Weighted composition followed and proceeded by differentiation operators fromZygmund spaces to Bloch-type spaces

Abstract

The boundedness and compactness of the weighted composition followed andproceeded by differentiation operators from Zygmund spaces to Bloch-type spacesand little Bloch-type spaces are characterized.

MSC: 47B38, 30H30, 32C15.

1 Introduction

Let $\mathrm{\Delta }=\{z:|z|<1\}$ be the open unit disc in the complex plane ℂ,and let $H(\mathrm{\Delta })$ be the class of all analytic functions on Δ. Theα-Bloch space $B^{\alpha }$ ($0<\alpha <\mathrm{\infty }$) is, by definition, the set of all functionsf in $H(\mathrm{\Delta })$ such that

$${\parallel f\parallel }_{B^{\alpha }}=|f(0)|+\underset{z\in \mathrm{\Delta }}{sup}{(1-{|z|}^2)}^{\alpha }|f^{\prime }(z)|<\mathrm{\infty }.$$

(1.1)

Under the above norm, $B^{\alpha }$ is a Banach space. When $\alpha =1$, $B^1=B$ is the well-known Bloch space. Let$B_0^{\alpha }$ denote the subspace of $B^{\alpha }$, i.e.,

$$B_0^{\alpha }=\{f:{(1-{|z|}^2)}^{\alpha }|f^{\prime }(z)|\to 0\text{ as }|z|\to 1,f\in B^{\alpha }\}.$$

This space is called the little α-Bloch space (see [1]).

Assume that μ is a positive continuous function on$[0,1)$, having the property that there exist positivenumbers s and t, $0<s<t$, and $\delta \in [0,1)$, such that

$$\begin{array}{r}\frac{\mu (r)}{{(1-r)}^s}\text{ is decreasing on }[\delta ,1),\underset{r\to 1}{lim}\frac{\mu (r)}{{(1-r)}^s}=0,\\ \frac{\mu (r)}{{(1-r)}^t}\text{ is increasing on }[\delta ,1),\underset{r\to 1}{lim}\frac{\mu (r)}{{(1-r)}^t}=\mathrm{\infty }.\end{array}$$

Then μ is called a normal function (see [2]).

Denote (see, e.g., [3–5])

$$B_{\mu }=\{f:{\parallel f\parallel }_{B_{\mu }}=|f(0)|+\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|f^{\prime }(z)|<\mathrm{\infty },f\in H(\mathrm{\Delta })\}.$$

(1.2)

It is well known that $B_{\mu }$ is a Banach space with the norm${\parallel \cdot \parallel }_{B_{\mu }}$ (see [4]).

Let $B_{\mu ,0}$ denote the subspace of $B_{\mu }$, i.e.,

$$B_{\mu ,0}=\{f:\mu (|z|)|f^{\prime }(z)|\to 0\text{ as }|z|\to 1,f\in B_{\mu }\}.$$

This space is called the little Bloch-type space. When $\mu (r)={(1-r^2)}^{\alpha }$, the induced space $B_{\mu }$ becomes the α-Bloch space$B^{\alpha }$.

An f in $H(\mathrm{\Delta })$ is said to belong to the Zygmund space, denoted by, if

$$sup\frac{|f(e^{i(\theta +h)})+f(e^{i(\theta -h)})-2f(e^{i\theta })|}{h}<\mathrm{\infty },$$

where the supremum is taken over all $e^{i\theta }\in \partial \mathrm{\Delta }$ and $h>0$. By Theorem 5.3 in [6], we see that $f\in \mathcal{Z}$ if and only if

$${\parallel f\parallel }_{\mathbb{Z}}=|f(0)|+|f^{\prime }(0)|+\underset{z\in \mathrm{\Delta }}{sup}(1-{|z|}^2)|f^{″}(z)|<\mathrm{\infty }.$$

(1.3)

It is easy to check that is a Banach space under the above norm. For every$f\in \mathcal{Z}$, by using a result in [7], we have

$$|f^{\prime }(z)|\le C{\parallel f\parallel }_{\mathcal{Z}}ln\frac{e}{1-{|z|}^2}.$$

(1.4)

Let ${\mathcal{Z}}_0$ denote the subspace of consisting of those$f\in \mathcal{Z}$ for which

$$\underset{|z|\to 1}{lim}(1-{|z|}^2)|f^{″}(z)|=0.$$

The space ${\mathcal{Z}}_0$ is called the little Zygmund space. For thecorresponding n-dimensional Zygmund space see, e.g., [8] and [9].

