) of a complex Banach space \({\mathbb {X}}.\) In Lemma 2.4, we define two test functions, which play a key role in the proof of our main theorems.
Lemma 2.3
([9], Lemma 2.1) Let \(\omega \) be a normal function. Denote \(k_0=\max (0,[\log _2\frac{1}{\omega (\delta )}])\), \(r_k=(\omega |_{[\delta ,1)})^{-1}(\frac{1}{2^k})\) and \(n_k=[\frac{1}{1-r_k}]\) for \(k> k_0\), where the symbol [x] means the greatest integer not more than x. Let
Then
(i) G is a holomorphic function on \({\mathbb {U}}\) such that G(r) is increasing on [0, 1) and
(ii) there exists a positive constant \(C_3\) such that the inequality
holds for all \(r\in [r_1,1)\), where \(r_1\in (0,1)\) is a constant such that
Lemma 2.4
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space \({\mathbb {X}}\). For any \(\nu \in {\mathbf {B}}_{{\mathbb {X}}}\backslash \{0\}\) and \(l_\nu \in T(\nu )\), let
where G is the function defined in Lemma 2.3 and \(0<k<+\infty \). Then
(a) \(f_{\nu ,k}\in {\mathcal {B}}_{{\mathcal {R}}, \omega _0}({\mathbf {B}}_{{\mathbb {X}}})\) and \(\Vert f_{\nu ,k}\Vert _{{\mathcal {R}},\omega }\leqslant 1+kC_2\), where \(C_2\) is the constant defined in Lemma 2.3.
(b) if \(\Vert \nu \Vert \geqslant r_1\) and
where \(r_1\) is defined in Lemma 2.3, then \(F_{\nu ,k}\in {\mathcal {B}}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\) and
where \(C_2,\)\(C_3\) are the constants defined in Lemma 2.3. Moreover, if \(\int _0^1\frac{1}{\omega (t)}\mathrm{{d}}t=\infty \), then \(F_{\nu ,k}\rightarrow 0\) uniformly on any closed ball strictly inside \({\mathbf {B}}_{{\mathbb {X}}}\) as \(\Vert \nu \Vert \rightarrow 1\).
Proof
(a) Using Lemma 2.3, we obtain
Therefore, \(f_{\nu ,k}\in {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) and \(\Vert f_{\nu ,k}\Vert _{{\mathcal {R}},\omega }\leqslant 1+kC_2.\) Moreover, since \({\mathcal {R}}f_{\nu ,k}\) is bounded on \({\mathbf {B}}_{{\mathbb {X}}}\), we have \(\lim \nolimits _{\Vert z\Vert \rightarrow 1}\omega (z)|{\mathcal {R}}f_{\nu ,k}(z)|=0,\) which implies \(f_{\nu ,k}\in \mathcal {B}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\).
(b) By some simple estimates and using the Lemma 2.3, we have
By (2.6), we obtain \(F_{\nu ,k}\in {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) and
Moreover, since \({\mathcal {R}}F_{\nu ,k}\) is bounded on \({\mathbf {B}}_{{\mathbb {X}}}\), we have \(\lim \nolimits _{\Vert z\Vert \rightarrow 1}\omega (z)|{\mathcal {R}}F_{\nu ,k}(z)|=0,\) which implies \(F_{\nu ,k}\in \mathcal {B}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\).
Next, if \(\int _0^1\frac{1}{\omega (t)}\mathrm{{d}}t=\infty \), then we can prove that \(F_{\nu ,k}\rightarrow 0\) uniformly for \(\Vert z\Vert \leqslant r\,\,(0<r<1)\) as \(\Vert \nu \Vert \rightarrow 1\) by a similar way in ([9], Lemma 2.7). The proof is finished. \(\square \)
Lemma 2.5
Suppose that \(u\in H({\mathbf {B}}_{{\mathbb {X}}}),\)\(\mu \) is normal and \(\varphi \in Aut({\mathbf {B}}_{{\mathbb {X}}})\). Then theoperator \(uC_\varphi \) : \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact if and only if \(uC_\varphi \): \({\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded, and for any bounded sequence \(\{f_k\}_{k\in {\mathbb {N}}}\) in \(\mathcal {B}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) converging to zero uniformly on compact subset of \({\mathbf {B}}_{{\mathbb {X}}}\) as \(k\rightarrow \infty \), we have that \(\lim \nolimits _{k\rightarrow \infty }\Vert uC_\varphi (f_k)\Vert _{H_{\mu }^{\infty }}=0\).
