Skip to content

Advertisement

  • Research
  • Open Access

Essential norm of generalized weighted composition operators from the Bloch space to the Zygmund space

Journal of Inequalities and Applications20162016:123

https://doi.org/10.1186/s13660-016-1066-4

  • Received: 19 January 2016
  • Accepted: 1 April 2016
  • Published:

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.

Keywords

  • Bloch space
  • Zygmund space
  • essential norm
  • generalized weighted composition operator

MSC

  • 30H30
  • 47B38

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,
$$\|T\|_{e, X\rightarrow Y}=\inf\bigl\{ \Vert T-K\Vert _{X\rightarrow Y}: K \mbox{ is compact} \bigr\} , $$
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
$$ \bigl(D^{n}_{\varphi, u} f\bigr) (z) =u(z) f^{(n)}\bigl( \varphi(z)\bigr) , \quad z\in\mathbb {D}. $$
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 [19]. 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, [1221]. Recently there has been a huge interest in the study of various related product-type operators containing composition operators; see, e.g., [2230] 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
$$\|f\|_{\mathcal{B}}=\bigl\vert f(0)\bigr\vert +\sup_{z \in\mathbb{D}} \bigl(1-|z|^{2}\bigr) \bigl\vert f'(z)\bigr\vert < \infty. $$
\(\mathcal{B}\) 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, 3245].
The Zygmund space, denoted by \(\mathcal{Z}\), is the space consisting of all \(f \in H(\mathbb{D})\) such that
$$\|f\|_{\mathcal{Z}}=\bigl\vert f(0)\bigr\vert +\bigl\vert f'(0)\bigr\vert +\sup_{z \in\mathbb{D}}\bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert f''(z)\bigr\vert < \infty. $$
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, 4650] 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])
$$\lim_{|\varphi(z)|\to1 } \frac{(1-|z|^{2}) }{(1-|\varphi(z)|^{2}) }\bigl\vert \varphi'(z) \bigr\vert =0. $$
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.,
$$\|C_{\varphi}\|_{e, \mathcal{B}\rightarrow\mathcal{B}}=\lim_{s\rightarrow1}\sup _{|\varphi(z)|>s}\frac{(1-|z |^{2}) |\varphi'(z)|}{(1-|\varphi(z)|^{2}) }. $$
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:
  1. (a)

    The operator \(D^{n}_{\varphi,u} :\mathcal{B}\rightarrow \mathcal{Z} \) is compact.

     
  2. (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
    $$\begin{aligned}& f_{\varphi (w)}(z)=\frac{1-|\varphi (w)|^{2}}{1-\overline{\varphi (w)}z}, \qquad g_{\varphi (w)}(z)= \frac{(1-|\varphi (w)|^{2})^{2}}{(1-\overline{\varphi (w)}z)^{2}}, \\& h_{\varphi (w)}(z)=\frac{ (1-|\varphi (w)|^{2})^{3}}{(1-\overline {\varphi (w)}z)^{3}},\quad z\in\mathbb{D}. \end{aligned}$$
     
  3. (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
$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \approx \max \{ A, B, C \}\approx\max \{ E, F, G \} , $$
where
$$\begin{aligned}& A:=\limsup_{|a|\to 1}\biggl\Vert D^{n}_{\varphi,u} \biggl(\frac{1-|a|^{2}}{1-\overline {a}z} \biggr)\biggr\Vert _{\mathcal{Z}}, \qquad B:= \limsup _{|a|\to 1}\biggl\Vert D^{n}_{\varphi,u} \biggl( \frac{(1-|a|^{2})^{2}}{(1-\overline {a}z)^{2}} \biggr)\biggr\Vert _{\mathcal{Z}}, \\& C:= \limsup_{|a|\to 1}\biggl\Vert D^{n}_{\varphi,u} \biggl(\frac{(1-|a|^{2})^{3}}{(1-\overline {a}z)^{3}} \biggr)\biggr\Vert _{\mathcal{Z}}, \qquad F:= \limsup _{|\varphi (z)|\rightarrow1}\frac{(1-|z|^{2}) |u''(z)| }{ (1-|\varphi(z)|^{2})^{n} }, \\& E:=\limsup_{ |\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |2u'(z)\varphi'(z)+u(z)\varphi''(z)| }{(1-|\varphi(z)|^{2})^{n+1} }, \end{aligned}$$
and
$$G:= \limsup_{ |\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |u(z)||\varphi'(z)|^{2} }{ (1-|\varphi(z)|^{2})^{n+2} }. $$

