- Research
- Open access
- Published:
Generalized weighted composition operators on Bloch-type spaces
Journal of Inequalities and Applications volume 2015, Article number: 59 (2015)
Abstract
In this paper, we give three different characterizations for the boundedness and compactness of generalized weighted composition operators on Bloch-type spaces, especially we characterize them in terms of the sequence of Bloch-type norms of the generalized weighted composition operator applied to the functions \(I^{j}(z)=z^{j}\).
1 Introduction
Let \(\mathbb{D}\) be an open unit disk in the complex plane ℂ and \(H(\mathbb{D})\) be the space of analytic functions on \(\mathbb {D}\). For \(0<\alpha <\infty\), the Bloch-type space (or α-Bloch space) \(\mathcal{B}^{\alpha}\) is the space that consists of all analytic functions f on \(\mathbb{D}\) such that
\(\mathcal{B}^{\alpha}\) becomes a Banach space under the norm \(\|f\|_{\mathcal{B}^{\alpha}}=|f(0)|+B_{\alpha }(f)\). When \(\alpha=1\), \(\mathcal{B}^{1}=\mathcal{B}\) is the well-known Bloch space. See [1, 2] for more information on Bloch-type spaces.
Throughout this paper, φ denotes a nonconstant analytic self-map of \(\mathbb{D}\). The composition operator \(C_{\varphi}\) induced by φ is defined by \(C_{\varphi}f = f \circ\varphi\) for \(f \in H(\mathbb{D})\). For a fixed \(u \in H(\mathbb{D})\), define a linear operator \(uC_{\varphi}\) as follows:
The operator \(uC_{\varphi}\) is called the weighted composition operator. The weighted composition operator is a generalization of the composition operator and the multiplication operator defined by \(M_{u}f=uf\).
A basic problem concerning composition operators on various Banach function spaces is to relate the operator theoretic properties of \(C_{\varphi}\) to the function theoretic properties of the symbol φ, which attracted a lot of attention recently; the reader can refer to [3].
The differentiation operator D is defined by \(Df=f'\), \(f\in H(\mathbb{D})\). For a nonnegative integer n, we define
Let φ be an analytic self-map of \(\mathbb{D}\), \(u \in H(\mathbb {D})\), and let n be a nonnegative integer. Define the linear operator \(D^{n}_{\varphi, u}\), called the generalized weighted composition operator, by (see [4–6])
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 \(n=1\), \(u(z)=\varphi'(z)\), then \(D^{n}_{\varphi, u}= DC_{\varphi}\), which was studied in [7–10]. For \(u(z)=1\), \(D^{n}_{\varphi, u}= C_{\varphi}D^{n}\), which was studied in [7, 11–14]. For the study of the generalized weighted composition operator on various function spaces, see, for example, [4–6, 15–19].
It is well known that the composition operator is bounded on the Bloch space by the Schwarz-Pick lemma. Composition operators and weighted composition operators on Bloch-type spaces were studied, for example, in [20–28]. The product-type operators on or into Bloch-type spaces have been studied in many papers recently, see [7–11, 13, 14, 18, 29–36] for example. In [27], Wulan et al. obtained a characterization for the compactness of the composition operators acting on the Bloch space as follows.
Theorem A
Let φ be an analytic self-map of \(\mathbb{D}\). Then \(C_{\varphi}: \mathcal{B}\rightarrow \mathcal{B}\) is compact if and only if
In [14], Wu and Wulan obtained two characterizations for the compactness of the product of differentiation and composition operators acting on the Bloch space as follows.
Theorem B
Let φ be an analytic self-map of \(\mathbb{D}\), \(n\in \mathbb {N}\). Then the following statements are equivalent.
-
(a)
\(C_{\varphi}D^{n}:\mathcal{B}\rightarrow \mathcal{B}\) is compact.
-
(b)
\(\lim_{j\rightarrow\infty}\|C_{\varphi}D^{n} I^{j} \|_{\mathcal{B}}=0\), where \(I^{j}(z)=z^{j}\).
-
(c)
\(\lim_{|a|\rightarrow1}\|C_{\varphi}D^{n}\sigma_{a}(z)\|_{\mathcal{B}}=0\), where \(\sigma_{a}(z)=(a-z)/(1-\overline{a}z)\) is the Möbius map on \(\mathbb{D}\).
