Abstract
In this paper, we give some estimates of the essential norm for generalized weighted composition operators from the Bloch space to the Zygmund space. Moreover, we give a new characterization for the boundedness and compactness of the operator.
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1 Introduction
Let X and Y be Banach spaces. The essential norm of a bounded linear operator \(T:X\rightarrow Y\) is its distance to the set of compact operators K mapping X into Y, that is,
where \(\|\cdot\|_{X\rightarrow Y}\) is the operator norm.
Let \(\mathbb{D}\) be the open unit disk in the complex plane \(\mathbb{C}\) and \(H(\mathbb{D})\) the space of analytic functions on \(\mathbb{D}\). Let φ be a nonconstant analytic self-map of \(\mathbb{D}\), \(u \in H(\mathbb{D})\), and n be a nonnegative integer. The generalized weighted composition operator, denoted by \(D^{n}_{\varphi, u}\), is defined on \(H(\mathbb{D})\) by
When \(n=0\), the generalized weighted composition operator \(D^{n}_{\varphi , u}\) is the weighted composition operator, denoted by \(uC_{\varphi}\). In particular, when \(n=0\) and \(u=1\), we get the composition operator \(C_{\varphi}\). If \(n=1\) and \(u(z)=\varphi'(z)\), then \(D^{n}_{\varphi, u}= DC_{\varphi}\), which was widely studied, for example, in [1–9]. If \(u(z)=1\), then \(D^{n}_{\varphi, u}= C_{\varphi}D^{n}\), which was studied, for example, in [1, 5, 10, 11]. For the study of the generalized weighted composition operator on various function spaces see, for example, [12–21]. Recently there has been a huge interest in the study of various related product-type operators containing composition operators; see, e.g., [22–30] and the references therein.
The Bloch space, denoted by \(\mathcal{B}\), is defined to be the set of all \(f \in H(\mathbb{D})\) such that
\(\mathcal{B}\) is a Banach space with the above norm. An \(f\in \mathcal{B}\) is said to belong to the little Bloch space \(\mathcal {B}_{0}\) if \(\lim_{|z|\rightarrow1}|f'(z)|(1-|z|^{2})=0\). See [31] for more information of Bloch spaces. Composition operators, as well as weighted composition operators mapping into Bloch-type spaces were studied a lot see, for example, [3, 6, 16, 32–45].
The Zygmund space, denoted by \(\mathcal{Z}\), is the space consisting of all \(f \in H(\mathbb{D})\) such that
It is easy to see that \(\mathcal{Z}\) is a Banach space with the above norm \(\| \cdot\|_{\mathcal{Z}}\). See [4, 7, 12, 15, 16, 22, 36, 46–50] for some results of the Zygmund space and related operators mapping into the Zygmund space or into some of its generalizations.
In 1995, Madigan and Matheson proved that \(C_{\varphi}:\mathcal {B}\rightarrow \mathcal{B}\) is compact if and only if (see [38])
In 1999, Montes-Rodrieguez in [40] obtained the exact value for the essential norm of the operator \(C_{\varphi}: \mathcal {B}\rightarrow\mathcal{B}\), i.e.,
Tjani in [43] proved that \(C_{\varphi}:\mathcal{B}\rightarrow\mathcal{B}\) is compact if and only if \(\lim_{|a|\rightarrow1} \| C_{\varphi}\sigma_{a} \| _{\mathcal{B}}=0\), where \(\sigma_{a} = \frac{a-z}{1-\overline{a}z}\). Wulan et al. in [44] showed that \(C_{\varphi}:\mathcal{B}\rightarrow\mathcal{B}\) is compact if and only if \(\lim_{j\rightarrow\infty}\| \varphi^{j} \|_{\mathcal{B}}=0\). Ohno et al. studied the boundedness and compactness of the operator \(u C_{\varphi}\) on the Bloch space in [41]. The estimate for the essential norm of the operator \(u C_{\varphi}\) on the Bloch space was given in [37]. Some new estimates for the essential norm of \(u C_{\varphi}\) on the Bloch space were given in [33, 39]. In [21], Zhu has obtained some estimates for the essential norm of \(D^{n}_{\varphi,u}\) on the Bloch space when n is a positive integer.