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
$$f_{a}(z)=\frac{1-|a|^{2}}{(1-\overline{a}z)}, \qquad g_{a}(z)= \frac {(1-|a|^{2})^{2}}{(1-\overline{a}z)^{2}}, \qquad h_{a}(z)=\frac{ (1-|a|^{2})^{3}}{(1-\overline{a}z)^{3}}, \quad z\in \mathbb{D}. $$
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
$$\lim_{|a|\to1}\|Kf_{a}\|_{\mathcal{Z}}=0, \qquad \lim _{|a|\to1}\|Kg_{a}\| _{\mathcal{Z}}=0, \qquad \lim _{|a|\to1}\|Kh_{a}\|_{\mathcal{Z}}=0. $$
Hence
$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{|a|\to 1}\bigl\Vert \bigl(D^{n}_{\varphi,u}-K \bigr)f_{a}\bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \geq \limsup_{|a|\to1}\bigl\Vert D^{n}_{\varphi,u} f_{a} \bigr\Vert _{\mathcal {Z}}- \limsup_{|a|\to1}\Vert Kf_{a}\Vert _{\mathcal{Z}}= A, \\& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{|a|\to 1}\bigl\Vert \bigl(D^{n}_{\varphi,u}-K \bigr)g_{a}\bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \geq \limsup_{|a|\to1}\bigl\Vert D^{n}_{\varphi,u} g_{a} \bigr\Vert _{\mathcal {Z}}- \limsup_{|a|\to1}\Vert Kg_{a}\Vert _{\mathcal{Z}}=B, \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim&\limsup_{|a|\to 1}\bigl\Vert \bigl(D^{n}_{\varphi,u}-K \bigr)h_{a}\bigr\Vert _{\mathcal{Z}} \\ \geq& \limsup_{|a|\to1}\bigl\Vert D^{n}_{\varphi,u} h_{a} \bigr\Vert _{\mathcal {Z}}-\limsup_{|a|\to1} \Vert Kh_{a}\Vert _{\mathcal{Z}}=C. \end{aligned}$$
Therefore, from the definition of the essential norm, we obtain
$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}}= \inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}} \gtrsim\max \{ A, B, C \} . $$
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
$$\begin{aligned}& k_{j}(z) = \frac{1-|\varphi(z_{j})|^{2}}{1-\overline{\varphi (z_{j})}z}-\frac{2n+5}{(n+1)(n+3)} \frac{(1-|\varphi (z_{j})|^{2})^{2}}{(1-\overline{\varphi(z_{j})}z)^{2}} +\frac{2}{(n+1)(n+3)} \frac{(1-|\varphi(z_{j})|^{2})^{3}}{(1-\overline {\varphi(z_{j})}z)^{3}}, \\& l_{j}(z) = \frac{1-|\varphi(z_{j})|^{2}}{1-\overline{\varphi (z_{j})}z}-\frac{2(n+3)}{2+(n+1)(n+4)} \frac{(1-|\varphi (z_{j})|^{2})^{2}}{(1-\overline{\varphi(z_{j})}z)^{2}} +\frac{2}{2+(n+1)(n+4)} \frac{(1-|\varphi(z_{j})|^{2})^{3}}{(1-\overline {\varphi(z_{j})}z)^{3}}, \end{aligned}$$
and
$$ m_{j}(z) = \frac{1-|\varphi(z_{j})|^{2}}{1-\overline{\varphi (z_{j})}z}-\frac{2}{n+1} \frac{(1-|\varphi(z_{j})|^{2})^{2}}{(1-\overline {\varphi(z_{j})}z)^{2}} +\frac{2}{(n+1)(n+2)} \frac{(1-|\varphi(z_{j})|^{2})^{3}}{(1-\overline {\varphi(z_{j})}z)^{3}}. $$
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,
$$\begin{aligned}& k^{(n)}_{j}\bigl(\varphi(z_{j})\bigr)=0, \qquad k^{(n+2)}_{j}\bigl(\varphi(z_{j})\bigr)=0,\qquad \bigl\vert k^{(n+1)}_{j}\bigl(\varphi(z_{j})\bigr)\bigr\vert =\frac{n!}{n+3}\frac{\vert \varphi (z_{j})\vert ^{n+1}}{(1-\vert \varphi(z_{j})\vert ^{2})^{n+1}}, \\& l^{(n+1)}_{j}\bigl(\varphi(z_{j})\bigr)=0, \qquad l^{(n+2)}_{j}\bigl(\varphi(z_{j})\bigr)=0,\qquad \bigl\vert l^{(n)}_{j}\bigl(\varphi(z_{j})\bigr) \bigr\vert = \frac{2 n!}{2+(n+1)(n+4)}\frac{\vert \varphi (z_{j})\vert ^{n}}{(1-\vert \varphi(z_{j})\vert ^{2})^{n}}, \\& m^{(n)}_{j}\bigl(\varphi(z_{j})\bigr)=0, \qquad m^{(n+1)}_{j}\bigl(\varphi(z_{j})\bigr)=0,\qquad \bigl\vert m^{(n+2)}_{j}\bigl(\varphi(z_{j})\bigr)\bigr\vert =2n!\frac{\vert \varphi (z_{j})\vert ^{n+2}}{(1-\vert \varphi(z_{j})\vert ^{2})^{n+2}}. \end{aligned}$$
Then for any compact operator \(K: \mathcal{B} \to \mathcal{Z} \), we obtain
$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(k_{j}) \bigr\Vert _{\mathcal{Z}}- \limsup_{j\rightarrow\infty} \bigl\Vert K(k_{j}) \bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \gtrsim \limsup_{j\rightarrow\infty} \frac{(1-|z_{j}|^{2}) |2u'(z_{j})\varphi'(z_{j})+u(z_{j})\varphi''(z_{j})||\varphi (z_{j})|^{n+1}}{(1-|\varphi(z_{j})|^{2})^{n+1}}, \\& \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(l_{j}) \bigr\Vert _{\mathcal{Z}} - \limsup_{j\rightarrow\infty} \bigl\Vert K(l_{j}) \bigr\Vert _{\mathcal{Z}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}}} \gtrsim \limsup_{j\rightarrow\infty} \frac {(1-|z_{j}|^{2})|u''(z_{j})||\varphi(z_{j})|^{n}}{(1-|\varphi(z_{j})|^{2})^{n}}, \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z}} \gtrsim& \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(m_{j}) \bigr\Vert _{\mathcal{Z}} - \limsup_{j\rightarrow\infty} \bigl\Vert K(m_{j}) \bigr\Vert _{\mathcal{Z}} \\ \gtrsim&\limsup_{j\rightarrow\infty} \frac {(1-|z_{j}|^{2})|u(z_{j})||\varphi'(z_{j})|^{2}|\varphi (z_{j})|^{n+2}}{(1-|\varphi(z_{j})|^{2})^{n+2}}. \end{aligned}$$
From the definition of the essential norm, we obtain