Let φ be a nonconstant analytic self-map of Δ, and letϕ be an analytic function in Δ. We define the linearoperators

$$\varphi C_{\phi }Df=\varphi (f^{\prime }\circ \phi )=\varphi f^{\prime }(\phi )\text{and}\varphi D{C}_{\phi }f=\varphi {(f\circ \phi )}^{\prime }=\varphi f^{\prime }(\phi ){\phi }^{\prime },\text{for }f\in H(\mathrm{\Delta }).$$

They are called weighted composition followed and proceeded by differentiationoperators, respectively, where $C_{\phi }$ and D are composition and differentiationoperators respectively. Associated with φ is the composition operator$C_{\phi }f=f\circ \phi$ and weighted composition operator$\varphi C_{\phi }f=\varphi f\circ \phi$ for $\varphi \in H(\mathrm{\Delta })$ and $f\in H(\mathrm{\Delta })$. It is interesting to provide a function theoreticcharacterization for φ inducing a bounded or compact compositionoperator, weighted composition operator and related ones on various spaces (see,e.g., [10–19]). For example, it is well known that $C_{\phi }$ is bounded on the classical Hardy, Bloch and Bergmanspaces. Operators $D{C}_{\phi }$ and $C_{\phi }D$ as well as some other products of linear operatorswere studied, for example, in [20–29] (see also the references therein). There has been some considerablerecent interest in investigation various type of operators from or to Zygmund typespaces (see, [7, 11, 23, 30–37]).

In this paper, we investigate the operators $\varphi D{C}_{\phi }$ and $\varphi C_{\phi }D$ from Zygmund spaces to Bloch-type spaces and littleBloch-type spaces. Some sufficient and necessary conditions for the boundedness andcompactness of these operators are given.

Throughout this paper, constants are denoted by C, they are positive and maydiffer from one occurrence to the other. The notation $A\approx B$ means that there is a positive constant Csuch that $\frac{B}{C}\le A\le CB$.

2 Main results and proofs

In this section, we state and prove our main results. In order to formulate our mainresults, we quote several lemmas which will be used in the proofs of the mainresults in this paper. The following lemma can be proved in a standard way (see,e.g., Proposition 3.11 in [10]). Hence we omit the details.

Lemma 2.1 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then$\varphi D{C}_{\phi }(\mathit{\text{or }}\varphi C_{\phi }D):\mathcal{Z}(\mathit{\text{or }}{\mathcal{Z}}_0)\to B_{\mu }$is compact if and only if$\varphi D{C}_{\phi }(\mathit{\text{or }}\varphi C_{\phi }D):\mathcal{Z}(\mathit{\text{or }}{\mathcal{Z}}_0)\to B_{\mu }$is bounded and for any bounded sequence${\{f_n\}}_{n\in N}$in(or${\mathcal{Z}}_0$) which converges to zero uniformly on compactsubsets of Δ as$n\to \mathrm{\infty }$, and${\parallel \varphi D{C}_{\phi }{f}_n\parallel }_{B_{\mu }}\to 0$ (or${\parallel \varphi C_{\phi }D{f}_n\parallel }_{B_{\mu }}\to 0$) as$n\to \mathrm{\infty }$.

Lemma 2.2[25]

A closed set of $B_{\mu ,0}$ is compact if and only if it is bounded and satisfied

$$\underset{|z|\to 1}{lim}\underset{f\in \mathbb{K}}{sup}\mu (|z|)|f^{\prime }(z)|=0.$$

Theorem 2.3 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.

• (i) $\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu }$ is bounded;

• (ii) $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is bounded;

• (iii)
  $$\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}<\mathrm{\infty }$$

  (2.1)

and

$$\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}<\mathrm{\infty }.$$

(2.2)

Proof of Theorem 2.3 (i) ⇒ (ii). This implication is obvious.

