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Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces
Journal of Inequalities and Applications volume 2015, Article number: 265 (2015)
Abstract
Some criteria for the boundedness, as well as for the compactness, of the generalized weighted composition operator \(D^{n}_{\varphi, u}\) from α-Bloch spaces into weighted-type spaces are given. Estimates for the norm and the essential norm of the operator are also given. Our results extend and complement some results in the literature.
1 Introduction
Let \(\mathbb{D}\) be the unit disk of the complex plane \(\Bbb{C}\), \(H(\mathbb{D})\) the class of functions analytic on \(\mathbb{D}\), and \(H^{\infty}=H^{\infty}(\mathbb{D})\) the space of bounded analytic functions on \(\mathbb{D}\). For \(0<\alpha <\infty\), an \(f\in H(\mathbb{D})\) is said to belong to the α-Bloch space \(\mathcal{B}^{\alpha}=\mathcal{B}^{\alpha}(\mathbb{D})\) if
It is easy to check that \(\mathcal{B}^{\alpha}\) becomes a Banach space with the norm \(\|f\|_{\mathcal{B}^{\alpha}}=|f(0)|+b_{\alpha}(f)\). The little α-Bloch space \(\mathcal{B}^{\alpha}_{0}=\mathcal{B}^{\alpha}_{0}(\mathbb{D})\), is a subspace of \(\mathcal{B}^{\alpha}\) consisting of all \(f\in H(\mathbb{D})\) such that
When \(\alpha=1\), \(\mathcal{B}^{1}=\mathcal{B}\) is the well-known Bloch space, while \(\mathcal{B}^{1}_{0}=\mathcal{B}_{0}\) is the well-known little Bloch space. For some results on the α-Bloch spaces and the little α-Bloch spaces, see, for example, [1].
A positive continuous function on \(\mathbb{D}\) is called a weight. Let \(\mu(z)\) be a weight. The weighted-type space on \(\mathbb{D}\) [2, 3], denoted by \(H^{\infty}_{\mu}=H^{\infty}_{\mu}(\mathbb{D})\), consists of all \(f\in H(\mathbb{D})\) such that
It is obvious that \(H^{\infty}_{0}=H^{\infty}\), while for \(\mu(z)=(1-|z|^{2})^{\beta}\), \(\beta>0\), is obtained the growth space \(H^{\infty}_{\beta}\) [4].
Let \(u \in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). The weighted composition operator \(uC_{\varphi}\), induced by φ and u, is defined by
When \(u(z)\equiv1\), then the weighted composition operator is reduced to the composition operator, usually denoted by \(C_{\varphi}\), while for \(\varphi(z)\equiv z\), it is reduced to the multiplication operator, usually denoted by \(M_{u}\).
A natural generalization of the weighted composition operator is the generalized weighted composition operator [5] or the weighted differentiation composition operator [6] \(D^{n}_{\varphi, u}\), which is defined as
where \(n\in\mathbb{N}_{0}\), \(u \in H(\mathbb{D})\), and φ is an analytic self-map of \(\mathbb{D}\). Clearly, when \(n=0\) and \(u(z)=1\), \(D^{n}_{\varphi,u}\) is the composition operator \(C_{\varphi}\), if \(n=0\), then \(D^{n}_{\varphi,u}\) is the weighted composition operator \(uC_{\varphi}\). If \(n=1\) and \(u(z)=\varphi'(z)\), then \(D^{n}_{\varphi, u}= DC_{\varphi}\), which was studied, for example, in [3, 7–15], while for \(u(z)=1\), \(D^{n}_{\varphi, u}= C_{\varphi}D^{n}\), which was studied in [3, 13, 15, 16]. For some other results on the generalized weighted composition operator on various spaces of holomorphic functions, see, for example, [17–22]. A fundamental problem concerning concrete operators is to relate function theoretic properties of their symbols to their operator theoretic properties (see, for example, [3, 5–29]).
It is well known that the composition operator is bounded on the Bloch space \(\mathcal{B}\). See, for example, [26, 28, 29] for the compactness and essential norm of the composition operator on \(\mathcal{B}\). In [28], it was shown that \(C_{\varphi}\) is compact on \(\mathcal{B}\) if and only if
where \(p_{j}(z)=z^{j}\), \(j\in\mathbb{N}_{0}\).