$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal {Z}} = \inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z} } \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal {Z}}}\gtrsim \limsup_{j\rightarrow\infty} \frac{(1-|z_{j}|^{2}) |2u'(z_{j})\varphi'(z_{j})+u(z_{j})\varphi''(z_{j})||\varphi (z_{j})|^{n+1}}{(1-|\varphi(z_{j})|^{2})^{n+1}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal {Z}}}= \limsup_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2}) |2u'(z)\varphi '(z)+u(z)\varphi''(z)|}{(1-|\varphi(z)|^{2})^{n+1}} = E , \\& \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} = \inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z} } \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}}}\gtrsim\limsup_{j\rightarrow\infty} \frac{(1-|z_{j}|^{2})|u''(z_{j})||\varphi (z_{j})|^{n}}{(1-|\varphi(z_{j})|^{2})^{n}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}}} = \limsup_{ |\varphi(z)|\rightarrow1}\frac {(1-|z|^{2})|u''(z)|}{(1-|\varphi(z)|^{2})^{n}}=F, \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} = &\inf _{K} \bigl\Vert D^{n}_{\varphi,u}-K\bigr\Vert _{\mathcal{B}\to\mathcal{Z} } \\ \gtrsim& \limsup_{j\rightarrow\infty} \frac {(1-|z_{j}|^{2})|u(z_{j})||\varphi'(z_{j})|^{2}|\varphi (z_{j})|^{n+2}}{(1-|\varphi(z_{j})|^{2})^{n+2}} \\ =& \limsup_{ |\varphi(z)|\rightarrow1}\frac {(1-|z|^{2})|u(z)||\varphi'(z)|^{2}}{(1-|\varphi (z)|^{2})^{n+2}}=G. \end{aligned}$$
Hence
$$\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \gtrsim \max \{ E, F , G \}. $$
Now, we prove that
$$\bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \lesssim \max \{ A, B, C \} \quad \mbox{and} \quad \bigl\Vert D^{n}_{\varphi,u} \bigr\Vert _{e,\mathcal {B}\to \mathcal{Z}} \lesssim\max \{ E, F, G \}. $$
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
$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \leq \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to \mathcal{Z} }. $$
(2.1)
Therefore, we only need to prove that
$$ \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}} \lesssim\max \{ A, B, C \} $$
and
$$ \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}} \lesssim\max \{ E, F, G \} . $$
For any \(f\in\mathcal{B}\) such that \(\|f\|_{\mathcal{B}}\leq1\), we consider
$$\begin{aligned}& \bigl\Vert \bigl( D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr)f\bigr\Vert _{\mathcal {Z}} \\& \quad = \bigl\vert u(0)f^{(n)}\bigl(\varphi(0)\bigr)-r_{j}^{n}u(0)f^{(n)} \bigl(r_{j}\varphi(0)\bigr)\bigr\vert \\& \qquad {}+\bigl\vert u'(0) (f-f_{r_{j}})^{(n)} \bigl(\varphi(0)\bigr)+u(0) (f-f_{r_{j}})^{(n+1)}\bigl(\varphi (0) \bigr)\varphi'(0)\bigr\vert \\& \qquad {}+\bigl\Vert u\cdot(f-f_{r_{j}})^{(n)}\circ\varphi\bigr\Vert _{*}, \end{aligned}$$
(2.2)
where \(\|f\|_{*}=\sup_{z \in\mathbb{D}}(1-|z|^{2}) |f''(z)|\).
It is obvious that
$$ \lim_{j\to\infty}\bigl\vert u(0)f^{(n)}\bigl(\varphi (0) \bigr)-r_{j}^{n}u(0)f^{(n)}\bigl(r_{j} \varphi(0)\bigr)\bigr\vert =0 $$
(2.3)
and
$$ \lim_{j\to\infty}\bigl\vert u'(0) (f-f_{r_{j}})^{(n)} \bigl(\varphi (0)\bigr)+u(0) (f-f_{r_{j}})^{(n+1)}\bigl(\varphi(0) \bigr)\varphi'(0)\bigr\vert =0. $$
(2.4)
Now, we consider
$$\begin{aligned}& \limsup_{j\to\infty}\bigl\Vert u\cdot(f-f_{r_{j}})^{(n)} \circ\varphi\bigr\Vert _{*} \\& \quad \le \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi ''(z)\bigr\vert \\& \qquad {}+ \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi ''(z)\bigr\vert \\& \qquad {}+\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert \\& \qquad {}+ \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert \\& \qquad {}+\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \\& \qquad {}+ \limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}}\bigl(1- \vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \\& \quad = Q_{1}+Q_{2}+Q_{3}+Q_{4}+Q_{5}+Q_{6}, \end{aligned}$$
(2.5)
where \(N\in\mathbb{N }\) is large enough such that \(r_{j}\geq\frac {1}{2}\) for all \(j\geq N\),
$$\begin{aligned}& Q_{1}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi ''(z)\bigr\vert , \\& Q_{2}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+1)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi''(z)\bigr\vert , \\& Q_{3}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert , \\& Q_{4}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z) \bigr\vert , \\& Q_{5}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert \leq r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert , \end{aligned}$$
and
$$Q_{6}:=\limsup_{j\to\infty}\sup_{\vert \varphi(z)\vert > r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert (f-f_{r_{j}})^{(n+2)} \bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert . $$
Since \(D^{n}_{\varphi,u}:\mathcal{B} \to\mathcal{Z} \) is bounded, by Theorem 1 of [12], we see that \(u\in\mathcal{Z} \),
$$\widetilde{K}_{1}:=\sup_{z\in\mathbb{D}}\bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert 2u'(z)\varphi '(z)+u(z)\varphi''(z)\bigr\vert < \infty $$
and
$$\widetilde{K}_{2}:=\sup_{z\in\mathbb{D}}\bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert \varphi '(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert < \infty. $$
