Generalized Weighted Composition Operators from Bloch-Type Spaces into Zygmund-Type Spaces

Abstract

A new characterization for the boundedness, compactness and the essential norm of generalized weighted composition operators from Bloch-type spaces into Zygmund-type spaces are given in this paper.

1 Introduction

Let $H(\mathbb {D})$ be the space of analytic functions on the open unit disk ${\mathbb D}$ in the complex plane ${\mathbb C}$. Let $\alpha >0$. The Bloch-type space, denoted by $\mathcal {B}^\alpha$, consists of all $f \in H(\mathbb {D})$ such that (see [42])

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}^\alpha }=|f(0)|+\sup _{z \in \mathbb {D}}(1-|z|^2)^\alpha |f'(z)|<\infty . \end{aligned}$$

$\mathcal {B}^\alpha$ is a Banach space with the above norm. Let $\mathcal {B}^\alpha _0$ denote the space of $f \in \mathcal {B}^\alpha$ for which $\lim _{|z| \rightarrow 1}(1-|z|^2)^\alpha |f'(z)|= 0.$$\mathcal {B}_0^\alpha$ is called the little Bloch-type space. $\mathcal {B}^1$ is the well-known Bloch space, which always denoted by $\mathcal {B}$.

Let $\beta > 0$. The Zygmund-type space, denoted by $\mathcal {Z}^\beta$, consists of all $f\in H({\mathbb D})$ such that

$$\begin{aligned} \Vert f\Vert _{\mathcal {Z}^\beta }=|f(0)|+|f'(0)|+\sup _{z \in \mathbb {D}}(1-|z|^2)^\beta |f''(z)|<\infty . \end{aligned}$$

It is easy to check that $\mathcal {Z}^\beta$ is a Banach space with the norm $\Vert \cdot \Vert _{\mathcal {Z}^\beta }$. When $\beta =1$, $\mathcal {Z}^1=\mathcal {Z}$ is the classical Zygmund space. When $\beta >1$, the space $\mathcal {Z}^\beta$ coincides with $\mathcal {B}^{\beta -1}$. See [3, 10, 11, 18, 30,31,32,33, 40] for the study of related operators mapping into the Zygmund space or into some of its generalizations.

Let $\varphi$ be an analytic self-maps of $\mathbb {D}$. The composition operator $C_\varphi$ with the symbol $\varphi$ is defined by

$$\begin{aligned} (C_\varphi f)(z) = f (\varphi (z)), ~f \in H(\mathbb {D}). \end{aligned}$$

See [4, 42] for the study of this operator. Let $u \in H(\mathbb {D})$ and n be a nonnegative integer. The generalized weighted composition operator, denoted by $D^n_{\varphi , u}$, is defined as follows (see [43, 44]).

$$\begin{aligned} (D^n_{\varphi , u} f)(z) =u(z)\cdot f^{(n)}(\varphi (z)) , \ \ f \in H(\mathbb {D}), ~~z\in \mathbb {D}. \end{aligned}$$

If $n=0$, then $D^n_{\varphi , u}$ is just the weighted composition operator and always denoted by $uC_\varphi$, recently, some important and interest results of the weighted composition operator have appeared, for example [1, 27]. When $n=0$ and $u(z)= 1$, then $D^n_{\varphi , u}$ is just the composition operator $C_\varphi$. If $n=1$, $u(z)=\varphi '(z)$, then $D^n_{\varphi , u}= DC_\varphi$, which was studied, for example, in [8, 12, 17, 18, 28, 29, 34, 35]. When $u(z)=1$, $D^n_{\varphi , u}= C_\varphi D^n$, which was studied in [8, 20, 37]. See, e.g., [9, 10, 19, 32, 33, 39, 43, 44] for the study of the operator $D^n_{\varphi , u}$ on various function spaces.

Various properties of composition operators, as well as weighted composition operators mapping into Bloch-type spaces were studied in [4,5,6, 13,14,15,16,17, 22,23,26, 33, 36, 38, 41]. In particular, Tjani [36] showed that $C_\varphi : \mathcal {B} \rightarrow \mathcal {B}$ is compact if and only if $\lim _{|a|\rightarrow 1} \Vert C_\varphi \sigma _a \Vert _ \mathcal {B} =0,$ where $\sigma _a(z)=(a-z)/(1-\overline{a}z)$ is a Möbius transformation of ${\mathbb D}$. Wulan et al. [38] proved that $C_\varphi :\mathcal {B}\rightarrow \mathcal {B}$ is compact if and only if

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert C_\varphi I^j \Vert _\mathcal {B}=\lim _{j\rightarrow \infty }\Vert \varphi ^j \Vert _\mathcal {B}=0. \end{aligned}$$

Here $I_j(z)=z^j$. Wu and Wulan extend the above two characterizations to the operator $C_\varphi D^m$ in [37]. Among others, they proved $C_\varphi D^m: \mathcal {B} \rightarrow \mathcal {B}$ is compact if and only if $\lim _{j\rightarrow \infty }\Vert C_\varphi D^m(I^j)\Vert _ \mathcal {B} =0$. In [9], the authors studied the boundedness, compactness and the essential norm of the operator $D^n_{\varphi ,u}: \mathcal {B} \rightarrow \mathcal {Z}$. For example, they proved that $D^n_{\varphi ,u}: \mathcal {B} \rightarrow \mathcal {Z}$ is bounded if and only if $u, u\varphi , u\varphi ^2\in \mathcal {Z}$ and

$$\begin{aligned} \max \big \{\sup \limits _{a\in \mathbb {D}}\Vert D^n_{\varphi ,u} k_{a}\Vert _{\mathcal {Z} }, \sup \limits _{a\in \mathbb {D}}\Vert D^n_{\varphi ,u} p_{a}\Vert _{\mathcal {Z} }, \sup \limits _{a\in \mathbb {D}}\Vert D^n_{\varphi ,u}q_{a}\Vert _{\mathcal {Z} }\big \} <\infty . \end{aligned}$$