• (ii) ⇒ (iii). Assume that $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is bounded, i.e., there exists a constant C such that

  $${\parallel \varphi D{C}_{\phi }f\parallel }_{B_{\mu }}\le C{\parallel f\parallel }_{\mathcal{Z}}$$

for all $f\in {\mathcal{Z}}_0$. Taking the functions $f(z)=z\in {\mathcal{Z}}_0$ and $f(z)=z^2\in {\mathcal{Z}}_0$ respectively, we get

$$\begin{array}{r}\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|<\mathrm{\infty },\\ \underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|(\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z))\phi (z)+\varphi (z){({\phi }^{\prime }(z))}^2|<\mathrm{\infty }.\end{array}$$

(2.3)

Using these facts and the boundedness of function φ, we have

$$\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2|<\mathrm{\infty }.$$

(2.4)

Set

$$h(z)=(z-1)[{(1+ln\frac{1}{1-z})}^2+1]$$

and

$$h_a(z)=\frac{h(\overline{a}z)}{\overline{a}}{(ln\frac{1}{1-{|a|}^2})}^{-1}$$

(2.5)

for $a\in \mathrm{\Delta }\setminus \{0\}$. It is known that $h_a\in {\mathcal{Z}}_0$ (see [7]). Since

$$h_a^{\prime }(z)={(ln\frac{1}{1-\overline{a}z})}^2{(ln\frac{1}{1-{|a|}^2})}^{-1}$$

(2.6)

and

$$h_a^{″}(z)=\frac{2\overline{a}}{1-\overline{a}z}(ln\frac{1}{1-\overline{a}z}){(ln\frac{1}{1-{|a|}^2})}^{-1},$$

(2.7)

for $|\phi (\lambda )|>\frac{1}{2}$, we have

$$\begin{array}{rl}C{\parallel \varphi D{C}_{\phi }\parallel }_{{\mathcal{Z}}_0\to B_{\mu }}\ge & {\parallel \varphi D{C}_{\phi }{h}_{\phi (\lambda )}\parallel }_{B_{\mu }}\ge \mu (|\lambda |)|\varphi (\lambda ){\phi }^{″}(\lambda )+{\varphi }^{\prime }(\lambda ){\phi }^{\prime }(\lambda )|ln\frac{1}{1-{|\phi (\lambda )|}^2}\\ -2\mu (|\lambda |)\frac{|\varphi (\lambda ){({\phi }^{\prime }(\lambda ))}^2\phi (\lambda )|}{1-{|\phi (\lambda )|}^2}.\end{array}$$

Hence

$$\begin{array}{r}\mu (|\lambda |)|\varphi (\lambda ){\phi }^{″}(\lambda )+{\varphi }^{\prime }(\lambda ){\phi }^{\prime }(\lambda )|ln\frac{1}{1-{|\phi (\lambda )|}^2}\\ \le C{\parallel \varphi D{C}_{\phi }\parallel }_{{\mathcal{Z}}_0\to B_{\mu }}+2\mu (|\lambda |)\frac{|\varphi (\lambda ){({\phi }^{\prime }(\lambda ))}^2\phi (\lambda )|}{1-{|\phi (\lambda )|}^2}.\end{array}$$

(2.8)

For $a\in \mathrm{\Delta }\setminus \{0\}$, set

$$f_a(z)=\frac{h(\overline{a}z)}{\overline{a}}{(ln\frac{1}{1-{|a|}^2})}^{-1}-{\int }_0^{z}ln\frac{1}{1-\overline{a}\omega }d\omega .$$

(2.9)

Then $f_a\in {\mathcal{Z}}_0$. It is easy to see that

$$f_a^{\prime }(z)={(ln\frac{1}{1-\overline{a}z})}^2{(ln\frac{1}{1-{|a|}^2})}^{-1}-ln\frac{1}{1-\overline{a}z},f_a^{\prime }(a)=0,$$

and

$$f_a^{″}(z)=\frac{2\overline{a}}{1-\overline{a}z}(ln\frac{1}{1-\overline{a}z}){(ln\frac{1}{1-{|a|}^2})}^{-1}-\frac{\overline{a}}{1-\overline{a}z},f_a^{″}(a)=\frac{\overline{a}}{1-{|a|}^2}.$$