Proof
It is similar to the proofs of the corresponding results [cf. ([4], Lemma 4.4) and ([12], Lemma 5)]. We omit the proof here.
Throughout this paper, the notation \(A\asymp B\) means that there is a positive constant C such that \(\frac{A}{C}\leqslant B\leqslant CA\). \(\square \)
3 Some Properties with Spaces \({\mathcal {B}}_{{\mathcal {R}},\mu }({\mathbf {B}}_{{\mathbb {X}}})\) and \({\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\)
In this section, we prove \({\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\) is a closed subset of \(\mathcal {B}_{{\mathcal {R}},\mu }({\mathbf {B}}_{{\mathbb {X}}})\) in Theorem 3.1, and the transform relationship between of them is given by delay function in Theorem 3.2, which generalize the previous works on \({\mathbb {B}}^n\) to the case of unit ball \({\mathbf {B}}_{{\mathbb {X}}}\) (see, Theorem 2 and Theorem 3 in Krantz and Stević [12]).
Theorem 3.1
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space \({\mathbb {X}}\). Then \({\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\) is a closed subset of \(\mathcal {B}_{{\mathcal {R}},\mu }({\mathbf {B}}_{{\mathbb {X}}})\), where \(\mu \) is normal on [0, 1).
Proof
Let \(\{f_j\}_{j\in {\mathbb {N}}}\) be a sequence in \(\mathcal {B}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\) such that
Using the (3.1), we have that, for every \(\varepsilon >0\), there is an \(j_0\in {\mathbb {N}}\) such that
for \(j\geqslant j_0\). In particular, taking \(j=j_0\) in (3.2), it gives that
On the other hand, \(f_{j_0}\in {\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\) makes sure that, for every \(\varepsilon >0\), there is a \(\delta >0\) such that
for \(\delta<\Vert z\Vert <1\). Thus, from (3.3) and (3.4), we have
for \(\delta<\Vert z\Vert <1\), which implies that \(f\in {\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\). The proof of this theorem is completed. \(\square \)
Theorem 3.2
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space \({\mathbb {X}}\). Assume that \(f\in {\mathcal {B}}_{{\mathcal {R}},\mu }({\mathbf {B}}_{{\mathbb {X}}})\) and \(f_r(z)=f(rz),\,\, r\in [0,1), \,\,z\in {\mathbf {B}}_{{\mathbb {X}}}.\) Then \(f\in {\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\) if and only if
where \(\mu \) is normal decreasing function of \(\Vert z\Vert \) with \(\mu (0)<+\infty .\)
Proof
After some simple computations, it is easy to see that
Since \(\mu \) is normal decreasing function of \(\Vert z\Vert \), by (3.6), it follows that
Assume that (3.5) holds. Let \(f\in {\mathcal {B}}_{{\mathcal {R}},\mu }({\mathbf {B}}_{{\mathbb {X}}})\), then by (3.7), we have
This implies that \(f_r\in {\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\). Furthermore, \(\mathcal {B}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\) is a closed subset of \({\mathcal {B}}_{{\mathcal {R}},\mu }({\mathbf {B}}_{{\mathbb {X}}})\) (see, Theorem 3.1), then (3.5) implies that \(f\in {\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\).