Here

$$\begin{aligned} k_a(z)= \frac{ 1-|a|^{2} }{ 1- \overline{a}z } ,\,~p_a(z)= \frac{ (1-|a|^2)^2}{(1-\overline{a}z)^2} , ~~q_a(z)=\frac{ (1-|a|^2)^3}{(1-\overline{a}z)^3},~~ z\in \mathbb {D}. \end{aligned}$$

Motivated by these observations, in this work we give a new characterization for the operator $D^n_{\varphi ,u}: \mathcal {B} \rightarrow \mathcal {Z}$. More generally, we study the boundedness, compactness and essential norm of the operator $D^n_{\varphi ,u} : \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$. For example, we show that $D^n_{\varphi ,u} : \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is bounded (respectively, compact) if and only if the sequence $(j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta })_{j=n}^\infty$ is bounded (respectively, convergent to 0 as $j\rightarrow \infty$).

Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. In addition, we say that $P\lesssim Q$ if there exists a constant C such that $P\le CQ$. The symbol $P\approx Q$ means that $P \lesssim Q \lesssim P$.

2 Boundedness of $D^n_{\varphi ,u}:\mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$

Now we are in a position to state and proof the main result in this section.

Theorem 1

Let $0<\alpha ,\beta <\infty$, $\varphi$ be an analytic self-map of ${\mathbb D}$, $u\in H(\mathbb {D})$ and n be a positive integer. Then the operator $D^n_{\varphi ,u}: \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is bounded if and only if

$$\begin{aligned} M:=\sup _{j\ge n} j^{ \alpha -1}\Vert D^n_{\varphi , u}I^j \Vert _{\mathcal {Z}^\beta }<\infty , \end{aligned}$$

where $I^j(z)=z^j.$

Proof

First we assume that $D^n_{\varphi ,u}: \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is bounded. For any $j\in \mathbb {N}$, the function $j^{ \alpha -1} I^j$ is bounded in $\mathcal {B}^\alpha$ and $j^{ \alpha -1}\Vert I^j\Vert _{\mathcal {B}^\alpha } \approx 1$ (see [41]). By the boundedness of $D^n_{\varphi ,u}: \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ we get the desired result.

Conversely, assume that $M<\infty$. Applying the operator $D^n_{\varphi , u}$ to $I^j$ with $j=n, n+1, n+2$, we see that $u, u\varphi , u\varphi ^2\in \mathcal {Z}^\beta$. For $a\in {\mathbb D}$, motivated by [2] (see also [7, 21]), we define

$$\begin{aligned} f_a(z)= \frac{ 1-|a|^{2} }{(1- \overline{a}z )^{\alpha } } ,\,~g_a(z)= \frac{ (1-|a|^2)^2}{(1-\overline{a}z)^{\alpha +1}} , ~~h_a(z)=\frac{ (1-|a|^2)^3}{(1-\overline{a}z)^{\alpha +2}},~~ z\in \mathbb {D}. \end{aligned}$$

It is easy to see that $f_a$, $g_a$ and $h_a$ have bounded norms in $\mathcal {B}^\alpha$. Moreover,

$$\begin{aligned} f_a(z)= & {} (1-|a|^2)\sum _{j=0}^\infty \frac{\Gamma (j+\alpha )}{j!\Gamma (\alpha )}\overline{a}^jz^j, ~~ g_a(z)=(1-|a|^2)^2\sum _{j=0}^\infty \frac{\Gamma (j+1+\alpha )}{j!\Gamma (\alpha +1)}\overline{a}^jz^j,\\ h_a(z)= & {} (1-|a|^2)^3\sum _{j=0}^\infty \frac{\Gamma (j+2+\alpha )}{j!\Gamma (\alpha +2)}\overline{a}^jz^j. \end{aligned}$$

By Stirling’s formula, we have $\frac{\Gamma (j+\alpha )}{j!\Gamma (\alpha )}\approx j^{\alpha -1}$ as $j\rightarrow \infty .$ Since a is fixed, by using pointwise estimate we get

$$\begin{aligned} \Vert D^n_{\varphi ,u}f_a\Vert _{\mathcal {Z}^\beta }\lesssim & {} (1-|a|^2)\sum _{j=0}^\infty |a|^j j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } \lesssim M , \\ \Vert D^n_{\varphi ,u}g_a\Vert _{\mathcal {Z}^\beta }\lesssim & {} (1-|a|^2)^2\sum _{j=0}^\infty j|a|^j j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta }\lesssim M \hbox { and }\\ \Vert D^n_{\varphi ,u}h_a\Vert _{\mathcal {Z}^\beta }\lesssim & {} (1-|a|^2)^3\sum _{j=0}^\infty j(j+1) |a|^j j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta }\lesssim M. \end{aligned}$$

Therefore, by the arbitrary of $a\in {\mathbb D}$, we get

$$\begin{aligned} Q_3:=\max \big \{\sup \limits _{a\in \mathbb {D}}\Vert D^n_{\varphi ,u} f_{a}\Vert _{\mathcal {Z}^\beta }, \sup \limits _{a\in \mathbb {D}}\Vert D^n_{\varphi ,u} g_{a}\Vert _{\mathcal {Z}^\beta }, \sup \limits _{a\in \mathbb {D}}\Vert D^n_{\varphi ,u}h_{a}\Vert _{\mathcal {Z}^\beta }\big \} <\infty . \end{aligned}$$