Stević studied the boundedness and compactness of \(D^{n}_{\varphi,u}: \mathcal{B} \rightarrow \mathcal{Z} \) in [16] (see also [50]). In [12], Li and Fu obtained a new characterization for the boundedness, as well as the compactness for \(D^{n}_{\varphi,u}: \mathcal{B} \rightarrow\mathcal{Z} \) by using three families of functions. We combine the results in [12] and [16] as follows.
Theorem A
Let n be a positive integer, \(u \in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Suppose that \(D^{n}_{\varphi,u} : \mathcal {B}\rightarrow \mathcal{Z} \) is bounded, then the following statements are equivalent:
-
(a)
The operator \(D^{n}_{\varphi,u} :\mathcal{B}\rightarrow \mathcal{Z} \) is compact.
-
(b)
$$\lim_{|\varphi(w)|\rightarrow1} \bigl\Vert D^{n}_{\varphi,u} f_{\varphi (w)}\bigr\Vert _{\mathcal{Z}}=\lim_{|\varphi(w)|\rightarrow1} \bigl\Vert D^{n}_{\varphi,u} g_{\varphi (w)}\bigr\Vert _{\mathcal{Z}}=\lim_{|\varphi (w)|\rightarrow1} \bigl\Vert D^{n}_{\varphi,u} h_{\varphi (w)}\bigr\Vert _{\mathcal{Z}}=0, $$
where
$$\begin{aligned}& f_{\varphi (w)}(z)=\frac{1-|\varphi (w)|^{2}}{1-\overline{\varphi (w)}z}, \qquad g_{\varphi (w)}(z)= \frac{(1-|\varphi (w)|^{2})^{2}}{(1-\overline{\varphi (w)}z)^{2}}, \\& h_{\varphi (w)}(z)=\frac{ (1-|\varphi (w)|^{2})^{3}}{(1-\overline {\varphi (w)}z)^{3}},\quad z\in\mathbb{D}. \end{aligned}$$ -
(c)
$$\begin{aligned} \lim_{|\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2}) |u''(z)| }{ (1-|\varphi(z)|^{2})^{n} } =& \lim_{|\varphi(z)|\rightarrow1} \frac{(1-|z|^{2}) |u(z)||\varphi'(z)|^{2} }{ (1-|\varphi(z)|^{2})^{n+2} } \\ =& \lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |2u'(z)\varphi'(z)+u(z)\varphi''(z)| }{(1-| \varphi(z)|^{2})^{1+n} } =0 . \end{aligned}$$
Motivated by these observations, the purpose of this paper is to give some estimates of the essential norm for the operator \(D^{n}_{\varphi ,u}:\mathcal{B}\rightarrow\mathcal{Z}\). Moreover, we give a new characterization for the boundedness, compactness, and essential norm of the operator \(D^{n}_{\varphi,u}:\mathcal{B}\rightarrow\mathcal{Z}\).
Throughout this paper, we say that \(P\lesssim Q\) if there exists a constant C such that \(P\leq CQ\). The symbol \(P\approx Q\) means that \(P\lesssim Q\lesssim P\).
2 Essential norm of \(D^{n}_{\varphi,u}:\mathcal{B} \to \mathcal{Z} \)
In this section, we give two estimates of the essential norm for the operator \(D^{n}_{\varphi,u}:\mathcal{B} \to\mathcal{Z} \).