Therefore

$$C{\parallel \varphi D{C}_{\phi }\parallel }_{{\mathcal{Z}}_0\to B_{\mu }}\ge {\parallel \varphi D{C}_{\phi }{f}_{\phi (\lambda )}\parallel }_{B_{\mu }}\ge \mu (|\lambda |)\frac{|\varphi (\lambda ){({\phi }^{\prime }(\lambda ))}^2\phi (\lambda )|}{1-{|\phi (\lambda )|}^2}.$$

(2.10)

From (2.8) and (2.10), we have

$$\underset{|\phi (\lambda )|>\frac{1}{2}}{sup}\mu (|\lambda |)|\varphi (\lambda ){\phi }^{″}(\lambda )+{\varphi }^{\prime }(\lambda ){\phi }^{\prime }(\lambda )|ln\frac{1}{1-{|\phi (\lambda )|}^2}\le C{\parallel \varphi D{C}_{\phi }\parallel }_{{\mathcal{Z}}_0\to B_{\mu }}<\mathrm{\infty }.$$

(2.11)

On the other hand, from the first inequality in (2.3), we have

$$\begin{array}{r}\underset{|\phi (\lambda )|\le \frac{1}{2}}{sup}\mu (|\lambda |)|\varphi (\lambda ){\phi }^{″}(\lambda )+{\varphi }^{\prime }(\lambda ){\phi }^{\prime }(\lambda )|ln\frac{1}{1-{|\phi (\lambda )|}^2}\\ \le \underset{\lambda \in \mathrm{\Delta }}{sup}\mu (|\lambda |)|\varphi (\lambda ){\phi }^{″}(\lambda )+{\varphi }^{\prime }(\lambda ){\phi }^{\prime }(\lambda )|ln\frac{4}{3}<\mathrm{\infty }.\end{array}$$

(2.12)

Hence, from (2.3), (2.11), and (2.12), we obtain (2.2). Further, from (2.10), wehave

$$\begin{array}{rl}\underset{|\phi (\lambda )|>\frac{1}{2}}{sup}\mu (|\lambda |)\frac{|\varphi (\lambda ){({\phi }^{\prime }(\lambda ))}^2|}{1-{|\phi (\lambda )|}^2}& \le \underset{|\phi (\lambda )|>\frac{1}{2}}{sup}2\mu (|\lambda |)\frac{|\varphi (\lambda ){({\phi }^{\prime }(\lambda ))}^2\phi (\lambda )|}{1-{|\phi (\lambda )|}^2}\\ \le C{\parallel \varphi D{C}_{\phi }\parallel }_{{\mathcal{Z}}_0\to B_{\mu }}<\mathrm{\infty }.\end{array}$$

(2.13)

On the other hand, by (2.4), we have

$$\underset{|\phi (\lambda )|\le \frac{1}{2}}{sup}\mu (|\lambda |)\frac{|\varphi (\lambda ){({\phi }^{\prime }(\lambda ))}^2|}{1-{|\phi (\lambda )|}^2}\le \underset{|\phi (\lambda )|\le \frac{1}{2}}{sup}\frac{4}{3}\mu (|\lambda |)|\varphi (\lambda ){({\phi }^{\prime }(\lambda ))}^2|<\mathrm{\infty }.$$

(2.14)

Combining (2.13) and (2.14), (2.1) follows.

• (iii) ⇒ (i). Assume that (2.1) and (2.2) hold. Then, for every $f\in \mathcal{Z}$, from (1.4), we have

  $$\begin{array}{r}\mu (|z|)|{(\varphi D{C}_{\phi }f)}^{\prime }(z)|\\ =\mu (|z|)|{\varphi }^{\prime }(z){\phi }^{\prime }(z)f^{\prime }(\phi (z))+\varphi (z)[f^{″}(\phi (z)){({\phi }^{\prime }(z))}^2+f^{\prime }(\phi (z)){\phi }^{″}(z)]|\\ \le \mu (|z|)|\varphi (z)f^{″}(\phi (z)){({\phi }^{\prime }(z))}^2|+\mu (|z|)|[\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)]f^{\prime }(\phi (z)|\\ \le \mu (|z|)C\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}{\parallel f\parallel }_{\mathcal{Z}}+\mu (|z|)C|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|\\ \times ln\frac{e}{1-{|\phi (z)|}^2}{\parallel f\parallel }_{\mathcal{Z}}.\end{array}$$

  (2.15)

Taking the supremum in (2.15) for $z\in \mathrm{\Delta }$, and employing (2.1) and (2.2), we deduce that$\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu }$ is bounded. The proof of Theorem 2.3 iscompleted. □

Theorem 2.4 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.