Set $q_0=\prod _{j=0}^{n-1}(\alpha +j ), ~q_1= \prod _{j=0}^{n}(\alpha +j ) ,~~q_2= \prod _{j=0}^{n+1}(\alpha +j ) ,$$v(z)=2u'(z)\varphi '(z)+u(z)\varphi ''(z)$. A calculation shows that

$$\begin{aligned} f_a^{(n)}(a)= & {} q_0\frac{\overline{a}^n}{(1-|a|^2)^{\alpha +n-1}}, \,\, g_a^{(n)}(a)= \frac{q_1}{\alpha } \frac{ \overline{a}^n}{(1-|a|^2)^{\alpha +n-1}} , \end{aligned}$$

(2.1)

$$\begin{aligned} h_a^{(n)}(a)= & {} \frac{q_2}{\alpha (\alpha +1)}\frac{ \overline{a}^n}{(1-|a|^2)^{\alpha +n-1}} . \end{aligned}$$

(2.2)

From (2.1) and (2.2), for $w\in \mathbb {D}$, we have

$$\begin{aligned} (D^n_{\varphi ,u} f_{\varphi (w)} )''(w)= & {} q_0 \frac{u''(w)\overline{\varphi (w)}^{n}}{(1-|\varphi (w)|^2)^{\alpha +n-1}}+ q_1 \frac{v(w)\overline{\varphi (w)}^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}} \nonumber \\&+q_2\frac{u(w)(\varphi '(w))^2\overline{\varphi (w)}^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}}, \end{aligned}$$

(2.3)

$$\begin{aligned} (D^n_{\varphi ,u} g_{\varphi (w)} )''(w)= & {} \frac{q_1}{\alpha }\frac{u''(w)\overline{\varphi (w)}^{n}}{(1-|\varphi (w)|^2)^{\alpha +n-1}} + \frac{q_2}{\alpha } \frac{v(w)\overline{\varphi (w)}^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}} \nonumber \\&+\frac{(\alpha +n+2)q_2}{\alpha } \frac{u(w)(\varphi '(w))^2\overline{\varphi (w)}^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}} \end{aligned}$$

(2.4)

and

$$\begin{aligned} (D^n_{\varphi ,u} h_{\varphi (w)} )''(w)= & {} \frac{q_2}{\alpha (\alpha +1)} \frac{u''(w)\overline{\varphi (w)}^{n}}{(1-|\varphi (w)|^2)^{\alpha +n-1}} +\frac{(\alpha +n+2)q_2}{\alpha (\alpha +1)}\frac{v(w)\overline{\varphi (w)}^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}} \nonumber \\&+\frac{(\alpha +n+2)(\alpha +n+3)q_2}{\alpha (\alpha +1)}\frac{u(w)(\varphi '(w))^2\overline{\varphi (w)}^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}}. \end{aligned}$$

(2.5)

Multiplying (2.3) by $-(\alpha +n)$ and (2.4) by $\alpha$, respectively, we obtain

$$\begin{aligned}&-(\alpha +n)(D^n_{\varphi ,u} f_{\varphi (w)} )''(w) +\alpha (D^n_{\varphi ,u} g_{\varphi (w)} )''(w)\nonumber \\&\quad =q_1\frac{v(w)\overline{\varphi (w)}^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}}+2q_2 \frac{u(w)(\varphi '(w))^2\overline{\varphi (w)}^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}}. \end{aligned}$$

(2.6)

Multiplying (2.3) by $-(\alpha +n)(\alpha +n+1)$ and (2.5) by $\alpha (\alpha +1)$, respectively, we obtain

$$\begin{aligned}&-(\alpha +n)(\alpha +n+1)(D^n_{\varphi ,u} f_{\varphi (w)} )''(w) +(\alpha +1)\alpha (D^n_{\varphi ,u} h_{\varphi (w)} )''(w)\nonumber \\&\quad =\frac{2q_2v(w)\overline{\varphi (w)}^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}}+ \frac{(4\alpha +4n+6)q_2u(w)(\varphi '(w))^2\overline{\varphi (w)}^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}}. \end{aligned}$$

(2.7)

Multiplying (2.6) by $2(\alpha +n+1)$, we get

$$\begin{aligned}&-2(\alpha +n)(\alpha +n+1)(D^n_{\varphi ,u} f_{\varphi (w)} )''(w) +2\alpha (\alpha +n+1)(D^n_{\varphi ,u} g_{\varphi (w)} )''(w)\nonumber \\&\quad =\frac{2q_2v(w)\overline{\varphi (w)}^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}}+ \frac{(4\alpha +4n+4)q_2u(w)(\varphi '(w))^2\overline{\varphi (w)}^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}}. \end{aligned}$$

(2.8)

Subtracting (2.8) from (2.7), we obtain

$$\begin{aligned} \frac{2q_2u(w)(\varphi '(w))^2\overline{\varphi (w)}^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}}= & {} (\alpha +n)(\alpha +n+1)(D^n_{\varphi ,u} f_{\varphi (w)} )''(w) \nonumber \\&-2\alpha (\alpha +n+1)(D^n_{\varphi ,u} g_{\varphi (w)} )''(w) \nonumber \\&+ \alpha (\alpha +1)(D^n_{\varphi ,u} h_{\varphi (w)} )''(w), \end{aligned}$$