Theorem 2.1
Let n be a positive integer, \(u \in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\) such that \(D^{n}_{\varphi,u}:\mathcal{B} \to\mathcal{Z} \) is bounded. Then
where
and
Proof
First we prove that \(\max \{ A, B, C \} \leq\|D^{n}_{\varphi,u}\|_{e,\mathcal {B}\to \mathcal{Z}} \). Let \(a\in\mathbb{D}\). Define
It is easy to check that \(f_{a}, g_{a}, h_{a}\in\mathcal{B}_{0}\) and \(\|f_{a}\| _{\mathcal{B}} \lesssim1\), \(\|g_{a}\|_{\mathcal{B}} \lesssim1\), \(\|h_{a}\|_{\mathcal{B}} \lesssim1\) for all \(a\in\mathbb{D}\) and \(f_{a}\), \(g_{a}\), \(h_{a}\) converge to 0 weakly in \(\mathcal{B}\) as \(|a|\to1\). This follows since a bounded sequence contained in \(\mathcal{B}_{0}\) which converges uniformly to 0 on compact subsets of \(\mathbb{D}\) converges weakly to 0 in \(\mathcal{B}\) (see [37, 42]). Thus, for any compact operator \(K: \mathcal{B}_{0}\to\mathcal{Z}\), we have
Hence
and
Therefore, from the definition of the essential norm, we obtain
Next, we prove that \(\|D^{n}_{\varphi,u}\|_{e,\mathcal{B}\to\mathcal{Z}} \gtrsim\max \{ E, F , G \}\). Let \(\{z_{j}\}_{j\in\mathbb{N}}\) be a sequence in \(\mathbb{D}\) such that \(|\varphi(z_{j})|\rightarrow1\) as \(j\rightarrow\infty\). Define
and
Similarly to the above we see that all \(k_{j}\), \(l_{j}\), and \(m_{j}\) belong to \(\mathcal{B}_{0}\) and converge to 0 weakly in \(\mathcal{B}\). Moreover,
Then for any compact operator \(K: \mathcal{B} \to \mathcal{Z} \), we obtain
and
From the definition of the essential norm, we obtain
and
Hence
Now, we prove that
For \(r\in[0,1)\), set \(K_{r}: H(\mathbb{D})\to H(\mathbb{D})\) by \((K_{r} f)(z)=f_{r}(z)=f(rz)\), \(f\in H(\mathbb{D})\). It is obvious that \(f_{r}\to f\) uniformly on compact subsets of \(\mathbb{D}\) as \(r\to1\). Moreover, the operator \(K_{r}\) is compact on \(\mathcal{B}\) and \(\|K_{r}\|_{\mathcal{B}\to\mathcal{B}}\leq1\) (see [37]). Let \(\{ r_{j}\}\subset(0,1)\) be a sequence such that \(r_{j}\to1\) as \(j\to\infty\). Then for all positive integer j, the operator \(D^{n}_{\varphi,u} K_{r_{j}}: \mathcal{B}\rightarrow \mathcal {Z}\) is compact. By the definition of the essential norm, we get
Therefore, we only need to prove that
and
For any \(f\in\mathcal{B}\) such that \(\|f\|_{\mathcal{B}}\leq1\), we consider
where \(\|f\|_{*}=\sup_{z \in\mathbb{D}}(1-|z|^{2}) |f''(z)|\).