Motivated by Theorems A and B, in this work we show that \(D^{n}_{\varphi,u}:\mathcal {B}^{\alpha }\to\mathcal{B}^{\beta}\) is bounded (respectively, compact) if and only if the sequence \((j^{\alpha -1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal{B}^{\beta}})_{j=n}^{\infty}\) is bounded (respectively, convergent to 0 as \(j\to\infty\)), where \(I^{j}(z)=z^{j}\). Moreover, we use two families of functions to characterize the boundedness and compactness of the operator \(D^{n}_{\varphi, u}\).
Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. In addition, we say that \(A\preceq B\) if there exists a constant C such that \(A\leq CB\). The symbol \(A\approx B\) means that \(A \preceq B \preceq A\).
2 Main results and proofs
In this section, we give our main results and proofs. First we characterize the boundedness of the operator \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha }\to\mathcal{B}^{\beta}\).
Theorem 1
Let n be a positive integer, \(0<\alpha , \beta<\infty\), \(u \in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). Then the following statements are equivalent.
-
(a)
The operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to \mathcal{B}^{\beta}\) is bounded.
-
(b)
\(\sup_{j\geq n} j^{ \alpha-1}\|D^{n}_{\varphi, u} I^{j}(z)\|_{\mathcal{B}^{\beta}}<\infty\), where \(I^{j}(z)=z^{j}\).
-
(c)
\(u\in\mathcal{B}^{\beta}\), \(\sup_{z\in\mathbb {D}}(1-|z|^{2})^{\beta}|u(z) | |\varphi'(z)|<\infty\) and
$$\sup_{a\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}f_{a} \bigr\| _{\mathcal{B}^{\beta}} < \infty,\qquad \sup_{a\in \mathbb {D}}\bigl\| D^{n}_{\varphi,u}h_{a}\bigr\| _{\mathcal{B}^{\beta}} <\infty, $$where
$$f_{a}(z)=\frac{1-|a|^{2}}{(1-\overline{a} z)^{\alpha}} \quad\textit{and} \quad h_{a}(z)= \frac{(1-|a|^{2})^{2}}{(1-\overline{a} z)^{\alpha+1}},\quad z\in \mathbb {D}. $$ -
(d)
$$\sup_{z\in\mathbb{D} } \frac{(1-|z |^{2})^{\beta}|u(z)|| \varphi' (z) | }{(1-|\varphi(z)|^{2})^{\alpha +n}} < \infty \quad\textit{and}\quad \sup _{z\in\mathbb{D} } \frac{(1-|z|^{2})^{\beta}|u'(z)|}{(1-|\varphi(z)|^{2})^{\alpha +n-1}}<\infty . $$
Proof
(a) ⇒ (b) This implication is obvious, since for \(j\in\mathbb{N}\), the function \(j^{ \alpha-1} I^{j}\) is bounded in \(\mathcal{B}^{\alpha }\) and \(j^{ \alpha-1}\|I^{j}\|_{\mathcal{B}^{\alpha }} \approx1\).
(b) ⇒ (c) Assume that (b) holds and let \(Q=\sup_{j\ge n}j^{ \alpha-1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal{B}^{\beta}} \). For any \(a\in \mathbb {D}\), it is easy to see that \(f_{a}\) and \(h_{a}\) have bounded norms in \(\mathcal{B}^{\alpha}\). It is clear that
By Stirling’s formula, we have \(\frac{\Gamma(j+\alpha)}{j!\Gamma(\alpha)}\approx j^{\alpha-1} \) as \(j\rightarrow\infty\). Using linearity we get
Therefore, by the arbitrariness of \(a\in \mathbb {D}\),
In addition, applying the operator \(D^{n}_{\varphi, u}\) to \(I^{j}\) with \(j=n, n+1\), we obtain
while for \(j< n\), \((D^{n}_{\varphi,u}I^{j})'(z)=0\). Thus, using the boundedness of the function φ, we have \(u\in\mathcal{B}^{\beta}\) and \(\sup_{z\in \mathbb{D}}(1-|z|^{2})^{\beta}|u(z) | |\varphi'(z)|<\infty\).