(2.9)

which implies that

$$\begin{aligned}&\frac{ (1-|w|^2)^\beta |u(w)(\varphi '(w))^2||\varphi (w)|^{n+2}}{(1-|\varphi (w)|^2)^{\alpha +n+1}}\nonumber \\&\quad \le \frac{(\alpha +n)(\alpha +n+1)}{2q_2} \Vert D^n_{\varphi ,u} f_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } + \frac{\alpha (\alpha +n+1)}{q_2} \Vert D^n_{\varphi ,u} g_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } \nonumber \\&\qquad + \frac{\alpha (\alpha +1)}{2q_2} \Vert D^n_{\varphi ,u} h_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } \end{aligned}$$

(2.10)

$$\begin{aligned}&\quad \le \frac{(\alpha +n)(\alpha +n+1)+ \alpha (3\alpha +2n+3) }{2q_2}Q_3 . \end{aligned}$$

(2.11)

From (2.7) and (2.9), we get

$$\begin{aligned} q_2 \frac{v(w)\overline{\varphi (w)}^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}}= & {} - (\alpha +n)(\alpha +n+1)(\alpha +n+2)(D^n_{\varphi ,u} f_{\varphi (w)} )''(w) \nonumber \\&+ \alpha (\alpha +n+1)(2\alpha +2n+3) (D^n_{\varphi ,u} g_{\varphi (w)} )''(w) \nonumber \\&- (\alpha +1)\alpha (\alpha +n+1) (D^n_{\varphi ,u} h_{\varphi (w)} )''(w). \end{aligned}$$

(2.12)

It follows from (2.12) that

$$\begin{aligned}&\frac{ (1-|w|^2)^\beta |v(w) ||\varphi (w)|^{n+1}}{(1-|\varphi (w)|^2)^{\alpha +n}}\nonumber \\&\quad \le \frac{(\alpha +n)(\alpha +n+1)(\alpha +n+2)}{q_2} \Vert D^n_{\varphi ,u} f_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } \nonumber \\&\quad + \frac{\alpha (\alpha +n+1)(2\alpha +2n+3)}{q_2} \Vert D^n_{\varphi ,u} g_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } \nonumber \\&\quad + \frac{\alpha (\alpha +1)(\alpha +n+1)}{q_2} \Vert D^n_{\varphi ,u} h_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } \end{aligned}$$

(2.13)

$$\begin{aligned}&\quad \le \frac{(\alpha +n+1)[(\alpha +n+2)(\alpha +n)+\alpha (2\alpha +2n+3)+\alpha (\alpha +1) ]}{q_2}Q_3 .\nonumber \\ \end{aligned}$$

(2.14)

From (2.3), (2.9) and (2.12), we get

$$\begin{aligned} q_0 \frac{u''(w)\overline{\varphi (w)}^{n}}{(1-|\varphi (w)|^2)^{\alpha +n-1}}= & {} \frac{2+(\alpha +n)(\alpha +n+3)}{2} (D^n_{\varphi ,u} f_{\varphi (w)} )''(w) \nonumber \\&- \alpha (\alpha +n+2) (D^n_{\varphi ,u} g_{\varphi (w)} )''(w)\nonumber \\&+ \frac{\alpha (\alpha +1)}{2} (D^n_{\varphi ,u} h_{\varphi (w)} )''(w), \end{aligned}$$

(2.15)

which implies that

$$\begin{aligned}&\frac{ (1-|w|^2)^\beta |u''(w) ||\varphi (w)|^{n}}{(1-|\varphi (w)|^2)^{\alpha +n-1}}\le \frac{2+(\alpha +n)(\alpha +n+3)}{2q_0}\Vert D^n_{\varphi ,u} f_{\varphi (w)} \Vert _{\mathcal {Z}^\beta }\nonumber \\&\quad +\frac{\alpha (\alpha +n+2) }{ q_0} \Vert D^n_{\varphi ,u} g_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } + \frac{ \alpha (\alpha +1) }{2q_0} \Vert D^n_{\varphi ,u} h_{\varphi (w)} \Vert _{\mathcal {Z}^\beta } \end{aligned}$$

(2.16)

$$\begin{aligned}&\quad \le \frac{4\alpha ^2+4n\alpha +8\alpha +n^2+3n+2}{2q_0} Q_3. \end{aligned}$$

(2.17)

If $|\varphi (w)|\le 1/2$, from the fact that $u\in \mathcal {Z}^\beta$, we get

$$\begin{aligned} \frac{(1-|w|^2)^\beta |u''(w)| }{(1-|\varphi (w)|^2)^{ \alpha +n-1}} \le \left( \frac{4}{3}\right) ^{ \alpha +n-1} \sup _{z\in \mathbb {D}} (1-|z|^{2})^\beta |u''(z)| <\infty . \end{aligned}$$

(2.18)

On the other hand, if $|\varphi (w)|>1/2$, then from (2.17) we obtain

$$\begin{aligned} \frac{ (1-|w|^2)^\beta |u''(w)| }{(1-|\varphi (w)|^2)^{ \alpha +n-1}} \le \frac{2^{n-1}(4\alpha ^2+4n\alpha +8\alpha +n^2+3n+2)}{q_0}Q_3 < \infty .\nonumber \\ \end{aligned}$$

(2.19)

From (2.18) and (2.19) we see that

$$\begin{aligned} M_1:=\sup _{z\in \mathbb {D}}\frac{(1-|z|^{2})^\beta |u''(z)| }{(1-|\varphi (z)|^2)^{\alpha +n-1} } <\infty . \end{aligned}$$

(2.20)

By the fact that $u, u\varphi \in \mathcal {Z}^\beta$, we get

$$\begin{aligned} N_2:=\sup _{z\in \mathbb {D}} (1-|z|^2)^\beta |2u'(z)\varphi '(z)+u(z)\varphi ''(z)| <\infty . \end{aligned}$$