• (i) $\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu }$ is compact;

• (ii) $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is compact;

• (iii) $\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu }$ is bounded,

  $$\underset{|\phi (z)|\to 1}{lim}\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}=0$$

  (2.16)

and

$$\underset{|\phi (z)|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}=0.$$

(2.17)

Proof of Theorem 2.4 (i) ⇒ (ii). This implication is clear.

• (ii) ⇒ (iii). Assume that $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is compact. Then it is clear that $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is bounded. By Theorem 2.3 we know that $\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu }$ is bounded. Let ${(z_n)}_{n\in N}$ be a sequence in Δ such that $|\phi (z_n)|\to 1$ as $n\to \mathrm{\infty }$ and $\phi (z_n)\ne 0$, $n\in N$ (if such a sequence does not exist then (2.16) and (2.17) are vacuously satisfied). Set

  $$h_n(z)=\frac{h(\overline{\phi (z_n)}z)}{\overline{\phi (z_n)}}{(ln\frac{1}{1-{|\phi (z_n)|}^2})}^{-1},n\in N.$$

  (2.18)

Then from the proof of Theorem 2.3, we see that $h_n\in {\mathcal{Z}}_0$ for each $n\in N$. Moreover $h_n\to 0$ uniformly on compact subsets of Δ as$n\to \mathrm{\infty }$ and

$$h_n^{\prime }(\phi (z_n))=ln\frac{1}{1-{|\phi (z_n)|}^2},h_n^{″}(\phi (z_n))=ln\frac{2\overline{\phi (z_n)}}{1-{|\phi (z_n)|}^2}.$$

Since $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is compact, by Lemma 2.1, we have

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel \varphi D{C}_{\phi }{h}_n\parallel }_{B_{\mu }}=0.$$

On the other hand, similar to the proof of Theorem 2.3, we have

$$\begin{array}{rl}{\parallel \varphi D{C}_{\phi }{h}_n\parallel }_{B_{\mu }}\ge & |2\mu (|z_n|)\frac{|\varphi (z_n){({\phi }^{\prime }(z_n))}^2||\phi (z_n)|}{1-{|\phi (z_n)|}^2}-\mu (|z_n|)|\\ \times \varphi (z_n){\phi }^{″}(z_n)+{\varphi }^{\prime }(z_n){\phi }^{\prime }(z_n)|ln\frac{1}{1-{|\phi (z_n)|}^2}|,\end{array}$$

which implies that

$$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}2\mu (|z_n|)\frac{|\varphi (z_n){({\phi }^{\prime }(z_n))}^2||\phi (z_n)|}{1-{|\phi (z_n)|}^2}\\ =\underset{n\to \mathrm{\infty }}{lim}\mu (|z_n|)|\varphi (z_n){\phi }^{″}(z_n)+{\varphi }^{\prime }(z_n){\phi }^{\prime }(z_n)|ln\frac{1}{1-{|\phi (z_n)|}^2},\end{array}$$

(2.19)

if one of these two limits exists.