It is obvious that
and
Now, we consider
where \(N\in\mathbb{N }\) is large enough such that \(r_{j}\geq\frac {1}{2}\) for all \(j\geq N\),
and
Since \(D^{n}_{\varphi,u}:\mathcal{B} \to\mathcal{Z} \) is bounded, by Theorem 1 of [12], we see that \(u\in\mathcal{Z} \),
and
Since \(r^{n+1}_{j}f^{(n+1)}_{r_{j}}\to f^{(n+1)}\), as well as \(r^{n+2}_{j}f^{(n+2)}_{r_{j}}\to f^{(n+2)} \) uniformly on compact subsets of \(\mathbb{D}\) as \(j\to\infty\), we have
and
Similarly, from the fact that \(u \in\mathcal{Z}\) we have
Next we consider \(Q_{2}\). We have \(Q_{2}\leq\limsup_{j\to\infty }(S_{1}+S_{2})\), where
and
First we estimate \(S_{1}\). Using the fact that \(\|f\|_{\mathcal{B}}\leq 1\) and Theorem 5.4 in [31], we have
Taking the limit as \(N\to\infty\) we obtain
Similarly, we have \(\limsup_{j\to\infty}S_{2}\lesssim A+B+C\), i.e., we get
From (2.9), we see that
Similarly we have \(\limsup_{j\to\infty}S_{2} \lesssim E\). Therefore
Next we consider \(Q_{4}\). We have \(Q_{4}\leq\limsup_{j\to\infty }(S_{3}+S_{4})\), where
and
After some calculation, we have
Taking the limit as \(N\to\infty\) we obtain
Similarly, we have \(\limsup_{j\to\infty}S_{4} \lesssim A+B+C\), i.e., we get
From (2.12), we see that
Similarly we have \(\limsup_{j\to\infty}S_{4} \lesssim F\). Therefore
Finally we consider \(Q_{6}\). We have \(Q_{6}\leq\limsup_{j\to\infty }(S_{5}+S_{6})\), where
and
After some calculation, we have
Taking the limit as \(N\to\infty\) we obtain
Similarly, we have \(\limsup_{j\to\infty}S_{6} \lesssim A+B+C\), i.e., we get
From (2.15), we see that
Similarly we have \(\limsup_{j\to\infty}S_{6} \lesssim G\). Therefore
Hence, by (2.2), (2.3), (2.4), (2.5), (2.6), (2.7), (2.8), (2.10), (2.13), and (2.16) we get
Similarly, by (2.2), (2.3), (2.4), (2.5), (2.6), (2.7), (2.8), (2.11), (2.14), and (2.17) we get
Therefore, by (2.1), (2.18), and (2.19), we obtain
This completes the proof of Theorem 2.1. □
3 New characterization of \(D^{n}_{\varphi,u}: \mathcal{B}\to \mathcal{Z}\)
In this section, we give a new characterization for the boundedness, compactness, and essential norm of the operator \(D^{n}_{\varphi ,u}:\mathcal{B} \to\mathcal{Z} \). For this purpose, we present some definitions and some lemmas which will be used later.
The weighted space, denoted by \(H^{\infty}_{v}\), consists of all \(f\in H(\mathbb{D})\) such that
where \(v:\mathbb{D}\rightarrow R_{+}\) is a continuous, strictly positive, and bounded function. \(H^{\infty}_{v}\) is a Banach space under the norm \(\| \cdot\|_{v}\). The weighted v is called radial if \(v(z)=v(|z|)\) for all \(z\in\mathbb{D}\). The associated weight ṽ of v is as follows:
When \(v=v_{\alpha}(z)=(1-|z|^{2})^{\alpha}\) (\(0<\alpha <\infty\)), it is well known that \(\tilde{v}_{\alpha}(z)=v_{\alpha}(z)\). In this case, we denote \(H^{\infty}_{v}\) by \(H^{\infty}_{v_{\alpha}}\).
Lemma 3.1
[33]
For \(\alpha >0\), we have \(\lim_{k\rightarrow\infty}k^{\alpha}\|z^{k-1}\|_{v_{\alpha}}=(\frac {2\alpha }{e})^{\alpha}\).
Lemma 3.2
[51]
Let v and w be radial, non-increasing weights tending to zero at the boundary of \(\mathbb {D}\). Then the following statements hold.
-
(a)
The weighted composition operator \(uC_{\varphi}:H_{v}^{\infty }\rightarrow H_{w}^{\infty}\) is bounded if and only if \(\sup_{z\in \mathbb{D}}\frac{w(z)}{\tilde{v}(\varphi(z))}|u(z)|<\infty\). Moreover, the following holds:
$$ \|uC_{\varphi}\|_{H_{v}^{\infty}\rightarrow H_{w}^{\infty}}=\sup_{z\in\mathbb{D}} \frac{w(z)}{\tilde{v}(\varphi (z))}\bigl\vert u(z)\bigr\vert . $$ -
(b)
Suppose \(uC_{\varphi}:H_{v}^{\infty}\rightarrow H_{w}^{\infty}\) is bounded. Then
$$ \|uC_{\varphi}\|_{e, H_{v}^{\infty}\rightarrow H_{w}^{\infty }}=\lim_{s\to1^{-}}\sup _{|\varphi(z)|>s}\frac{w(z)}{\tilde {v}(\varphi(z))}\bigl\vert u(z)\bigr\vert . $$
Lemma 3.3
[52]
Let v and w be radial, non-increasing weights tending to zero at the boundary of \(\mathbb {D}\). Then the following statements hold.