(2.21)

Using (2.14), (2.21) and similar arguments as above, we see that

$$\begin{aligned} M_2:=\sup _{z\in \mathbb {D}}\frac{(1-|z|^2)^\beta |2u'(z)\varphi '(z)+u(z)\varphi ''(z)|}{(1-|\varphi (z)|^2)^{\alpha +n} } <\infty . \end{aligned}$$

(2.22)

By the fact that $u, u\varphi , u\varphi ^2 \in \mathcal {Z}^\beta$, we get

$$\begin{aligned} N_1:=\sup _{z\in \mathbb {D}}(1-|z|^{2})^\beta |u(z)||\varphi '(z)|^2 <\infty . \end{aligned}$$

(2.23)

Using (2.11), (2.23) and similar arguments, we get

$$\begin{aligned} M_3:=\sup _{z\in \mathbb {D}}\frac{(1-|z|^{2})^\beta |u(z)||\varphi '(z)|^2 }{ (1-|\varphi (z)|^2)^{\alpha +n+1} }<\infty . \end{aligned}$$

(2.24)

From [42], we see that for any positive integer n and $f\in \mathcal {B}^\alpha$, there is a positive constant C independent of f such that

$$\begin{aligned} | f^{(n)}(z)|\le C\frac{\Vert f\Vert _{\mathcal {B}^\alpha }}{(1-|z|^2)^{\alpha +n-1}}. \end{aligned}$$

(2.25)

Hence for any $f\in \mathcal {B}^\alpha$ and arbitrary $z\in \mathbb {D}$, we have $|(D^n_{\varphi ,u} f)(0)| \lesssim \frac{ |u(0)|\Vert f\Vert _{ \mathcal {B}^\alpha }}{ (1-|\varphi (0)|^2)^{\alpha +n-1} },$

$$\begin{aligned} |(D^n_{\varphi ,u} f)'(0)| \lesssim \Big (\frac{ |u'(0)| }{ (1-|\varphi (0)|^2)^{\alpha +n-1} } + \frac{ |u(0)\varphi '(0)|}{ (1-|\varphi (0)|^2)^{\alpha +n} }\Big ) \Vert f\Vert _{\mathcal {B}^\alpha }, \end{aligned}$$

and

$$\begin{aligned}&(1-|z|^{2})^\beta |(D^n_{\varphi ,u} f)''(z)|\nonumber \\&\quad \le (1-|z|^{2})^\beta |u''(z)||f^{(n)}(\varphi (z))|+(1-|z|^{2})^\beta |f^{(n+2)}(\varphi (z))||u(z)(\varphi '(z) )^2 |\nonumber \\&\quad + (1-|z|^{2})^\beta |f^{(n+1)}(\varphi (z))||2u'(z)\varphi '(z)+u(z)\varphi ''(z) |\nonumber \\&\quad \lesssim \frac{(1-|z|^{2})^\beta |u''(z)|}{(1-|\varphi (z)|^2)^{\alpha +n-1} }\Vert f\Vert _{ \mathcal {B}^\alpha } + \frac{ (1-|z|^2)^\beta |2u'(z)\varphi '(z)+u(z)\varphi ''(z) |}{(1-|\varphi (z)|^2)^{\alpha +n} }\Vert f\Vert _{\mathcal {B}^\alpha } \nonumber \\&\quad + \frac{(1-|z|^{2})^\beta |u(z)||\varphi '(z)|^2 }{(1-|\varphi (z)|^2)^{\alpha +n+1} }\Vert f\Vert _{ \mathcal {B}^\alpha }\nonumber \\&\quad \lesssim (M_1+M_2+M_3) \Vert f\Vert _{ \mathcal {B}^\alpha }. \end{aligned}$$

(2.26)

Taking the supremum in (2.26) over $\mathbb {D}$ we see that $D^n_{\varphi ,u} :\mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is bounded. The proof is complete. $\square$

3 Essential Norm and Compactness of $D^n_{\varphi ,u}:\mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$

Theorem 2

Let $0<\alpha ,\beta <\infty$, $\varphi$ be an analytic self-map of ${\mathbb D}$, $u\in H(\mathbb {D})$ and n be a positive integer such that $D^n_{\varphi ,u}:\mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is bounded. Then

$$\begin{aligned} \Vert D^n_{\varphi ,u}\Vert _{e, \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta } \approx \limsup _{j \rightarrow \infty }j^{\alpha -1} \Vert D^n_{\varphi , u} I^j\Vert _{\mathcal {Z}^\beta } . \end{aligned}$$

Proof

For each nonnegative integer j, then $\{j^{\alpha -1}I^j\} \in \mathcal {B} ^\alpha$ and $\Vert j^{\alpha -1}I^j\Vert _{ \mathcal {B} ^\alpha } \lesssim 1$ for all j and converges uniformly to 0 on compact subsets of ${\mathbb D}$. By Proposition 1.2 in [4] we see that $\{j^{\alpha -1}I^j\}$ converges to 0 weakly in $\mathcal {B} ^\alpha _0$ as $j \rightarrow \infty$. By Hahn–Banach theorem we see that $\{j^{\alpha -1}I^j\}$ converges to 0 weakly in $\mathcal {B} ^\alpha$ as $j \rightarrow \infty$. Thus for any compact operator $K: \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$, we have that $\lim _{j \rightarrow \infty } \Vert K(j^{\alpha -1}I^j)\Vert _{\mathcal {Z}^\beta }=0.$ Hence

$$\begin{aligned} \Vert D^n_{\varphi , u}\Vert _{e, \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta }= & {} \inf _{K}\Vert D^n_{\varphi , u}-K\Vert _{\mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta }\\\gtrsim & {} \inf _{K}\limsup _{j \rightarrow \infty }\Vert (D^n_{\varphi , u}-K)(j^{\alpha -1}I^j)\Vert _{\mathcal {Z}^\beta }\\\gtrsim & {} \limsup _{j \rightarrow \infty }j^{\alpha -1} \Vert D^n_{\varphi , u} I^j\Vert _{\mathcal {Z}^\beta }. \end{aligned}$$