Next, set

$$\begin{array}{rl}{f}_n(z)=& \frac{h(\overline{\phi (z_n)}z)}{\overline{\phi (z_n)}}{(ln\frac{1}{1-{|\phi (z_n)|}^2})}^{-1}\\ -{\int }_0^z{ln}^3\frac{1}{1-\overline{\phi (z_n)}\omega }d\omega {(ln\frac{1}{1-{|\phi (z_n)|}^2})}^{-2}.\end{array}$$

(2.20)

Then $f_n\in {\mathcal{Z}}_0$ and $f_n$ converges to 0 uniformly on compact subsets of Δas $n\to \mathrm{\infty }$ (see [7]). Since

$$f_n^{\prime }(z)={(ln\frac{1}{1-\overline{\phi (z_n)}z})}^2{(ln\frac{1}{1-{|\phi (z_n)|}^2})}^{-1}-{(ln\frac{1}{1-\overline{\phi (z_n)}z})}^3{(ln\frac{1}{1-{|\phi (z_n)|}^2})}^{-2},$$

we have $f_n^{\prime }(\phi (z_n))=0$, for every $n\in N$ and

$$f_n^{″}(\phi (z_n))=-\frac{\overline{\phi (z_n)}}{1-{|\phi (z_n)|}^2}.$$

By using these facts, since $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is compact, and from Lemma 2.1, we find that

$$0\le \underset{n\to \mathrm{\infty }}{lim}\mu (|z_n|)\frac{|\varphi (z_n){({\phi }^{\prime }(z_n))}^2||\phi (z_n)|}{1-{|\phi (z_n)|}^2}\le \underset{n\to \mathrm{\infty }}{lim}{\parallel \varphi D{C}_{\phi }{f}_n\parallel }_{B_{\mu }}=0.$$

Therefore

$$\underset{n\to \mathrm{\infty }}{lim}\mu (|z_n|)\frac{|\varphi (z_n){({\phi }^{\prime }(z_n))}^2|}{1-{|\phi (z_n)|}^2}=\underset{n\to \mathrm{\infty }}{lim}\mu (|z_n|)\frac{|\varphi (z_n){({\phi }^{\prime }(z_n))}^2\phi (z_n)|}{1-{|\phi (z_n)|}^2}=0,$$

which implies (2.16). From this and (2.19), we have

$$\underset{n\to \mathrm{\infty }}{lim}\mu (|z_n|)|\varphi (z_n){\phi }^{″}(z_n)+{\varphi }^{\prime }(z_n){\phi }^{\prime }(z_n)|ln\frac{1}{1-{|\phi (z_n)|}^2}=0.$$

(2.21)

From (2.21), it follows that ${lim}_{n\to \mathrm{\infty }}\mu (|z_n|)|\varphi (z_n){\phi }^{″}(z_n)+{\varphi }^{\prime }(z_n){\phi }^{\prime }(z_n)|=0$, which altogether imply (2.17).

• (iii) ⇒ (i). Suppose that $\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu }$ is bounded and conditions (2.16) and (2.17) hold. From Theorem 2.3, it follows that

  $$\begin{array}{r}{C}_1=\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|<\mathrm{\infty },\\ C_2=\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2|<\mathrm{\infty }.\end{array}$$

  (2.22)

By the assumption, for every $\epsilon >0$, there is a $\delta \in (0,1)$, such that

$$\begin{array}{r}\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}<\epsilon \text{and}\\ \mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}<\epsilon ,\end{array}$$

(2.23)

whenever $\delta <|\phi (z)|<1$.

Assume that ${(f_k)}_{k\in N}$ is a sequence in such that${sup}_{k\in N}{\parallel f_k\parallel }_{\mathbb{Z}}\le L$ and $f_k$ converges to 0 uniformly on compact subsets of Δas $k\to \mathrm{\infty }$. Let $K=\{z\in \mathrm{\Delta }:|\phi (z)|\le \delta \}$. Then by (1.4), (2.22), and (2.23), we have