Now assume \(f\in {\mathcal {B}}_{{\mathcal {R}},\mu _0}({\mathbf {B}}_{{\mathbb {X}}})\). Then, for every \(\varepsilon >0\), there is a \(\delta \in (0,1)\) such that
as \(\delta ^2<\Vert z\Vert <1\). By (3.6), we have
It is obviously
and
Thus, using (3.10) and (3.11), it follows that
On the other hand, by (3.8), for all \(z\in {\mathbf {B}}_{{\mathbb {X}}}\) and \(r\in [0,1)\) such that \(\delta<\Vert z\Vert<1, \delta<r<1\), we have
Hence, by (3.8) and (3.13), it follows that
for every \(r\in (\delta ,1)\). Using (3.12) and (3.14) in (3.9), we obtain (3.5). The proof is finished. \(\square \)
4 \(uC_{\varphi }:{\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\)(or \({\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}}))\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\)(or \(H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}}))\)
In this section, we study the boundedness and compactness of operator \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) (resp. \(\mathcal {B}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}}))\), which generalize thecorresponding results on \({\mathbb {B}}^n\) to the case of unit ball \({\mathbf {B}}_{{\mathbb {X}}}\) of a infinite dimensional complex Banach space \({\mathbb {X}}\) (see, Theorems 16–19 in Krantz and Stević [12]). Thefollowing set \(E_{\varepsilon ,\rho }\) is needed when consider the compactness of the operator \(uC_{\varphi }\). For \(\forall \varepsilon >0\) and \(\rho \in (0,1)\), we define
Theorem 4.1
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space. Assume that \(u\in H({\mathbf {B}})\), \(\varphi \in \) Aut\(({\mathbf {B}}_{{\mathbb {X}}})\), \(\omega \) and \(\mu \) are normal on [0, 1). Then \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded if and only if
Moreover, if \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded, then
Proof
Assume that (4.2) holds and \(f\in {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\), \(z\in {\mathbf {B}}_{{\mathbb {X}}}\). Using Proposition 2.4 in [9], we have
From (4.2) and (4.4), it follows that
and \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded.
Conversely, assume that \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded. For \(\nu \in {\mathbf {B}}_{{\mathbb {X}}}, k\geqslant 0\) and \(l_\nu \in T(\nu )\), we give the test function
where G is defined as Lemma 2.3. By Lemma 2.4, then \(f_{\nu ,k}\in {\mathcal {B}}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\subset {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) and \(\Vert f_{\nu ,k}\Vert _{{\mathcal {R}},\omega }\leqslant C_2\). Let \(r_1\) be the constant in Lemma 2.3. Thus, for \(\nu \in {\mathbf {B}}_{{\mathbb {X}}}\) with \(\Vert \varphi (\nu )\Vert \geqslant r_1\), we get
If \(\Vert \varphi (\nu )\Vert <r_1\), then by Lemma 2.3, we have
The inequalities (4.7) and (4.8) yield (4.2), as desired. Moreover, from (4.4), (4.7) and (4.8), we obtain (4.3). This completes the proof. \(\square \)
Theorem 4.2
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space \({\mathbb {X}}\). Assume \(u\in H({\mathbf {B}})\), \(\varphi \in \) Aut\(({\mathbf {B}}_{{\mathbb {X}}})\), \(\omega \) and \(\mu \) are normal functions on [0, 1). Let the set \(E_{\varepsilon ,\rho }\) is relatively compact in \({\mathbf {B}}_{{\mathbb {X}}}\) for any \(\varepsilon >0\) and \(\rho \in (0,1)\). The following statements are true.
- (a):
-
If \(\int _{0}^{1}\frac{1}{\omega (t)}\mathrm{{d}}t<\infty \), then \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact if and only if \(u\in H_{\mu }^\infty ({\mathbf {B}}_{{\mathbb {X}}})\).