(c) ⇒ (d) Assume that (c) holds. Let
For \(w\in\mathbb{D}\), set
It is easy to check that \(g_{w}\in\mathcal{B}^{\alpha }\), \(\|g_{w}\|_{\mathcal{B}^{\alpha }} <\infty\) for every \(w\in\mathbb{D}\). Moreover,
In addition,
It follows that
for any \(\lambda\in \mathbb {D}\). For any fixed \(r\in (0,1)\), from (2.1) we have
From the assumption that \(\sup_{z\in\mathbb{D}}(1-|z|^{2})^{\beta}|u(z) | |\varphi'(z)|<\infty\), we get
Therefore, (2.2) and (2.3) yield the first inequality of (d).
Next, note that
for any \(\lambda\in \mathbb {D}\). From (2.1) we get
By arbitrary \(\lambda\in\mathbb{D} \), we get
Combining (2.4) with the fact that \(u \in\mathcal{B}^{\beta}\), similarly to the former proof, we get the second inequality of (d).
(d) ⇒ (a) For any \(f \in\mathcal{B}^{\alpha }\), we have
where in the last inequality we used the fact that for \(f \in \mathcal{B}^{\alpha }\) (see [2])
Moreover
From (d) we see that
Therefore the operator \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha }\rightarrow\mathcal{B}^{\beta}\) is bounded. The proof is complete. □
For the study of the compactness of \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\rightarrow\mathcal{B}^{\beta}\), we need the following lemma, which can be proved in a standard way; see, for example, Proposition 3.11 in [3].
Lemma 2
Let n be a positive integer, \(0<\alpha , \beta<\infty\), \(u \in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). Then \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\rightarrow\mathcal{B}^{\beta}\) is compact if and only if \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\rightarrow\mathcal{B}^{\beta}\) is bounded and for any bounded sequence \((f_{j})_{j\in{ \mathbb {N}}}\) in \(\mathcal{B}^{\alpha}\) which converges to zero uniformly on compact subsets of \(\mathbb{D}\), \(\|D^{n}_{\varphi,u} f_{j} \|_{\mathcal {B}^{\beta}}\to0\) as \(j\to\infty\).
Theorem 3
Let n be a positive integer, \(0<\alpha , \beta<\infty\), \(u \in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\) such that \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\to\mathcal{B}^{\beta}\) is bounded. Then the following statements are equivalent.
-
(a)
\(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\to\mathcal{B}^{\beta}\) is compact.
-
(b)
\(\lim_{j\rightarrow\infty} j^{\alpha -1}\|D^{n}_{\varphi, u} I^{j} \|_{\mathcal{B}^{\beta}}=0\), where \(I^{j}(z)=z^{j}\).
-
(c)
\(\lim_{|\varphi (a)|\to1}\|D^{n}_{\varphi,u}f_{\varphi(a)}\|_{\mathcal{B}^{\beta}}=0\) and \(\lim_{|\varphi (a)|\to1}\|D^{n}_{\varphi,u}h_{\varphi(a)}\|_{\mathcal{B}^{\beta}}=0\).
-
(d)
$$\lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z |^{2})^{\beta}|u(z)||\varphi'(z)|}{(1-|\varphi(z)|^{2})^{n+\alpha }}=0 \quad \textit{and}\quad \lim _{ |\varphi (z)|\rightarrow1}\frac{(1-|z|^{2})^{\beta}|u'(z)|}{(1-|\varphi(z)|^{2})^{n+\alpha -1}}=0. $$
Proof
(a) ⇒ (b) Assume that \(D^{n}_{\varphi,u}:\mathcal {B}^{\alpha}\to\mathcal{B}^{\beta}\) is compact. Since the sequence \(\{j^{\alpha -1}I^{j}\}\) is bounded in \(\mathcal{B}^{\alpha}\) and converges to 0 uniformly on compact subsets, by Lemma 2 it follows that \(j^{\alpha -1}\|D^{n}_{\varphi, u} I^{j}\| _{\mathcal{B}^{\beta}} \to0\) as \(j\to\infty\).