Next we prove that

$$\begin{aligned} \Vert D^n_{\varphi , u}\Vert _{e, \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta } \lesssim \limsup _{j \rightarrow \infty }j^{\alpha -1} \Vert D^n_{\varphi , u} I^j\Vert _{\mathcal {Z}^\beta }. \end{aligned}$$

For this purpose, we first prove that

$$\begin{aligned} \Vert D^n_{\varphi ,u}\Vert _{e, \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta } \lesssim \max \big \{ A, B, E \big \} , \end{aligned}$$

where

$$\begin{aligned} A:= & {} \limsup _{|a|\rightarrow 1}\left\| D^n_{\varphi ,u}f_a \right\| _{\mathcal {Z}^\beta }, \,\,\, B:= \limsup _{|a|\rightarrow 1}\left\| D^n_{\varphi ,u}g_a\right\| _{\mathcal {Z}^\beta },\\ E:= & {} \limsup _{|a|\rightarrow 1}\left\| D^n_{\varphi ,u}h_a \right\| _{\mathcal {Z}^\beta }. \end{aligned}$$

For $r\in [0,1)$, define

$$\begin{aligned} (K_r f)(z)=f_r(z)=f(rz), ~~f\in H(\mathbb {D}). \end{aligned}$$

It is clear that $f_r-f\rightarrow 0$ uniformly on compact subsets of $\mathbb {D}$ as $r\rightarrow 1$. Moreover, the operator $K_r$ is compact on $\mathcal {B} ^\alpha$ and $\Vert K_r\Vert _{ \mathcal {B} ^\alpha \rightarrow \mathcal {B} ^\alpha }\le 1$ (see [22]). Let $\{r_j\}\subset (0,1)$ be a sequence such that $r_j\rightarrow 1$ as $j\rightarrow \infty$. Then for all positive integer j, the operator $D^n_{\varphi ,u} K_{r_j}: \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta$ is compact. Hence

$$\begin{aligned} \Vert D^n_{\varphi ,u}\Vert _{e, \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta } \le \limsup _{j\rightarrow \infty }\Vert D^n_{\varphi ,u}- D^n_{\varphi ,u} K_{r_j}\Vert _{ \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta }. \end{aligned}$$

(3.1)

Therefore, we only need to prove that

$$\begin{aligned} \limsup _{j\rightarrow \infty }\Vert D^n_{\varphi ,u}- D^n_{\varphi ,u} K_{r_j}\Vert _{ \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta }\lesssim \max \big \{ A, B, E \big \}. \end{aligned}$$

(3.2)

Let $f\in \mathcal {B} ^\alpha$ with $\Vert f\Vert _{ \mathcal {B} ^\alpha }\le 1$. It is clear that

$$\begin{aligned} \lim _{j\rightarrow \infty }|u(0)f^{(n)}(\varphi (0))-r_j^nu(0)f^{(n)}(r_j\varphi (0))|=0 \end{aligned}$$

(3.3)

and

$$\begin{aligned} \lim _{j\rightarrow \infty }|u'(0)(f-f_{r_j})^{(n)}(\varphi (0))+u(0)(f-f_{r_j})^{(n+1)}(\varphi (0))\varphi '(0)|=0. \end{aligned}$$

(3.4)

Hence

$$\begin{aligned}&\Vert ( D^n_{\varphi ,u}- D^n_{\varphi ,u} K_{r_j})f\Vert _{\mathcal {Z}^\beta }\nonumber \\&\quad =\Vert u\cdot (f-f_{r_j})^{(n)}\circ \varphi \Vert _*+ |u(0)f^{(n)}(\varphi (0))-r_j^nu(0)f^{(n)}(r_j\varphi (0))|\nonumber \\&\quad +|u'(0)(f-f_{r_j})^{(n)}(\varphi (0))+u(0)(f-f_{r_j})^{(n+1)}(\varphi (0))\varphi '(0)|\nonumber \\&\quad = \limsup _{j\rightarrow \infty }\Vert u\cdot (f-f_{r_j})^{(n)} \circ \varphi \Vert _*\nonumber \\&\quad \le \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+1)}(\varphi (z))||v(z)|\nonumber \\&\quad + \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+1)}(\varphi (z))||v(z)| \nonumber \\&\quad +\limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n)}(\varphi (z))||u''(z)| \nonumber \\&\quad + \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n)}(\varphi (z))||u''(z)|\nonumber \\&\quad +\limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+2)}(\varphi (z))||\varphi '(z)|^2|u(z)| \nonumber \\&\quad + \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+2)}(\varphi (z))||\varphi '(z)|^2|u(z)|\nonumber \\&\quad := P_1+P_2+P_3+P_4+P_5+P_6, \end{aligned}$$