- (b):
-
If \(\int _{0}^{1}\frac{1}{\omega (t)}\mathrm{{d}}t=\infty \), then \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact if and only if \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded and
$$\begin{aligned} \lim \limits _{\Vert \varphi (z)\Vert \rightarrow 1}\mu (z)|u(z)|\Bigg (1+\int _0^{\Vert \varphi (z)\Vert }\frac{1}{\omega (t)}\mathrm{{d}}t\Bigg )=0. \end{aligned}$$(4.9)
Proof
(a) Suppose that \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact. Then it is clear that \(uC_{\varphi }\) is bounded. Take the test function \({\widehat{f}}(z)\equiv 1, z\in {\mathbf {B}}_{{\mathbb {X}}}.\) It is easy to know that \({\widehat{f}}\in {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\). We have
which implies \(u\in H_{\mu }^\infty ({\mathbf {B}}_{{\mathbb {X}}}).\)
Conversely, if \(u\in H_{\mu }^\infty ({\mathbf {B}}_{{\mathbb {X}}})\), then \(\int _{0}^{1}\frac{1}{\omega (t)}\mathrm{{d}}t<\infty \) makes sure that (4.2) holds. Thus, \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded by Theorem 4.1. For any \(\varepsilon >0\), there exists \(\rho \in (\frac{1}{2},1)\) such that
when \(z\in {\mathbf {B}}_{{\mathbb {X}}}\) with \(\rho<\Vert \varphi (z)\Vert <1.\) Let \(\{f_j\}_{j\in {\mathbb {N}}}\) be a bounded sequence in \(\mathcal {B}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) which converges to 0 uniformly on any compact subset of \({\mathbf {B}}_{{\mathbb {X}}}\). We may assume that \(\Vert f_j\Vert _{{\mathcal {R}},\omega }\leqslant 1\). Then \(|f_j|\leqslant C_\rho \) for all j and \(\Vert z\Vert \leqslant \rho \) by Proposition 2.4 in [9], where
There exists a positive N such that
Therefore, for \(\Vert \varphi (z)\Vert \leqslant \rho \) and \(t=1\) or for \(\rho<\Vert \varphi (z)\Vert <1\) and \(t=\frac{\rho }{\Vert \varphi (z)\Vert }\), we have
Next, we will finish the proof through a case by case check:
Case I: if \(z\in {\mathbf {B}}_{{\mathbb {X}}}\) with \(\Vert \varphi (z)\Vert \leqslant \rho \), by (4.11), then we have
holds.
Case II: if \(z\in {\mathbf {B}}_{{\mathbb {X}}}\) with \(\rho<\Vert \varphi (z)\Vert <1\), by (4.10), (4.11) and some simple calculation, we have
for \(j>N.\)
For any \(\varepsilon >0\), employing the (4.12) and (4.13), we obtain
Using Lemma 2.5 and (4.14), \(uC_{\varphi }: \mathcal {B}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact.
(b) If \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact, then \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded. Now, we assume that conditions (4.9) dose not hold, then there exist \(\varepsilon >0\) and a sequence \(\{\varphi (z_k)\}_{k\in {\mathbb {N}}}\subset {\mathbf {B}}_{{\mathbb {X}}}\) such that
and
For \(\nu \in {\mathbf {B}}_{{\mathbb {X}}}\) and \(l_\nu \in T(\nu )\), we give the test function
where G is defined as Lemma 2.3 and \(z\in {\mathbf {B}}_{{\mathbb {X}}}\). In fact, (4.15) implies that we can assume that \(\Vert \varphi (z_k)\Vert >r_1\), where \(r_1\) is the constant in Lemma 2.3. Furthermore, if we take \(f_k(z)=F_{\varphi (z_k),\frac{C_3}{C_1}}(z)\) for \(z\in {\mathbf {B}}_{{\mathbb {X}}}\), from Lemma 2.4, then \(\{f_k\}_{k\in N}\) is a bounded sequence in \({\mathcal {B}}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\) and \(f_k\rightarrow 0\) uniformly on any compact subset of \({\mathbf {B}}_{{\mathbb {X}}}\). Therefore, we have
On the other hand, using (4.16) and Lemma 2.3 with \(\Vert \varphi (z_k)\Vert >r_1\), we have
(4.19) is a contradiction compare with (4.18). Thus, we obtain (4.9).