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
$$ Q_{1} \leq\widetilde{K}_{1} \limsup_{j\to\infty} \sup_{\vert w\vert \leq r_{N}} \bigl\vert f^{(n+1)}(w)- r^{n+1}_{j}f^{(n+1)} (r_{j} w)\bigr\vert =0 $$
(2.6)
and
$$ Q_{5} \leq\widetilde{K}_{2} \limsup_{j\to\infty} \sup_{\vert w\vert \leq r_{N}} \bigl\vert f^{(n+2)}(w)- r^{n+2}_{j}f^{(n+2)} (r_{j} w)\bigr\vert =0. $$
(2.7)
Similarly, from the fact that \(u \in\mathcal{Z}\) we have
$$ Q_{3} \leq\|u\|_{\mathcal{Z}}\limsup_{j\to\infty}\sup _{\vert w\vert \leq r_{N}} \bigl\vert f^{(n)}(w)- r^{n}_{j}f^{(n)} (r_{j} w)\bigr\vert =0. $$
(2.8)
Next we consider \(Q_{2}\). We have \(Q_{2}\leq\limsup_{j\to\infty }(S_{1}+S_{2})\), where
$$ S_{1}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n+1)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z)\varphi'(z)+u(z) \varphi''(z)\bigr\vert $$
and
$$ S_{2}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) r^{n+1}_{j}\bigl\vert f^{(n+1)} \bigl(r_{j}\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z) \varphi'(z)+u(z)\varphi''(z)\bigr\vert . $$
First we estimate \(S_{1}\). Using the fact that \(\|f\|_{\mathcal{B}}\leq 1\) and Theorem 5.4 in [31], we have
$$\begin{aligned} S_{1} =&\sup_{\vert \varphi(z) \vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n+1)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert 2u'(z)\varphi'(z)+u(z) \varphi''(z)\bigr\vert \\ &{} \times\frac{(1-\vert \varphi(z)\vert ^{2})^{n+1}(n+3)}{ \vert \varphi (z)\vert ^{n+1}n!} \frac{ \vert \varphi(z)\vert ^{n+1} n! }{(n+3)(1-\vert \varphi (z)\vert ^{2})^{n+1}} \\ \lesssim&\frac{(n+3) \|f\|_{\mathcal{B}} }{n! r_{N}^{n+1}}\sup_{\vert \varphi(z) \vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert 2u'(z)\varphi'(z)+u(z) \varphi''(z)\bigr\vert \\ &{} \times\frac{n! \vert \varphi(z)\vert ^{n+1} }{(n+3)(1-\vert \varphi (z)\vert ^{2})^{n+1}} \\ \lesssim& \sup_{\vert \varphi(z) \vert >r_{N}} \bigl(1-\vert z\vert ^{2} \bigr) \bigl\vert 2u'(z)\varphi '(z)+u(z) \varphi''(z)\bigr\vert \frac{n! \vert \varphi(z)\vert ^{n+1} }{(n+3)(1-\vert \varphi (z)\vert ^{2})^{n+1}} \\ \lesssim& \sup_{\vert a \vert >r_{N}} \biggl\Vert D^{n}_{\varphi,u} \biggl(f_{a}-\frac {(2n+5) g_{a}}{(n+1)(n+3)}+\frac{2h_{a}}{(n+1)(n+3)} \biggr)\biggr\Vert _{\mathcal{Z}} \\ \lesssim& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \frac{2n+5}{(n+1)(n+3)} \sup _{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} +\frac{2}{(n+1)(n+3)} \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}}. \end{aligned}$$
(2.9)
Taking the limit as \(N\to\infty\) we obtain
$$\begin{aligned} \begin{aligned} \limsup_{j\to\infty}S_{1} \lesssim{}& \limsup _{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \limsup_{|a|\to 1} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal{Z}} \\ &{} + \limsup_{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal{Z}} \\ = {}& A+B+C. \end{aligned} \end{aligned}$$
Similarly, we have \(\limsup_{j\to\infty}S_{2}\lesssim A+B+C\), i.e., we get
$$ Q_{2}\lesssim A+ B+ C \lesssim\max \{ A, B , C \} . $$
(2.10)
From (2.9), we see that
$$\limsup_{j\to\infty}S_{1} \lesssim\limsup _{|\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2}) |2u'(z)\varphi'(z)+u(z)\varphi''(z)|}{(1-|\varphi (z)|^{2})^{n+1}}=E. $$
Similarly we have \(\limsup_{j\to\infty}S_{2} \lesssim E\). Therefore
$$ Q_{2} \lesssim E . $$
(2.11)
Next we consider \(Q_{4}\). We have \(Q_{4}\leq\limsup_{j\to\infty }(S_{3}+S_{4})\), where
$$ S_{3}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z)\bigr\vert $$
and
$$ S_{4}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) r^{n}_{j}\bigl\vert f^{(n)} \bigl(r_{j}\varphi (z)\bigr)\bigr\vert \bigl\vert u''(z)\bigr\vert . $$
After some calculation, we have
$$\begin{aligned} S_{3} =& \sup_{\vert \varphi(z) \vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \bigl\vert u''(z)\bigr\vert \\ &{} \times\frac{(1-\vert \varphi(z)\vert ^{2})^{n}(2+(n+1)(n+4))}{2n! \vert \varphi(z)\vert ^{n}} \frac{2}{2+(n+1)(n+4)} \frac{n! \vert \varphi(z)\vert ^{n} }{(1-\vert \varphi (z)\vert ^{2})^{n}} \\ \lesssim&\frac{ 2^{n}(2+(n+1)(n+4))}{2n! } \|f\|_{\mathcal{B}} \sup_{\vert \varphi (z) \vert >r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert u''(z) \bigr\vert \\ &{} \times \frac{2}{2+(n+1)(n+4)} \frac{n!\vert \varphi(z)\vert ^{n} }{(1-\vert \varphi (z)\vert ^{2})^{n}} \\ \lesssim& \sup_{\vert \varphi(z) \vert >r_{N}} \frac{2n! }{2+(n+1)(n+4)} \frac {(1-\vert z\vert ^{2}) \vert u''(z)\vert \vert \varphi(z)\vert ^{n} }{(1-\vert \varphi(z)\vert ^{2})^{n}} \\ \lesssim& \sup_{\vert a\vert >r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \frac{2(n+3)}{2+(n+1)(n+4)} \sup _{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} + \frac{2}{2+(n+1)(n+4)} \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ \lesssim& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}}. \end{aligned}$$