Motivated by this result, in [22], the author proved that \(D^{n}_{\varphi,u}: \mathcal{B} \to H^{\infty}_{\beta}\) is compact if and only if it is bounded and
Following the line of the above mentioned investigations, in this work, we consider the operators \(D^{n}_{\varphi, u} :\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\), and show that \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\) is bounded (respectively, compact) if and only if the sequence \((j^{\alpha -1}\|D^{n}_{\varphi,u} (p_{j})\|_{H_{\mu}^{\infty}})_{j=n}^{\infty}\) is bounded (respectively, convergent to 0 as \(j\to\infty\)). Moreover, we give some estimates for the norm, as well as for the essential norm of the operator \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal {B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\). Recall that the essential norm of the operator \(T:X\rightarrow Y\) is its distance to the set of compact operators K mapping X to Y, that is,
where X and Y are Banach spaces and \(\|\cdot\|_{X\rightarrow Y}\) is the operator norm. Consequently, \(\|T\|_{e,X\rightarrow Y}=0\) if and only if T is compact.
Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. We write \(P\preceq Q\) if there exists a positive constant C independent of the quantities P and Q such that \(P\leq CQ\). The symbol \(P\approx Q\) means that \(P\preceq Q\preceq P\).
2 Boundedness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\to H^{\infty}_{\mu}\)
For \(w\in\mathbb{D}\), set
Note that
In this section, we will use this family of functions, as well as the sequence of functions \((j^{\alpha -1} p_{j})_{j\in\mathbb{N}}\) to characterize the boundedness and compactness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\to H^{\infty}_{\mu}\).
Theorem 2.1
Let n be a positive integer, \(\alpha >0\), μ a weight, \(u\in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Then the following statements are equivalent.
-
(a)
The operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded.
-
(b)
The operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}\) is bounded.
-
(c)
\(M_{1}:= \sup_{j \geq n} j^{\alpha -1} \| D^{n}_{\varphi, u}(p_{j})\|_{H^{\infty}_{\mu}}<\infty\).
-
(d)
\(M_{2}:=\sup_{w\in\mathbb{D}}\|D^{n}_{\varphi,u} f_{\varphi (w)}\|_{H^{\infty}_{\mu}}<\infty\) and \(u\in H^{\infty}_{\mu}\).
-
(e)
\(M_{3}:= \sup_{z\in\mathbb{D} } \frac{\mu(z)|u(z) |}{(1-|\varphi(z)|^{2})^{ n+\alpha -1}} <\infty \) and \(u\in H^{\infty}_{\mu}\).
Moreover, if the operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded, then the following asymptotic relations hold:
Proof
(a) ⇒ (b) Since \(\mathcal{B}^{\alpha}_{0}\subset \mathcal{B}^{\alpha}\), this implication, as well as the inequality
is obvious.
(b) ⇒ (c) It is easy to see that the sequence \((j^{\alpha -1} p_{j})_{j\in\mathbb{N}}\) is bounded in \(\mathcal {B}^{\alpha}_{0}\) and
which implies that \(\|j^{\alpha -1}p_{j}\|_{\mathcal{B}^{\alpha}}\approx1\). Notice that \((D^{n}_{\varphi, u} p_{n})(z)=u(z)n! \), \(z\in\mathbb{D}\), while for \(j< n\), \(D^{n}_{\varphi, u} (p_{j}) =0\). Therefore, by the boundedness of \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}\), we get
for every \(j\in \mathbb{N}\), proving (c), as well as the asymptotic relation
(c) ⇒ (a) If \(\|\varphi\|_{\infty}=\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1\), then by Proposition 8 in [1], we have
from which the boundedness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) follows in this case.
Now assume that \(\|\varphi\|_{\infty}=1\). Let \(\mathbb{D}_{j}=\{z\in \mathbb{D}: r_{j} \leq |\varphi(z)|< r_{j+1}\}\) where \(r_{j}=(j-n)/(j+\alpha-1)\) for \(j\ge n\). Then from Lemma 1 in [16], which also holds for \(m=0\), i.e., \(n=1\) in our case, we have that there is a \(\delta>0\) such that
for every \(j\ge k+1\), where k is the smallest natural number such that \(\mathbb{D}_{k}\ne\emptyset\).