(3.5)

where $N\in \mathbb {N }$ is large enough such that $r_j\ge \frac{1}{2}$ for all $j\ge N$ and $v(z)=2u'(z)\varphi '(z)+u(z)\varphi ''(z)$, $\Vert f\Vert _{*}=\sup _{z \in \mathbb {D}}(1-|z|^2)^\beta |f''(z)|.$ Since $r^{n}_jf^{(n)}_{r_j}-f^{(n)}\rightarrow 0$, uniformly on compact subsets of ${\mathbb D}$ as $j\rightarrow \infty$, by the fact that $u, u\varphi , u\varphi ^2\in \mathcal {Z}^\beta$, we have

$$\begin{aligned} P_1= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+1)}(\varphi (z))||v(z)|\nonumber \\\le & {} (\Vert u\Vert _{\mathcal {Z}^\beta }+\Vert u\varphi \Vert _{\mathcal {Z}^\beta }) \limsup _{j\rightarrow \infty }\sup _{|w|\le r_N} |f^{(n+1)}(w)- r^{n+1}_jf^{(n+1)} (r_j w)|\nonumber \\= & {} 0, \end{aligned}$$

(3.6)

$$\begin{aligned} P_3= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n)}(\varphi (z))||u''(z)|\nonumber \\\le & {} \Vert u\Vert _{\mathcal {Z}^\beta }\limsup _{j\rightarrow \infty }\sup _{|w|\le r_N} |f^{(n)}(w)- r^{n}_jf^{(n)} (r_j w)| =0 \end{aligned}$$

(3.7)

and

$$\begin{aligned} P_5= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|\le r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+2)}(\varphi (z))||\varphi '(z)|^2|u(z)|\nonumber \\\le & {} (\Vert u\Vert _{\mathcal {Z}^\beta }\!+\!\Vert u\varphi \Vert _{\mathcal {Z}^\beta }+\Vert u\varphi ^2 \Vert _{\mathcal {Z}^\beta }) \limsup _{j\rightarrow \infty }\sup _{|w|\le r_N} |f^{(n+2)}(w)- r^{n+2}_jf^{(n+2)} (r_j w)| \nonumber \\= & {} 0. \end{aligned}$$

(3.8)

To estimate $P_2$, using the fact that $\Vert f\Vert _{ \mathcal {B} ^\alpha }\le 1$ and (2.25), we have

$$\begin{aligned} P_2= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+1)}(\varphi (z))||v(z)|\\\lesssim & {} \limsup _{j\rightarrow \infty } \frac{(\alpha +n+2) \Vert f-f_{r_j}\Vert _{ \mathcal {B} ^\alpha } }{q_0r_N^{n+1}}\sup _{|\varphi (z) |>r_N} \frac{q_0 |\varphi (z)|^{n+1} (1-|z|^2)^\beta |v(z)|}{(\alpha +n+2)(1-|\varphi (z)|^2)^{n+\alpha }} \\\lesssim & {} \sup _{|\varphi (z) |>r_N} \frac{q_0 |\varphi (z)|^{n+1}(1-|z|^2)^\beta |v(z)| }{(\alpha +n+2)(1-|\varphi (z)|^2)^{n+\alpha }} \\\lesssim & {} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u}\left( f_a- \frac{\alpha (2\alpha +2n+3)g_a}{(\alpha +n)(\alpha +n+2)} +\frac{\alpha (\alpha +1)}{(\alpha +n)(\alpha +n+2)} h_a \right) \right\| _{\mathcal {Z}^\beta } \\\lesssim & {} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} f_a \right\| _{\mathcal {Z}^\beta }+ \frac{\alpha (2\alpha +2n+3)}{(\alpha +n)(\alpha +n+2)} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} g_a \right\| _{\mathcal {Z}^\beta } \\&+\frac{\alpha (\alpha +1)}{(\alpha +n)(\alpha +n+2)} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} h_a \right\| _{\mathcal {Z}^\beta }. \end{aligned}$$

Taking the limit as $N\rightarrow \infty$ we obtain

$$\begin{aligned} P_2\lesssim & {} \limsup _{|a|\rightarrow 1}\left\| D^n_{\varphi ,u} f_a \right\| _{\mathcal {Z}^\beta }+ \limsup _{|a|\rightarrow 1}\left\| D^n_{\varphi ,u} g_a \right\| _{\mathcal {Z}^\beta } + \limsup _{|a|\rightarrow 1}\left\| D^n_{\varphi ,u} h_a \right\| _{\mathcal {Z}^\beta } \nonumber \\= & {} A+B+E \lesssim \max \big \{ A, B , E\big \} . \end{aligned}$$

(3.9)

Similarly,

$$\begin{aligned} P_4= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n)}(\varphi (z))||u''(z)| \\\lesssim & {} \sup _{|\varphi (z) |>r_N} \frac{2q_0}{2+(\alpha +n)(\alpha +n+3)} \frac{(1-|z|^2)^\beta |u''(z)| |\varphi (z)|^n }{(1-|\varphi (z)|^2)^{\alpha +n-1} } \\\lesssim & {} \sup _{|a|>r_N} \left\| D^n_{\varphi ,u} f_a \right\| _{\mathcal {Z}^\beta }+ \frac{2\alpha (\alpha +n+2)}{2+(\alpha +n)(\alpha +n+3)} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} g_a \right\| _{\mathcal {Z}^\beta } \\&+ \frac{\alpha (\alpha +1)}{2+(\alpha +n)(\alpha +n+3)} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} h_a \right\| _{\mathcal {Z}^\beta }, \end{aligned}$$

which yields

$$\begin{aligned} P_4 \lesssim A+B+E\lesssim \max \big \{ A, B , E\big \} . \end{aligned}$$

(3.10)