Conversely, assume that \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded and (4.9) holds. By Theorem 4.1, we know that condition (4.2) holds. Furthermore, since \(\int _{0}^{1}\frac{1}{\omega (t)}\mathrm{{d}}t=\infty \), we get (4.2) implies \(u\in H_\mu ^\infty ({\mathbf {B}}_{{\mathbb {X}}})\). Moreover, according to (4.9), there exists \(\rho \in (\frac{1}{2},1)\) such that
where \(\varepsilon \) is any small positive number. The rest of the proof is similar to the case (a), we omit it. This proof is completed. \(\square \)
Theorem 4.3
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space. Assume \(u\in H({\mathbf {B}}_{{\mathbb {X}}})\), \(\varphi \in \) Aut\(({\mathbf {B}}_{{\mathbb {X}}})\), \(\omega \) and \(\mu \) are normal functions on [0, 1). Then \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu ,0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded if and only if \(u\in H_{\mu _0}^\infty ({\mathbf {B}}_{{\mathbb {X}}})\) and
Proof
First assume that \(u\in H_{\mu ,0}^\infty ({\mathbf {B}}_{{\mathbb {X}}})\) and (4.20) hold. By Theorem 4.1, we know that \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu }^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded. Therefore, we only need to prove that \(uC_{\varphi }(f)\in H_{\mu ,0}^\infty ({\mathbf {B}}_{{\mathbb {X}}})\) for any \(f\in \mathcal {B}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\). Now, let \(f\in {\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\). Applying (4.20), then there exists \(\rho _0\in (\frac{1}{2},1)\) such that
where \(\varepsilon \) is an arbitrarily small positive number.
Taking \({\widetilde{z}}=\rho _0\frac{\varphi (z)}{\Vert \varphi (z)\Vert }\), then \(\Vert {\widetilde{z}}\Vert =\rho _0<1\). For any \(z\in {\mathbf {B}}_{{\mathbb {X}}}\) with \(\rho _0<\Vert z\Vert <1\), using (4.21), we have
In the above equality (4.22), we use the fact that
Set \(K=\sup \nolimits _{\Vert z\Vert \leqslant \rho _0}|f(z)|\). By Proposition 2.4 in [9], \(K<\infty .\) Since \(u\in H_{\mu ,0}^\infty ({\mathbf {B}}_{{\mathbb {X}}})\), we have, for any \(\varepsilon >0\), there is \(\rho _1\in (\rho _0,1)\) such that
whenever \(\rho _1<\Vert z\Vert <1\). Combining (4.22) and (4.23), for \(\rho _1<\Vert z\Vert <1\), we obtain
Hence, (4.24) implies that \(uC_{\varphi }(f)\in H_{\mu ,0}^\infty ({\mathbf {B}}_{{\mathbb {X}}})\).
Conversely, assume that \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu ,0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is bounded. Taking \(f(z)\equiv 1\), then \(f\in {\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}}).\) Thus,
which implies that \(u\in H_{\mu ,0}^\infty ({\mathbf {B}}_{{\mathbb {X}}})\). On the other hand, For \(\nu \in {\mathbf {B}}_{{\mathbb {X}}}, k\geqslant 0\) and \(l_\nu \in T(\nu )\), we give the test function
where G is defined as Lemma 2.3. Applying the facts \({\mathcal {B}}_{{\mathcal {R}},\omega _0}({\mathbf {B}}_{{\mathbb {X}}})\subset {\mathcal {B}}_{{\mathcal {R}},\omega }({\mathbf {B}}_{{\mathbb {X}}})\) and \(H_{\mu ,0}^\infty ({\mathbf {B}}_{{\mathbb {X}}})\subset H_{\mu }^\infty ({\mathbf {B}}_{{\mathbb {X}}})\), we can obtain (4.20) by (4.7) and (4.8). The proof is finished. \(\square \)
Theorem 4.4
Let \({\mathbf {B}}_{{\mathbb {X}}}\) be the unit ball of a complex Banach space \({\mathbb {X}}\). Assume \(u\in H({\mathbf {B}}_{{\mathbb {X}}}),\)\(\varphi \in \) Aut\(({\mathbf {B}}_{{\mathbb {X}}})\), \(\omega \) and \(\mu \) are normal functions on [0, 1). The set \(E_{\varepsilon ,\rho }\) is relatively compact in \({\mathbf {B}}_{{\mathbb {X}}}\) for any \(\varepsilon >0\) and \(\rho \in (0,1)\). Then \(uC_{\varphi }: \mathcal {B}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact if and only if
Proof
Assume that (4.26) holds. Taking any \(f\in \mathcal {B}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\), by Proposition 2.4 in [9], we have
which implies
Therefore, \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact.