(2.12)
Taking the limit as \(N\to\infty\) we obtain
$$\begin{aligned} \limsup_{j\to\infty}S_{3} \lesssim& \limsup _{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \limsup_{|a|\to 1} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} + \limsup_{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ =& A+B+C. \end{aligned}$$
Similarly, we have \(\limsup_{j\to\infty}S_{4} \lesssim A+B+C\), i.e., we get
$$ Q_{4} \lesssim A + B + C\lesssim\max \{ A, B , C \} . $$
(2.13)
From (2.12), we see that
$$\limsup_{j\to\infty}S_{3} \lesssim\limsup _{ |\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2})^{\beta}|u''(z)|}{(1-|\varphi(z)|^{2})^{n}}=F. $$
Similarly we have \(\limsup_{j\to\infty}S_{4} \lesssim F\). Therefore
$$ Q_{4} \lesssim F . $$
(2.14)
Finally we consider \(Q_{6}\). We have \(Q_{6}\leq\limsup_{j\to\infty }(S_{5}+S_{6})\), where
$$ S_{5}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert f^{(n+2)}\bigl(\varphi (z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert $$
and
$$ S_{6}:=\sup_{\vert \varphi(z)\vert >r_{N}}\bigl(1-\vert z\vert ^{2}\bigr) r^{n+2}_{j}\bigl\vert f^{(n+2)} \bigl(r_{j}\varphi (z)\bigr)\bigr\vert \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert . $$
After some calculation, we have
$$\begin{aligned} S_{5} \lesssim& \frac{ 2^{n+2} \|f\|_{\mathcal{B}} }{2n! } \sup _{\vert \varphi(z) \vert >r_{N}} \bigl(1-\vert z\vert ^{2}\bigr) \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \frac{ 2n!\vert \varphi(z)\vert ^{n+2} }{(1-\vert \varphi(z)\vert ^{2})^{n+2}} \\ \lesssim& \frac{ 2^{n+2}}{2n! } \sup_{\vert \varphi(z) \vert >r_{N}} \bigl(1-\vert z \vert ^{2}\bigr) \bigl\vert \varphi'(z)\bigr\vert ^{2}\bigl\vert u(z)\bigr\vert \frac{ 2n!\vert \varphi(z)\vert ^{n+2} }{(1-\vert \varphi (z)\vert ^{2})^{n+2}} \\ \lesssim& \sup_{\vert a \vert >r_{N}} \biggl\Vert D^{n}_{\varphi,u} \biggl(f_{a}- \frac{2}{n+1} g_{a}+ \frac {2}{(n+1)(n+2)} h_{a} \biggr)\biggr\Vert _{\mathcal{Z}} \\ \lesssim& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \frac{2}{n+1} \sup _{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} + \frac{2}{(n+1)(n+2)} \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ \leq& \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}}+ \sup_{|a |>r_{N}} \bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}}. \end{aligned}$$
(2.15)
Taking the limit as \(N\to\infty\) we obtain
$$\begin{aligned} \begin{aligned} \limsup_{j\to\infty}S_{5} \lesssim{}& \limsup _{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (f_{a} )\bigr\Vert _{\mathcal{Z}}+ \limsup_{|a|\to 1} \bigl\Vert D^{n}_{\varphi,u} (g_{a} )\bigr\Vert _{\mathcal {Z}} \\ &{} + \limsup_{|a|\to 1}\bigl\Vert D^{n}_{\varphi,u} (h_{a} )\bigr\Vert _{\mathcal {Z}} \\ ={}& A+B+C. \end{aligned} \end{aligned}$$
Similarly, we have \(\limsup_{j\to\infty}S_{6} \lesssim A+B+C\), i.e., we get
$$ Q_{6} \lesssim A + B + C\lesssim\max \{ A, B , C \} . $$
(2.16)
From (2.15), we see that
$$\limsup_{j\to\infty}S_{5} \lesssim\limsup _{ |\varphi(z)|\rightarrow 1}\frac{(1-|z|^{2})|\varphi'(z)|^{2}|u(z)|}{(1-|\varphi(z)|^{2})^{n+2}}=G. $$
Similarly we have \(\limsup_{j\to\infty}S_{6} \lesssim G\). Therefore
$$ Q_{6} \lesssim G . $$
(2.17)
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
$$\begin{aligned}& \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z} } \\& \quad = \limsup_{j\to\infty}\sup_{ \Vert f\Vert _{\mathcal{B}} \leq1}\bigl\Vert \bigl(D^{n}_{\varphi ,u}- D^{n}_{\varphi,u} K_{r_{j}}\bigr)f\bigr\Vert _{\mathcal{Z} } \\& \quad = \limsup_{j\to\infty}\sup_{ \Vert f\Vert _{\mathcal{B}} \leq1} \bigl\Vert u\cdot (f-f_{r_{j}})^{(n)}\circ\varphi\bigr\Vert _{*} \lesssim\max \{ A, B, C \}. \end{aligned}$$
(2.18)
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
$$ \limsup_{j\to\infty}\bigl\Vert D^{n}_{\varphi,u}-D^{n}_{\varphi,u} K_{r_{j}}\bigr\Vert _{ \mathcal{B}\to\mathcal{Z}}\lesssim\max \{ E, F, G \} . $$
(2.19)
Therefore, by (2.1), (2.18), and (2.19), we obtain
$$ \bigl\Vert D^{n}_{\varphi,u}\bigr\Vert _{e,\mathcal{B}\to\mathcal{Z}} \lesssim \max \{ A, B, C \} \quad \mbox{and} \quad \bigl\Vert D^{n}_{\varphi,u} \bigr\Vert _{e,\mathcal {B}\to \mathcal{Z}} \lesssim\max \{ E, F, G \} . $$
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
$$\|f\|_{v}=\sup_{z \in\mathbb{D}}v(z)\bigl\vert f(z)\bigr\vert < \infty, $$
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 of v is as follows:
$$\tilde{v}=\bigl(\sup\bigl\{ \bigl\vert f(z)\bigr\vert : f\in H^{\infty}_{v}, \|f\|_{v} \leq1 \bigr\} \bigr)^{-1},\quad z\in\mathbb{D}. $$
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.
  1. (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:
    $$ \|uC_{\varphi}\|_{H_{v}^{\infty}\rightarrow H_{w}^{\infty}}=\sup_{z\in\mathbb{D}} \frac{w(z)}{\tilde{v}(\varphi (z))}\bigl\vert u(z)\bigr\vert . $$
     