(b) ⇒ (c) Suppose that (b) holds. Fix \(\varepsilon >0\) and choose \(N\in \mathbb {N}\) such that \(j^{\alpha -1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal {B}^{\beta}} <\varepsilon \) for all \(j\ge N\). Let \(z_{k} \in \mathbb{D}\) such that \(|\varphi (z_{k})|\to1\) as \(k\to\infty\). Arguing as in the proof of Theorem 1, we have
where \(Q=\sup_{j\ge n}j^{\alpha -1}\|D^{n}_{\varphi,u}I^{j}\|_{\mathcal{B}^{\beta}} \). Since \(|\varphi (z_{k})|\to1\) as \(k\to\infty\), from the last inequality and the arbitrariness of ε, we get \(\lim_{k\rightarrow\infty}\|D^{n}_{\varphi,u}f_{\varphi(z_{k})}\|_{\mathcal{B}^{\beta}} =0\), i.e., \(\lim_{|\varphi (a)|\to 1}\|D^{n}_{\varphi,u}f_{\varphi(a)}\|_{\mathcal{B}^{\beta}} =0\).
Notice that
arguing as in the proof of Theorem 1, we get
Therefore, \(\lim_{k\to\infty}\|D^{n}_{\varphi,u}h_{\varphi (z_{k})}\|_{\mathcal{B}^{\beta}} \le C\varepsilon \). By the arbitrariness of ε, we obtain the desired result.
(c) ⇒ (d) To prove (d) we only need to show that if \((z_{k})_{k\in \mathbb {N}}\) is a sequence in \(\mathbb{D}\) such that \(|\varphi(z_{k})| \rightarrow1\) as \(k\rightarrow\infty\), then
Let \((z_{k})_{k\in \mathbb {N}}\) be such a sequence that \(|\varphi(z_{k})| \rightarrow1\) as \(k\rightarrow\infty\). Arguing as in the proof of Theorem 1, we obtain
Hence \(\lim_{k\to\infty}\|D^{n}_{\varphi,u}g_{\varphi (z_{k})}\|_{\mathcal{B}^{\beta}} = 0\). Similarly to the proof of Theorem 1, we have
which implies
In addition,
From (2.6) and the assumption that \(\|D^{n}_{\varphi,u} f_{\varphi (z_{k})} \|_{{\mathcal{B}^{\beta}} }\to0\) as \(k\to\infty\), we have
as desired.
(d) ⇒ (a) Assume that \((f_{k})_{k\in \mathbb {N}}\) 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 \(\delta\in(0,1)\) such that
when \(\delta<|\varphi(z)|<1\). Suppose that \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha }\to\mathcal{B}^{\beta}\) is bounded, by Theorem 1, we have
and
Let \(K=\{ z\in\mathbb{D}:|\varphi(z)| \leq\delta\}\). Then by (2.8) and (2.9) we have that
i.e., we get
Since \(f_{k}\) converges to 0 uniformly on compact subsets of \(\mathbb{D}\) as \(k\to\infty\), Cauchy’s estimate gives that \(f^{(n)}_{k} \to0\) as \(k\to\infty\) on compact subsets of \(\mathbb{D}\). Hence, letting \(k\to\infty\) in (2.10) and using the fact that ε is an arbitrary positive number, we obtain \(\|D^{n}_{\varphi,u} f_{k}\|_{\mathcal{B}^{\beta}}\rightarrow0\) as \(k\to\infty\). Applying Lemma 2 the result follows. □
References
Zhu, K: Operator Theory in Function Spaces. Dekker, New York (1990)
Zhu, K: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23, 1143-1177 (1993)
Cowen, CC, MacCluer, BD: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)
Zhu, X: Products of differentiation, composition and multiplication from Bergman type spaces to Bers type space. Integral Transforms Spec. Funct. 18, 223-231 (2007)
Zhu, X: Generalized weighted composition operators on weighted Bergman spaces. Numer. Funct. Anal. Optim. 30, 881-893 (2009)
Zhu, X: Generalized weighted composition operators from Bloch spaces into Bers-type spaces. Filomat 26, 1163-1169 (2012)
Hibschweiler, R, Portnoy, N: Composition followed by differentiation between Bergman and Hardy spaces. Rocky Mt. J. Math. 35, 843-855 (2005)
Li, S, Stević, S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 9, 195-205 (2007)
Li, S, Stević, S: Composition followed by differentiation between \(H^{\infty}\) and α-Bloch spaces. Houst. J. Math. 35, 327-340 (2009)
Yang, W: Products of composition differentiation operators from \(Q_{K}(p,q)\) spaces to Bloch-type spaces. Abstr. Appl. Anal. 2009, Article ID 741920 (2009)
Liang, Y, Zhou, Z: Essential norm of the product of differentiation and composition operators between Bloch-type space. Arch. Math. 100, 347-360 (2013)