Conversely, assume that \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact, then \(uC_{\varphi }\) is bounded. Taking the test function \({\widehat{f}}(z)=1\), we have
Now we prove
holds. In fact, if (4.30) does not hold, then there exist \(\varepsilon >0\) and a sequence \(\{\varphi (z_j)\}\subset {\mathbf {B}}_{{\mathbb {X}}}\) such that \(\lim \nolimits _{j\rightarrow \infty }\Vert \varphi (z_j)\Vert =1\) and
Taking the same test functions \(f_k(z)\) as (4.17) and using the similar ways inTheorem 4.2, we note that \(\{f_k\}\) is a bounded sequence in \({\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\) and \(f_k\rightarrow 0\)uniformly on any compact subset of \({\mathbf {B}}_{{\mathbb {X}}}\). Since \(uC_{\varphi }: {\mathcal {B}}_{{\mathcal {R}},{\omega _0}}({\mathbf {B}}_{{\mathbb {X}}})\rightarrow H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) is compact, we may assume that there exists some \(h\in H_{\mu _0}^{\infty }({\mathbf {B}}_{{\mathbb {X}}})\) such that \(\Vert uC_{\varphi }(f_k)-h\Vert _{H_{\mu ,0}^{\infty }}\rightarrow 0\) as \(k\rightarrow \infty \). Then for each \(z\in {\mathbf {B}}_{{\mathbb {X}}}\), we have
Thus, we have \(\Vert uC_{\varphi }(f_k)\Vert _{H_{\mu ,0}^{\infty }}\rightarrow 0\) as \(k\rightarrow \infty \). This contradicts with (4.31). Thus, (4.30) holds. By (4.30) we have that for any \(\varepsilon >0\) there exists \(r\in (0,1)\) such that
with \(r<\Vert \varphi (z)\Vert <1.\) On the other hand, using (4.29), there exists \(\rho \in (0,1)\) such that
with \(\rho<\Vert z\Vert <1.\) Therefore, when \(\rho<\Vert z\Vert <1\) and \(r<\Vert \varphi (z)\Vert <1\), from (4.32) and (4.33), we have
If \(\rho<\Vert z\Vert <1\) and \(\Vert \varphi (z)\Vert \leqslant r\), from (4.33), then
Combining (4.34) and (4.35), we obtain
This proof is completed. \(\square \)
Remark 4.5
This assumption that the set \(E_{\varepsilon ,\rho }\) is relatively compact in \({\mathbf {B}}_{{\mathbb {X}}}\) isautomatically satisfied when dim(\({\mathbb {X}})<+\infty \) (see Theorems 4.2, 4.4).
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The project is supported by the National Natural Science Foundation of China (no. 11671306).
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Tu, Z., Xiong, L. Weighted Space and Bloch-Type Space on the Unit Ball of an Infinite Dimensional Complex Banach Space. Bull. Iran. Math. Soc. 45, 1389–1406 (2019). https://doi.org/10.1007/s41980-019-00204-8
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DOI: https://doi.org/10.1007/s41980-019-00204-8