  2. (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.
  1. (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.

     
  2. (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
$$ \left \{ \textstyle\begin{array}{l} \sup_{j\geq1} j^{n+1}\|(2u'\varphi'+u\varphi'')\varphi^{j-1}\| _{v_{1}}< \infty, \\ \sup_{j\geq1} j^{n}\|u''\varphi^{j-1}\|_{v_{1}}< \infty, \\ \sup_{j\geq1} j^{n+2}\|u\varphi^{\prime 2}\varphi^{j-1}\|_{v_{1}}< \infty. \end{array}\displaystyle \right . $$
(3.1)

Proof

By [16], \(D^{n}_{\varphi, u} : \mathcal{B} \to\mathcal {Z} \) is bounded if and only if
$$ \left \{ \textstyle\begin{array}{l} \sup_{z\in\mathbb{D}}\frac{(1-|z|^{2}) |2u'(z)\varphi '(z)+u(z)\varphi''(z)|}{(1-|\varphi(z)|^{2})^{n+1}}< \infty, \\ \sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|u''(z)|}{(1-|\varphi (z)|^{2})^{n}}< \infty, \\ \sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|u(z)||\varphi '(z)|^{2}}{(1-|\varphi(z)|^{2})^{n+2}}< \infty. \end{array}\displaystyle \right . $$
(3.2)
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
$$\sup_{j\geq1}\frac{\|(2u'\varphi'+u\varphi'') \varphi^{j-1}\| _{v_{1}}}{\|z^{j-1}\|_{v_{1+n}}}< \infty. $$
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
$$\sup_{j\geq1}\frac{\|u'' \varphi^{j-1}\|_{v_{1}}}{\|z^{j-1}\| _{v_{n}}}< \infty. $$
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
$$\sup_{j\geq1}\frac{\|u\varphi^{\prime 2} \varphi^{j-1}\|_{v_{1}}}{\|z^{j-1}\| _{v_{n+2}}}< \infty. $$
By Lemma 3.1, we see that \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \) is bounded if and only if
$$\begin{aligned}& \sup_{j\geq1} j^{n+1}\bigl\Vert \bigl(2u' \varphi'+u\varphi''\bigr) \varphi^{j-1}\bigr\Vert _{v_{1}} \approx\sup _{j\geq1}\frac{j^{n+1}\Vert (2u'\varphi'+u\varphi '') \varphi^{j-1}\Vert _{v_{1}}}{j^{n+1}\Vert z^{j-1}\Vert _{v_{1+n}}}< \infty, \\& \sup_{j\geq1} j^{n}\bigl\Vert u'' \varphi^{j-1}\bigr\Vert _{v_{1}} \approx\sup _{j\geq1}\frac{j^{n}\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{j^{n}\Vert z^{j-1}\Vert _{v_{n}}} < \infty, \end{aligned}$$
and
$$\sup_{j\geq1} j^{ n+2}\bigl\Vert u \varphi^{\prime 2} \varphi^{j-1}\bigr\Vert _{v_{1}} \approx \sup_{j\geq1}\frac{j^{ n+2}\Vert u\varphi^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{j^{ n+2}\Vert z^{j-1}\Vert _{v_{ n+2}}}< \infty. $$
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
$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \approx \max \{M_{1}, M_{2}, M_{3} \}, $$
where
$$\begin{aligned}& M_{1}:=\limsup_{j\rightarrow\infty} j^{1+n}\bigl\Vert \bigl(2u'\varphi'+u\varphi '' \bigr)\varphi^{j-1}\bigr\Vert _{v_{1}}, \\& M_{2}:= \limsup_{j\rightarrow\infty} j^{n}\bigl\Vert u''\varphi^{j-1}\bigr\Vert _{v_{1}},\qquad M_{3}:=\limsup_{j\rightarrow\infty} j^{ n+2}\bigl\Vert u \bigl(\varphi'\bigr)^{2}\varphi ^{j-1}\bigr\Vert _{v_{1}}. \end{aligned}$$

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
$$\begin{aligned}& \bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}} = \limsup _{j\rightarrow \infty}\frac{\Vert (2u'\varphi'+u\varphi'') \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{1+n}}} \\& \hphantom{\bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}}}= \limsup_{j\rightarrow\infty}\frac{j^{1+n}\Vert (2u'\varphi '+u\varphi'') \varphi^{j-1}\Vert _{v_{1}}}{j^{1+n}\Vert z^{j-1}\Vert _{v_{1+n}}} \\& \hphantom{\bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}}}\approx \limsup_{j\rightarrow\infty}j^{1+n}\bigl\Vert \bigl(2u'\varphi'+u\varphi'' \bigr) \varphi^{j-1}\bigr\Vert _{v_{1}}, \\& \bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}} = \limsup_{j\rightarrow\infty}\frac{\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{n}}}=\limsup _{j\rightarrow\infty}\frac{j^{\alpha +n-1}\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{j^{\alpha +n-1}\Vert z^{j-1}\Vert _{v_{n}}} \\& \hphantom{\bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}}}\approx \limsup_{j\rightarrow\infty}j^{n}\bigl\Vert u'' \varphi^{j-1}\bigr\Vert _{v_{1}}, \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert u\varphi^{\prime 2}C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}} =& \limsup_{j\rightarrow\infty}\frac{\Vert u\varphi ^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{ n+2}}}=\limsup _{j\rightarrow\infty}\frac{j^{ n+2}\Vert u\varphi^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{j^{ n+2}\Vert z^{j-1}\Vert _{v_{ n+2}}} \\ \approx& \limsup_{j\rightarrow\infty}j^{ n+2}\bigl\Vert u \varphi^{\prime 2} \varphi^{j-1}\bigr\Vert _{v_{1}}. \end{aligned}$$
It follows that
$$\begin{aligned} \begin{aligned} \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \lesssim{}& \bigl\Vert \bigl(2u'\varphi'+u \varphi''\bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}}+ \bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}} \\ &{}+ \bigl\Vert u\varphi^{\prime 2}C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}} \\ \lesssim{}& \max \{M_{1}, M_{2}, M_{3} \}. \end{aligned} \end{aligned}$$
The lower estimate. From Theorem 2.1, and Lemmas 3.1 and 3.2, we have
$$\begin{aligned}& \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \gtrsim E = \bigl\Vert \bigl(2u'\varphi'+u \varphi''\bigr)C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{1+n}}\rightarrow H^{\infty}_{v_{1}}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} }} = \limsup_{j\rightarrow\infty}\frac{\Vert (2u'\varphi'+u\varphi '')\varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{1+n}}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} }} \approx \limsup_{j\rightarrow\infty}j^{\alpha +n}\bigl\Vert \bigl(2u'\varphi '+u\varphi'' \bigr) \varphi^{j-1}\bigr\Vert _{v_{1}} , \\& \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \gtrsim F =\bigl\Vert u''C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{n}}\rightarrow H^{\infty}_{v_{1}}}=\limsup_{j\rightarrow\infty }\frac{\Vert u'' \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{n}}} \\& \hphantom{\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} }} \approx \limsup_{j\rightarrow\infty}j^{\alpha +n-1}\bigl\Vert u'' \varphi^{j-1}\bigr\Vert _{v_{1}}, \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e, \mathcal{B} \to \mathcal{Z} } \gtrsim&G = \bigl\Vert u\varphi^{\prime 2}C_{\varphi}\bigr\Vert _{e, H^{\infty}_{v_{ n+2}}\rightarrow H^{\infty}_{v_{1}}}= \limsup_{j\rightarrow\infty }\frac{\Vert u\varphi^{\prime 2} \varphi^{j-1}\Vert _{v_{1}}}{\Vert z^{j-1}\Vert _{v_{ n+2}}} \\ \approx& \limsup_{j\rightarrow\infty}j^{ n+2}\bigl\Vert u \varphi^{\prime 2} \varphi^{j-1}\bigr\Vert _{v_{1}}. \end{aligned}$$
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
$$\begin{aligned}& \limsup_{j\rightarrow\infty} j^{1+n}\bigl\Vert \bigl(2u'\varphi'+u\varphi '' \bigr)\varphi^{j-1}\bigr\Vert _{v_{1}}=0, \qquad \limsup_{j\rightarrow\infty} j^{n}\bigl\Vert u''\varphi^{j-1}\bigr\Vert _{v_{1}}=0, \end{aligned}$$
and
$$\limsup_{j\rightarrow\infty} j^{ n+2}\bigl\Vert u\bigl( \varphi'\bigr)^{2}\varphi^{j-1}\bigr\Vert _{v_{1}}=0. $$