Fix \(N\ge k+1\). Then, clearly \(N\ge n+1\) and we have
The finiteness of \(M_{1}\) implies \(u\in H_{\mu}^{\infty}\). Hence, as in the first case, we have
On the other hand, since \(\mathbb{D}\setminus \{|\varphi(z)|<\frac{N-n}{N+\alpha-1}\}= \bigcup_{j\ge N}\mathbb{D}_{j}\), we get
From (5), (6) and (7), the boundedness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) follows.
(c) ⇒ (d) First note that (c) implies that \(u\in H^{\infty}_{\mu}\). Further, since
the family of functions \((f_{w})_{w\in\mathbb{D}}\) is uniformly bounded in \(\mathcal{B}^{\alpha}\). Furthermore
By Stirling’s formula, we have \(\frac{\Gamma(j+\alpha)}{j!\Gamma(\alpha)}\approx j^{\alpha-1} \) as \(j\rightarrow\infty\). Using this fact, the linearity and continuity of the operator, we get
Consequently, \(\sup_{w\in\mathbb{D}} \|D^{n}_{\varphi, u} f_{\varphi(w)}\|_{H^{\infty}_{\mu}}\preceq M_{1}\), and along with the inequality \(n^{\alpha -1}n!\|u\|_{H^{\infty}_{\mu}}\leq M_{1}\), obtained by considering \(\|D^{n}_{\varphi, u}(n^{\alpha -1}p_{n})\|_{H^{\infty}_{\mu}}\), we also have
(d) ⇒ (e) For \(\lambda\in\mathbb{D}\), it follows from (d) and (1) that
For any fixed \(r\in(0,1)\), from (9), we have
On the other hand, from \(u \in H^{\infty}_{\mu}\), we have
Therefore, (10) and (11) yield the inequality of (e), as well as the asymptotic relation
(e) ⇒ (a) By Proposition 8 in [1], if \(f \in \mathcal{B}^{\alpha}\) and \(k\in\mathbb{N}\), we see that
for some constant C independent of f. Therefore, for \(z\in\mathbb{D}\), we have
where C is independent of f. Taking the supremum in (13) over \(\mathbb{D}\) and then using the first condition in (e) we see that \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is bounded, and
If the operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded, then from (3), (4), (8), (12), and (14), we obtain (2), completing the proof. □
3 Compactness and essential norm of \(D^{n}_{\varphi,u}: \mathcal {B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\rightarrow H^{\infty}_{\mu}\)
In this section we will give an estimate for the essential norm of the operator \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\), as well as of \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}\). For this purpose, we state several lemmas, which will be used in the proof of the main result.
Lemma 3.1
[16]
Let \(\alpha >0\), \(m\geq n+1\), where \(n\in\mathbb{N}\). Define the function \(H_{m,\alpha }:[0,1]\rightarrow[0,\infty)\) by
Then the following statements hold:
-
(i)
$$\max_{0\leq x\leq1}H_{m, \alpha}(x)=H_{m,\alpha}(r_{m})= \left \{ \textstyle\begin{array}{l@{\quad}l} (n+1)!, & m=n+1, \\ \frac{m!}{(m-n-1)!} (\frac{m-n-1}{m+\alpha-1} )^{m-n-1} (\frac{n+\alpha}{m+\alpha-1} )^{\alpha+n} , & m > n+1 , \end{array}\displaystyle \right . $$
where
$$r_{m}= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, & m=n+1, \\ \frac{m-n-1}{m+\alpha-1} , & m > n+1 . \end{array}\displaystyle \right . $$ -
(ii)
For \(m>n+1\), \(H_{m,\alpha }\) is decreasing on \([r_{m}, r_{m+1}]\), and so
$$ \min_{r_{m}\leq x\leq r_{m+1}}H_{m, \alpha}(x)=H_{m,\alpha}(r_{m+1})= \frac{m!}{(m-n-1)!} \biggl(\frac {m-n}{m+\alpha} \biggr)^{m-n-1} \biggl( \frac{n+\alpha}{m+\alpha} \biggr)^{\alpha+n} . $$Consequently,
$$ \lim_{m\rightarrow\infty} m^{\alpha-1} \min_{r_{m}\leq x\leq r_{m+1}}H_{m, \alpha}(x) = \frac{(n+\alpha)^{n+\alpha}}{e^{n+\alpha}}. $$
Denote by \(K_{r}f(z)=f(rz)\) for \(r\in(0,1)\) and \(z\in\mathbb{D}\). Then \(K_{r}\) is a compact operator on \({\mathcal{B}}^{\alpha}\) for every \(\alpha >0\), and \(\|K_{r}\|\leq1\) (see, e.g., Proposition 1.3 in [24] and [27]). Let I denote the identity operator. The following three lemmas can be found in [25] (see also [16]).