-
(a)
\(uC_{\varphi}:H_{v}^{\infty}\rightarrow H_{w}^{\infty}\) is bounded if and only if \(\sup_{k\geq0}\frac{\|u \varphi^{k}\|_{w}}{\|z^{k}\|_{v}}<\infty\), with the norm comparable to the above supremum.
-
(b)
Suppose \(uC_{\varphi}:H_{v}^{\infty}\rightarrow H_{w}^{\infty}\) is bounded. Then
$$ \|uC_{\varphi}\|_{e,H_{v}^{\infty}\rightarrow H_{w}^{\infty} }=\limsup_{k\to \infty} \frac{\|u \varphi^{k}\|_{w}}{\|z^{k}\|_{v}}. $$
Theorem 3.1
Let n be a positive integer, \(u \in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Then the operator \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) is bounded if and only if
Proof
By [16], \(D^{n}_{\varphi, u} : \mathcal{B} \to\mathcal {Z} \) is bounded if and only if
By Lemma 3.2, the first inequality in (3.2) is equivalent to the weighted composition operator \((2u'\varphi'+u\varphi'')C_{\varphi}: H^{\infty}_{v_{n+1}}\rightarrow H^{\infty}_{v_{1}}\) is bounded. By Lemma 3.3, this is equivalent to
The second inequality in (3.2) is equivalent to the operator \(u''C_{\varphi}: H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}\) is bounded. By Lemma 3.3, this is equivalent to
The third inequality in (3.2) is equivalent to the operator \(u\varphi^{\prime 2}C_{\varphi}: H^{\infty}_{v_{n+2}}\rightarrow H^{\infty}_{v_{1}}\) is bounded. By Lemma 3.3, this is equivalent to
By Lemma 3.1, we see that \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) is bounded if and only if
and
The proof is completed. □
Theorem 3.2
Let n be a positive integer, \(u \in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\) such that \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) is bounded. Then
where
Proof
From the proof of Theorem 3.1 we know that the boundedness of \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) is equivalent to the boundedness of the operators \((2u'\varphi'+u\varphi'')C_{\varphi}: H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}\), \(u''C_{\varphi}: H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}} \), and \(u\varphi^{\prime 2}C_{\varphi}: H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}\).
The upper estimate. By Lemmas 3.1 and 3.3, we get
and
It follows that
The lower estimate. From Theorem 2.1, and Lemmas 3.1 and 3.2, we have
and
Therefore \(\| D^{n}_{\varphi, u}\|_{e, \mathcal{B} \to \mathcal{Z} } \gtrsim\max \{M_{1}, M_{2}, M_{3} \}\). This completes the proof. □
From Theorem 3.2, we immediately get the following result.
Theorem 3.3
Let n be a positive integer, \(u \in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\) such that \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) is bounded. Then \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) is compact if and only if
and
4 Conclusion
The boundedness and compactness of \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) were characterized in [12] and [16]. In this paper, we give a new characterization for the boundedness and compactness of \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \). Moreover, using the method in [21], we completely characterize the essential norm of \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \).
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This project is partially supported by the Macao Science and Technology Development Fund (No. 083/2014/A2).
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Hu, Q., Shi, Y., Shi, Y. et al. Essential norm of generalized weighted composition operators from the Bloch space to the Zygmund space. J Inequal Appl 2016, 123 (2016). https://doi.org/10.1186/s13660-016-1066-4
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DOI: https://doi.org/10.1186/s13660-016-1066-4