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 \(D^{n}_{\varphi, u} : \mathcal{B} \to \mathcal{Z} \).

Declarations

Acknowledgements

This project is partially supported by the Macao Science and Technology Development Fund (No. 083/2014/A2).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China
(2)
School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, 200433, China
(3)
Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau
(4)
Department of Mathematics, Jiaying University, Meizhou, Guangdong, 515063, China

References

  1. Hibschweiler, R, Portnoy, N: Composition followed by differentiation between Bergman and Hardy spaces. Rocky Mt. J. Math. 35, 843-855 (2005) MathSciNetView ArticleMATHGoogle Scholar
  2. Li, S, Stević, S: Composition followed by differentiation between Bloch type spaces. J. Comput. Anal. Appl. 9, 195-205 (2007) MathSciNetMATHGoogle Scholar
  3. Li, S, Stević, S: Composition followed by differentiation between \(H^{\infty}\) and α-Bloch spaces. Houst. J. Math. 35, 327-340 (2009) MathSciNetMATHGoogle Scholar
  4. Li, S, Stević, S: Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Appl. Math. Comput. 217, 3144-3154 (2010) MathSciNetView ArticleMATHGoogle Scholar
  5. 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) MathSciNetView ArticleMATHGoogle Scholar
  6. Stević, S: Products of composition and differentiation operators on the weighted Bergman space. Bull. Belg. Math. Soc. Simon Stevin 16, 623-635 (2009) MathSciNetMATHGoogle Scholar
  7. Stević, S: Composition followed by differentiation from \(H^{\infty}\) and the Bloch space to nth weighted-type spaces on the unit disk. Appl. Math. Comput. 216, 3450-3458 (2010) MathSciNetView ArticleMATHGoogle Scholar
  8. Stević, S: Characterizations of composition followed by differentiation between Bloch-type spaces. Appl. Math. Comput. 218, 4312-4316 (2011) MathSciNetView ArticleMATHGoogle Scholar
  9. Yang, W: Products of composition differentiation operators from \(\mathcal{Q}_{K}(p,q)\) spaces to Bloch-type spaces. Abstr. Appl. Anal. 2009, Article ID 741920 (2009) MathSciNetMATHGoogle Scholar
  10. Stević, S, Sharma, A: Iterated differentiation followed by composition from Bloch-type spaces to weighted BMOA spaces. Appl. Math. Comput. 218, 3574-3580 (2011) MathSciNetView ArticleMATHGoogle Scholar
  11. Wu, Y, Wulan, H: Products of differentiation and composition operators on the Bloch space. Collect. Math. 63, 93-107 (2012) MathSciNetView ArticleMATHGoogle Scholar
  12. 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) MathSciNetMATHGoogle Scholar
  13. Li, S, Stević, S: Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces. J. Inequal. Appl. 2015, Article ID 265 (2015) MathSciNetView ArticleMATHGoogle Scholar
  14. Stević, S: Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211, 222-233 (2009) MathSciNetView ArticleMATHGoogle Scholar
  15. Stević, S: Weighted differentiation composition operators from mixed-norm spaces to the nth weighted-type space on the unit disk. Abstr. Appl. Anal. 2010, Article ID 246287 (2010) MATHGoogle Scholar
  16. Stević, S: Weighted differentiation composition operators from \(H^{\infty}\) and Bloch spaces to nth weighted-type spaces on the unit disk. Appl. Math. Comput. 216, 3634-3641 (2010) MathSciNetView ArticleMATHGoogle Scholar
  17. Yang, W, Zhu, X: Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces. Taiwan. J. Math. 16, 869-883 (2012) MathSciNetMATHGoogle Scholar
  18. Zhu, X: Products of differentiation, composition and multiplication from Bergman type spaces to Bers type space. Integral Transforms Spec. Funct. 18, 223-231 (2007) MathSciNetView ArticleMATHGoogle Scholar
  19. Zhu, X: Generalized weighted composition operators on weighted Bergman spaces. Numer. Funct. Anal. Optim. 30, 881-893 (2009) MathSciNetView ArticleMATHGoogle Scholar
  20. Zhu, X: Generalized weighted composition operators on Bloch-type spaces. J. Inequal. Appl. 2015, Article ID 59 (2015) MathSciNetView ArticleMATHGoogle Scholar
  21. Zhu, X: Essential norm of generalized weighted composition operators on Bloch-type spaces. Appl. Math. Comput. 274, 133-142 (2016) MathSciNetView ArticleGoogle Scholar
  22. Li, S, Stević, S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 338, 1282-1295 (2008) MathSciNetView ArticleMATHGoogle Scholar
  23. 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) MathSciNetView ArticleMATHGoogle Scholar
  24. Li, S, Stević, S: Composition followed by differentiation from mixed-norm spaces to α-Bloch spaces. Sb. Math. 199(12), 1847-1857 (2008) MathSciNetView ArticleMATHGoogle Scholar