Lemma 3.2
Let \(0<\alpha<1\). Then there is a sequence \((r_{k})_{k\in\mathbb{N}}\), with \(0< r_{k}<1\) tending to 1, such that the sequence of compact operators \(L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}\), \(j\in\mathbb{N}\), on \(\mathcal {B}_{0}^{\alpha}\) satisfies the following.
-
(i)
For any \(t\in(0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0\).
-
(ii)
\(\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}^{\alpha}}\leq1}\sup_{z\in\mathbb{D}}|(I-L_{j})f(z)|=0\).
-
(iii)
\(\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1\).
Furthermore, these statements hold as well for the sequence of biadjoints \(L^{**}_{j}\) on \(\mathcal{B}^{\alpha}\).
Lemma 3.3
Let \(\alpha=1\). Then there is a sequence \((r_{k})_{k\in\mathbb{N}}\), with \(0< r_{k}<1\) tending to 1, such that the sequence of compact operators \(L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}\), \(j\in\mathbb{N}\), on \(\mathcal{B}_{0}\) satisfies the following.
-
(i)
For any \(t\in[0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0\).
-
(iia)
\(\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}}\leq1}\sup_{|z|>s}|(I-L_{j})f(z)| (\log\frac {1}{1-|z|^{2}} )^{-1}\leq1\), for s sufficiently close to 1.
-
(iib)
\(\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}}\leq1}\sup_{|z|\leq s}|(I-L_{j})f(z)|=0\) for the above s.
-
(iii)
\(\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1\).
Furthermore, these statements hold as well for the sequence of biadjoints \(L^{**}_{j}\) on \(\mathcal{B}\).
Lemma 3.4
Let \(\alpha>1\). Then there is a sequence \((r_{k})_{k\in\mathbb{N}}\), with \(0< r_{k}<1\) tending to 1, such that the sequence of compact operators \(L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}\), \(j\in\mathbb{N}\), on \(\mathcal {B}_{0}^{\alpha}\) satisfies the following.
-
(i)
For any \(t\in[0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0\).
-
(ii)
For any \(t\in[0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|(I-L_{j})f(z)|=0\).
-
(iii)
\(\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1\).
Furthermore, these statements hold as well for the sequence of biadjoints \(L^{**}_{j}\) on \(\mathcal{B}^{\alpha}\).
To study the compactness, we also need the following lemma, which can be proved in a standard way (see, for example, Proposition 3.11 in [23]).
Lemma 3.5
Let n be a nonnegative integer, \(\alpha >0\), μ a weight, \(u \in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). Then \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\ (\textit{or } \mathcal{B}^{\alpha}_{0})\rightarrow H^{\infty}_{\mu}\) is compact if and only if \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\ (\textit{or } \mathcal {B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\) is bounded and for any bounded sequence \((f_{k})_{k\in{\mathbb{N}}}\) in \(\mathcal{B}^{\alpha}\), which converges to zero uniformly on compact subsets of \(\mathbb{D}\),
Now we are ready to state and prove the main results in this section.
Theorem 3.6
Let n be a positive integer, \(\alpha >0\), μ a weight, \(u\in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Suppose that \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is bounded. Then
Proof
First note that the inequality
obviously holds.