Similarly,

$$\begin{aligned} P_6= & {} \limsup _{j\rightarrow \infty }\sup _{|\varphi (z)|> r_N}(1-|z|^2)^\beta |(f-f_{r_j})^{(n+2)}(\varphi (z))||\varphi '(z)|^2|u(z)|\\\lesssim & {} \sup _{|\varphi (z) |>r_N} (1-|z|^2)^\beta |\varphi '(z)|^2|u(z)| \frac{ 2q_0|\varphi (z)|^{n+2} }{(1-|\varphi (z)|^2)^{\alpha +n+1}} \\\lesssim & {} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u}\left( f_a- \frac{2\alpha }{\alpha +n} g_a+ \frac{\alpha (\alpha +1)}{(\alpha +n)(\alpha +n+1)} h_a \right) \right\| _{\mathcal {Z}^\beta } \\\lesssim & {} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} f_a \right\| _{\mathcal {Z}^\beta }+ \frac{2\alpha }{\alpha +n} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} g_a \right\| _{\mathcal {Z}^\beta } \\&+ \frac{\alpha (\alpha +1)}{(\alpha +n)(\alpha +n+1)} \sup _{|a |>r_N} \left\| D^n_{\varphi ,u} h_a \right\| _{\mathcal {Z}^\beta } , \end{aligned}$$

which implies that

$$\begin{aligned} P_6 \lesssim A+B+E \lesssim \max \big \{ A, B , E\big \} . \end{aligned}$$

(3.11)

Hence, by (3.5)–(3.11) we obtain

$$\begin{aligned}&\limsup _{j\rightarrow \infty }\Vert D^n_{\varphi ,u}- D^n_{\varphi ,u} K_{r_j}\Vert _{ \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta } \nonumber \\&\quad =\limsup _{j\rightarrow \infty }\sup _{ \Vert f\Vert _{ \mathcal {B} ^\alpha } \le 1}\Vert (D^n_{\varphi ,u}- D^n_{\varphi ,u} K_{r_j})f\Vert _{\mathcal {Z}^\beta } \nonumber \\&\quad =\limsup _{j\rightarrow \infty }\sup _{ \Vert f\Vert _{ \mathcal {B} ^\alpha } \le 1} \Vert u\cdot (f-f_{r_j})^{(n)}\circ \varphi \Vert _* \lesssim \max \big \{ A, B, E \big \}. \end{aligned}$$

(3.12)

Therefore, by (3.1) and (3.12), we obtain $\Vert D^n_{\varphi ,u}\Vert _{e, \mathcal {B} ^\alpha \rightarrow \mathcal {Z}^\beta } \lesssim \max \big \{ A, B, E \big \} .$

For $a\in \mathbb {D}$, we know that

$$\begin{aligned} f_a(z)=(1-|a|^2)\sum _{j=0}^\infty \frac{\Gamma (j+\alpha )}{j!\Gamma (\alpha )} \overline{a}^jz^j . \end{aligned}$$

Since $D^n_{\varphi , u} : \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is bounded, from Theorem 1 we known that $M=\displaystyle \sup _{j\ge n} j^{ \alpha -1}\Vert D^n_{\varphi , u} I^j \Vert _{\mathcal {Z}^\beta }<\infty$. For fixed positive integer $N\ge 1$, we obtain

$$\begin{aligned}&\Vert D^n_{\varphi ,u}f_{a}\Vert _{\mathcal {Z}^\beta } \lesssim (1-|a|^2)\sum _{j=0}^\infty |a|^jj^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } \\&\quad \lesssim (1-|a|^2)\bigg (\sum _{j=0}^{N-1} |a|^jj^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } + \sum _{j=N}^{\infty } |a|^jj^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } \bigg ) \\&\quad \lesssim M(1-|a|^N) +\sup _{j\ge N} j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } . \end{aligned}$$

Letting $|a| \rightarrow 1$ in the last inequality. We get

$$\begin{aligned} \limsup _{|a| \rightarrow 1}\Vert D^n_{\varphi ,u}f_{a}\Vert _{\mathcal {Z}^\beta }\lesssim \sup _{j\ge N} j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta }. \end{aligned}$$

Thus,

$$\begin{aligned} A=\limsup _{|a| \rightarrow 1}\Vert D^n_{\varphi ,u}f_{a}\Vert _{\mathcal {Z}^\beta } \lesssim \limsup _{j \rightarrow \infty }j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } . \end{aligned}$$

Similarly, we have

$$\begin{aligned} B \lesssim \limsup _{j \rightarrow \infty }j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } , \, \, \, E \lesssim \limsup _{j \rightarrow \infty }j^{\alpha -1}\Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } . \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert D^n_{\varphi ,u}\Vert _{e, \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta } \lesssim \max \big \{ A, B, E \big \}\lesssim \limsup _{j \rightarrow \infty }j^{\alpha -1} \Vert D^n_{\varphi ,u}I^j\Vert _{\mathcal {Z}^\beta } . \end{aligned}$$

The proof is complete. $\square$

From Theorem 2, we get the following characterization of compactness of the operator $D^n_{\varphi ,u} : \mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$.

Corollary 1

Let $0<\alpha ,\beta <\infty$, $\varphi$ be an analytic self-map of ${\mathbb D}$, $u\in H(\mathbb {D})$ and n be a positive integer such that $D^n_{\varphi ,u}:\mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is bounded. Then the operator $D^n_{\varphi ,u} :\mathcal {B}^\alpha \rightarrow \mathcal {Z}^\beta$ is compact if and only if

$$\begin{aligned} \limsup \limits _{j\rightarrow \infty }j^{\alpha -1}\Vert D^n_{\varphi , u} I^j \Vert _{\mathcal {Z}^\beta }=0 . \end{aligned}$$