  25. Li, S, Stević, S: Products of integral-type operators and composition operators between Bloch-type spaces. J. Math. Anal. Appl. 349, 596-610 (2009) MathSciNetView ArticleMATHGoogle Scholar
  26. Stević, S: On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces. Nonlinear Anal. TMA 71, 6323-6342 (2009) MathSciNetView ArticleMATHGoogle Scholar
  27. Stević, S: Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. Sib. Math. J. 50, 726-736 (2009) MathSciNetView ArticleMATHGoogle Scholar
  28. Stević, S: On some integral-type operators between a general space and Bloch-type spaces. Appl. Math. Comput. 218, 2600-2618 (2011) MathSciNetView ArticleMATHGoogle Scholar
  29. Stević, S, Sharma, A, Bhat, A: Products of multiplication composition and differentiation operators on weighted Bergman spaces. Appl. Math. Comput. 217, 8115-8125 (2011) MathSciNetView ArticleMATHGoogle Scholar
  30. Stević, S, Sharma, A, Bhat, A: Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces. Appl. Math. Comput. 218, 2386-2397 (2011) MathSciNetView ArticleMATHGoogle Scholar
  31. Zhu, K: Operator Theory in Function Spaces, 2nd edn. Am. Math. Soc., Providence (2007) View ArticleMATHGoogle Scholar
  32. Cowen, C, Maccluer, B: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995) MATHGoogle Scholar
  33. Hyvärinen, O, Lindström, M: Estimates of essential norm of weighted composition operators between Bloch-type spaces. J. Math. Anal. Appl. 393, 38-44 (2012) MathSciNetView ArticleMATHGoogle Scholar
  34. Li, S, Stević, S: Weighted composition operators from Bergman-type spaces into Bloch spaces. Proc. Indian Acad. Sci. Math. Sci. 117(3), 371-385 (2007) MathSciNetView ArticleMATHGoogle Scholar
  35. Li, S, Stević, S: Weighted composition operators between \(H^{\infty}\) and α-Bloch spaces in the unit ball. Taiwan. J. Math. 12, 1625-1639 (2008) MathSciNetMATHGoogle Scholar
  36. Li, S, Stević, S: Weighted composition operators from Zygmund spaces into Bloch spaces. Appl. Math. Comput. 206, 825-831 (2008) MathSciNetView ArticleMATHGoogle Scholar
  37. MacCluer, B, Zhao, R: Essential norm of weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33, 1437-1458 (2003) MathSciNetView ArticleMATHGoogle Scholar
  38. Madigan, K, Matheson, A: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347, 2679-2687 (1995) MathSciNetView ArticleMATHGoogle Scholar
  39. 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) MathSciNetView ArticleMATHGoogle Scholar
  40. Montes-Rodriguez, A: The essential norm of composition operators on the Bloch space. Pac. J. Math. 188, 339-351 (1999) MathSciNetView ArticleMATHGoogle Scholar
  41. Ohno, S, Stroethoff, K, Zhao, R: Weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33, 191-215 (2003) MathSciNetView ArticleMATHGoogle Scholar
  42. Stević, S: Essential norms of weighted composition operators from the α-Bloch space to a weighted-type space on the unit ball. Abstr. Appl. Anal. 2008, Article ID 279691 (2008) MathSciNetMATHGoogle Scholar
  43. Tjani, M: Compact composition operators on some Möbius invariant Banach space. PhD dissertation, Michigan State University (1996) Google Scholar
  44. Wulan, H, Zheng, D, Zhu, K: Compact composition operators on BMOA and the Bloch space. Proc. Am. Math. Soc. 137, 3861-3868 (2009) MathSciNetView ArticleMATHGoogle Scholar
  45. Zhao, R: Essential norms of composition operators between Bloch type spaces. Proc. Am. Math. Soc. 138, 2537-2546 (2010) MathSciNetView ArticleMATHGoogle Scholar
  46. Choe, B, Koo, H, Smith, W: Composition operators on small spaces. Integral Equ. Oper. Theory 56, 357-380 (2006) MathSciNetView ArticleMATHGoogle Scholar
  47. Duren, P: Theory of H p Spaces. Academic Press, New York (1970) MATHGoogle Scholar
  48. Esmaeili, K, Lindström, M: Weighted composition operators between Zygmund type spaces and their essential norms. Integral Equ. Oper. Theory 75, 473-490 (2013) MathSciNetView ArticleMATHGoogle Scholar
  49. Li, S, Stević, S: Volterra type operators on Zygmund spaces. J. Inequal. Appl. 2007, Article ID 32124 (2007) MathSciNetMATHGoogle Scholar
  50. Yu, Y, Liu, Y: Weighted differentiation composition operators from \(H^{\infty}\) to Zygmund spaces. Integral Transforms Spec. Funct. 22, 507-520 (2011) MathSciNetView ArticleMATHGoogle Scholar
  51. Montes-Rodriguez, A: Weighed composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61, 872-884 (2000) MathSciNetView ArticleMATHGoogle Scholar
  52. Hyvärinen, O, Kemppainen, M, Lindström, M, Rautio, A, Saukko, E: The essential norm of weighted composition operators on weighted Banach spaces of analytic functions. Integral Equ. Oper. Theory 72, 151-157 (2012) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Hu et al. 2016

Advertisement