Now we give a lower estimate for the essential norm \(\|D^{n}_{\varphi, u}\|_{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}\). Without loss of generality, we assume that \(j\geq n\). Choose the sequence of functions \(q_{j}=j^{\alpha-1} p_{j}\in\mathcal{B}^{\alpha}_{0} \), \(j\in\mathbb{N}\). Then \(\|q_{j}\|_{\mathcal{B}^{\alpha}}\approx1\), and \((q_{j})_{j\in\mathbb{N}}\) converges to zero weakly on \(\mathcal{B}^{\alpha}_{0}\) as \(j\rightarrow\infty\) (see, for example, Theorem 7.5 in [30]). Since by a well-known theorem, for any compact operator \(\widehat{K}:X\to Y\), where X and Y are Banach spaces, the weak convergence \(x_{n}\stackrel{w}{\to}x_{0}\) implies the norm convergence \(\widehat{K}x_{n}\to\widehat{K}x_{0}\) [31], we have
for any given compact operator K from \(\mathcal{B}^{\alpha}_{0}\) to \(H^{\infty}_{\mu}\).
Hence
Letting \(j\to\infty\) in the last relation and using (17), we obtain
and consequently
Now, we give the upper estimates for the essential norm \(\|D^{n}_{\varphi, u}\|_{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}\). For the case of \(\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1\), there is a number \(\delta\in(0,1)\) such that \(\sup_{z\in{\mathbb{D}}}|\varphi (z)|<\delta\). In this case, the operator \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is compact. Indeed, choose a bounded sequence \((f_{j})_{j\in\mathbb{N}}\) in \(\mathcal {B}^{\alpha}\) which converges to zero uniformly on compact subsets of \({\mathbb {D}}\). From Cauchy’s integral formula, \((f^{(n)}_{j})_{j\in\mathbb{N}}\) also converges to zero on compact subsets of \({\mathbb{D}}\) as \(j\rightarrow\infty \). Hence
From this and by Lemma 3.5 we see that the operator \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is compact. This also shows that
From (16), (18), and (19), we get the desired result in the case \(\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1\).
Next, we assume that \(\sup_{z\in{\mathbb{D}}}|\varphi(z)|=1\). Let \((L_{j})_{j\in\mathbb{N}}\) be the sequence of operators given in Lemmas 3.2-3.4. Since \(L_{j}^{**}\) is compact on \(\mathcal{B}^{\alpha}\), for every \(j\in\mathbb{N}\), and \(D^{n}_{\varphi, u}\) is bounded from \(\mathcal {B}^{\alpha}\) to \(H^{\infty}_{\mu}\), then \(D^{n}_{\varphi, u} L_{j}^{**}\) is also compact from \(\mathcal{B}^{\alpha}\) to \(H^{\infty}_{\mu}\). Hence
For each positive integer \(i\geq n\), we define \({\mathbb{D}}_{i}=\{z\in {\mathbb{D}}: r_{i}\leq|\varphi(z)|< r_{i+1}\}\), where \(r_{i}\) is given in Lemma 3.1. Let k be the smallest positive integer such that \({\mathbb{D}}_{k}\neq\emptyset\). Since \(\sup_{z\in{\mathbb {D}}}|\varphi(z)|=1\), \({\mathbb{D}}_{i}\) is not empty for every integer \(i\geq k\) and \(\mathbb{D}=\bigcup^{\infty}_{i=k}{\mathbb{D}}_{i}\), we have
where
and
Here N is a positive integer determined as follows.
By Lemma 3.1, \(\lim_{i\rightarrow\infty} \frac{i^{1-\alpha }}{H_{i,\alpha}(r_{i+1})}= \frac{e^{n+\alpha}}{(n+\alpha)^{n+\alpha}} \). Hence, for any given \(\varepsilon>0\), there exists an \(N\in\mathbb{N}\) such that
when \(i\geq N\). For such N it follows that
Thus
By Lemmas 3.2, 3.3, 3.4, and Cauchy’s integral formula, we have
which together with (20) implies that
Therefore
From the last relation we get
From (16), (18), and (21), the asymptotic relations in (15) follow, completing the proof of the theorem. □
From Theorem 3.6, letting \(\alpha =1\) we deduce the following result.