Stević, S: Products of composition and differentiation operators on the weighted Bergman space. Bull. Belg. Math. Soc. Simon Stevin 16, 623-635 (2009)
Stević, S: Norm and essential norm of composition followed by differentiation from α-Bloch spaces to \(H^{\infty}_{\mu}\). Appl. Math. Comput. 207, 225-229 (2009)
Wu, Y, Wulan, H: Products of differentiation and composition operators on the Bloch space. Collect. Math. 63, 93-107 (2012)
Li, H, Fu, X: A new characterization of generalized weighted composition operators from the Bloch space into the Zygmund space. J. Funct. Spaces Appl. 2013, Article ID 925901 (2013)
Stević, S: Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211, 222-233 (2009)
Stević, S: Weighted differentiation composition operators from mixed-norm spaces to the n-th weighted-type space on the unit disk. Abstr. Appl. Anal. 2010, Article ID 246287 (2010)
Stević, S: Weighted differentiation composition operators from \(H^{\infty}\) and Bloch spaces to n-th weighted-type spaces on the unit disk. Appl. Math. Comput. 216, 3634-3641 (2010)
Yang, W, Zhu, X: Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces. Taiwan. J. Math. 3, 869-883 (2012)
Lou, Z: Composition operators on Bloch type spaces. Analysis 23, 81-95 (2003)
Maccluer, B, Zhao, R: Essential norm of weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33, 1437-1458 (2003)
Madigan, K, Matheson, A: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347, 2679-2687 (1995)
Manhas, J, Zhao, R: New estimates of essential norms of weighted composition operators between Bloch type spaces. J. Math. Anal. Appl. 389, 32-47 (2012)
Ohno, S: Weighted composition operators between \(H^{\infty}\) and the Bloch space. Taiwan. J. Math. 5, 555-563 (2001)
Ohno, S, Stroethoff, K, Zhao, R: Weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33, 191-215 (2003)
Tjani, M: Compact composition operators on some Möbius invariant Banach space. Ph.D. dissertation, Michigan State University (1996)
Wulan, H, Zheng, D, Zhu, K: Compact composition operators on BMOA and the Bloch space. Proc. Am. Math. Soc. 137, 3861-3868 (2009)
Zhao, R: Essential norms of composition operators between Bloch type spaces. Proc. Am. Math. Soc. 138, 2537-2546 (2010)
Li, S, Stević, S: Weighted composition operators from Bergman-type spaces into Bloch spaces. Proc. Indian Acad. Sci. Math. Sci. 117, 371-385 (2007)
Li, S, Stević, S: Weighted composition operators from \(H^{\infty}\) to the Bloch space on the polydisc. Abstr. Appl. Anal. 2007, Article ID 48478 (2007)
Li, S, Stević, S: Products of composition and integral type operators from \(H^{\infty}\) to the Bloch space. Complex Var. Elliptic Equ. 53(5), 463-474 (2008)
Li, S, Stević, S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput. 206(2), 825-831 (2008)
Li, S, Stević, S: Products of integral-type operators and composition operators between Bloch-type spaces. J. Math. Anal. Appl. 349, 596-610 (2009)
Stević, S: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 354, 426-434 (2009)
Stević, S: Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Sib. Math. J. 50(4), 726-736 (2009)
Stević, S: On an integral operator between Bloch-type spaces on the unit ball. Bull. Sci. Math. 134, 329-339 (2010)
Acknowledgements
The author was partially supported by the Macao Science and Technology Development Fund (No. 098/2013/A3), NSF of Guangdong Province (No. S2013010011978) and NNSF of China (No. 11471143).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Zhu, X. Generalized weighted composition operators on Bloch-type spaces. J Inequal Appl 2015, 59 (2015). https://doi.org/10.1186/s13660-015-0580-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0580-0