Corollary 3.7
Let n be a positive integer, μ a weight, \(u \in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Suppose that \(D^{n}_{\varphi, u}:\mathcal{B} \rightarrow H^{\infty}_{\mu}\) is bounded. Then
Theorem 3.8
Let n be a positive integer, \(\alpha >0\), μ a weight, \(u\in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). If \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded, then the following statements are equivalent.
-
(a)
The operator \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is compact.
-
(b)
The operator \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}\) is compact.
-
(c)
\(\lim_{j\rightarrow\infty} j^{\alpha -1}\|D^{n}_{\varphi, u} (p_{j})\|_{H^{\infty}_{\mu}}=0\).
-
(d)
\(\lim_{|\varphi(w)|\rightarrow1} \|D^{n}_{\varphi,u} f_{\varphi (w)}\|_{H^{\infty}_{\mu}}=0\).
-
(e)
\(\lim_{|\varphi(z)|\rightarrow 1}\frac{\mu(z)|u(z)| }{(1-|\varphi(z)|^{2})^{n+\alpha -1}}=0\).
Proof
The equivalence of statements (a)-(c) follows from Theorem 3.6.
(c) ⇒ (d) From (c), we see that, for every \(\varepsilon>0\), there is an \(N\in\mathbb{N}\) such that
for all \(j\geq N\).
Let \((z_{k})_{k\in\mathbb{N}}\subset{\mathbb{D}}\) be an arbitrary sequence such that \(|\varphi (z_{k})|\to1\) as \(k\to\infty\) (if such a sequence does not exist then the equality in (d) vacuously holds). Similarly to the proof of Theorem 2.1, we have
for \(k\in\mathbb{N}\), where \(M_{0}=\max_{n\leq j \leq N-1}j^{\alpha -1}\|D^{n}_{\varphi, u} (p_{j}) \|_{H^{\infty}_{\mu}}\).
Since \(|\varphi (z_{k})|\to1\) as \(k\to\infty\), from (22), we deduce that
Since ε is an arbitrary positive number, the implication follows.
(d) ⇒ (e) Let \((z_{k})_{k\in\mathbb{N}}\) be a sequence in \(\mathbb{D}\) such that \(\lim_{k\to\infty}|\varphi (z_{k})|=1\) (if such a sequence does not exist then the implication vacuously holds). Since the sequence \((f_{\varphi (z_{k})})_{k\in\mathbb{N}}\) is bounded in \(\mathcal{B}^{\alpha}\) and converges to 0 uniformly on compact subsets of \(\mathbb{D}\), by (9) and Lemma 3.5, we have
Therefore
which implies (e).
(e) ⇒ (a) Assume \((f_{k})_{k\in\mathbb{N}}\) is a bounded sequence in \(\mathcal{B}^{\alpha}\) converging to 0 uniformly on compact subsets of \(\mathbb{D}\). By the assumption, for any \(\varepsilon>0\), there exists a \(\delta\in(0,1)\) such that
when \(\delta<|\varphi(z)|<1\).
Therefore, since \(u\in H^{\infty}_{\mu}\) we have
where \(\Omega_{\delta}=\{ z\in\mathbb{D}:|\varphi(z)| \leq\delta\}\).
Since \((f_{k})_{k\in\mathbb{N}}\) converges to 0 uniformly on compact subsets of \(\mathbb{D}\), by Cauchy’s estimate so do the sequences \((f^{(n)}_{k})_{k\in\mathbb{N}}\) for every \(n\in\mathbb{N}\). Letting \(k\to \infty\) in (25) and using the fact that ε is an arbitrary positive number, we obtain \(\lim_{k\to\infty}\|D^{n}_{\varphi,u} f_{k}\|_{H^{\infty}_{\mu}}=0\). By Lemma 3.5, we deduce that \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\to H_{\mu}^{\infty}\) is compact. □
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Acknowledgements
Songxiao Li was supported by the Macao Science and Technology Development Fund (No. 083/2014/A2). Stevo Stević was supported by the Serbian Ministry of Science projects III 41025 and III 44006.
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Li, S., Stević, S. Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces. J Inequal Appl 2015, 265 (2015). https://doi.org/10.1186/s13660-015-0770-9
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DOI: https://doi.org/10.1186/s13660-015-0770-9