$$\begin{array}{r}\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|{(\varphi D{C}_{\phi }{f}_k)}^{\prime }(z)|\\ =\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|{\varphi }^{\prime }(z){\phi }^{\prime }(z)f_k^{\prime }(\phi (z))+\varphi (z)[f_k^{″}(\phi (z)){({\phi }^{\prime }(z))}^2+f_k^{\prime }(\phi (z)){\phi }^{″}(z)]|\\ \le \underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2{f}^{″}(\phi (z))|+\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)\left|[\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)]f^{\prime }(\phi (z))\right|\\ \le \underset{z\in K}{sup}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2{f}^{″}(\phi (z))|+\underset{z\in K}{sup}\mu (|z|)\left|[\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)]f^{\prime }(\phi (z))\right|\\ +\underset{z\in \mathrm{\Delta }\setminus K}{sup}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2{f}^{″}(\phi (z))|\\ +\underset{z\in \mathrm{\Delta }\setminus K}{sup}\mu (|z|)\left|[\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)]f^{\prime }(\phi (z))\right|\\ \le \underset{z\in K}{sup}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2{f}^{″}(\phi (z))|+\underset{z\in K}{sup}\mu (|z|)\left|[\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)]f^{\prime }(\phi (z))\right|\\ +\underset{z\in \mathrm{\Delta }\setminus K}{sup}\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}{\parallel f_k\parallel }_{\mathcal{Z}}\\ +C\underset{z\in \mathrm{\Delta }\setminus K}{sup}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}{\parallel f_k\parallel }_{\mathcal{Z}}\\ \le C_2\underset{|\omega |\le \delta }{sup}|f_k^{″}(\omega )|+C_1\underset{|\omega |\le \delta }{sup}|f_k^{\prime }(\omega )|+(C+1)\epsilon {\parallel f_k\parallel }_{\mathcal{Z}},\end{array}$$

i.e. we obtain

$$\begin{array}{rl}{\parallel \varphi D{C}_{\phi }{f}_k\parallel }_{B_{\mu }}\le & C_2\underset{|\omega |\le \delta }{sup}|f_k^{″}(\omega )|+C_1\underset{|\omega |\le \delta }{sup}|f_k^{\prime }(\omega )|\\ +(C+1)\epsilon {\parallel f_k\parallel }_{\mathcal{Z}}+|\varphi (0)|\left|f_k^{\prime }(\phi (0))\right||{\phi }^{\prime }(0)|.\end{array}$$

(2.24)

Since $f_k$ converges to 0 uniformly on compact subsets of Δas $k\to \mathrm{\infty }$, from Cauchy’s estimate, it follows that$f_k^{\prime }\to 0$ and $f_k^{″}\to 0$ as $k\to \mathrm{\infty }$ on compact subsets of Δ. Hence, letting$k\to \mathrm{\infty }$ in (2.24), and using the fact that ε isan arbitrary positive number, we obtain

$$\underset{k\to \mathrm{\infty }}{lim}{\parallel \varphi D{C}_{\phi }{f}_k\parallel }_{B_{\mu }}=0.$$

By combining this with Lemma 2.1 the result easily follows. The proof ofTheorem 2.4 is completed. □

Theorem 2.5 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then$\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu ,0}$is bounded if and only if$\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$is bounded and

$$\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2|=0\mathit{\text{and}}\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|=0.$$

(2.25)

Proof of Theorem 2.5 Assume that $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu ,0}$ is bounded. Then, it is clear that$\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is bounded. Taking the test functions$f(z)=z$ and $f(z)=z^2$ respectively, we obtain (2.25).

Conversely, assume that $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is bounded and (2.25) holds. Then for each polynomialp, we have

$$\begin{array}{r}\mu (|z|)|{(\varphi D{C}_{\phi }p)}^{\prime }(z)|\\ \le \mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2{p}^{″}(\phi (z))|+\mu (|z|)\left|[\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)]p^{\prime }(\phi (z))\right|.\end{array}$$

(2.26)

In view of the facts

$$\underset{\omega \in \mathrm{\Delta }}{sup}|p^{″}(\omega )|<\mathrm{\infty },\underset{\omega \in \mathrm{\Delta }}{sup}|p^{\prime }(\omega )|<\mathrm{\infty },$$

from (2.25) and (2.26), it follows that $\varphi D{C}_{\phi }p\in B_{\mu ,0}$. Since the set of all polynomials is dense in${\mathcal{Z}}_0$ (see [23]), it follows that for every $f\in {\mathcal{Z}}_0$, there is a sequence of polynomials${(p_n)}_{n\in N}$ such that ${\parallel f-p_n\parallel }_{\mathcal{Z}}\to 0$ as $n\to \mathrm{\infty }$. Hence

$${\parallel \varphi D{C}_{\phi }f-\varphi D{C}_{\phi }{p}_n\parallel }_{B_{\mu }}\le {\parallel \varphi D{C}_{\phi }\parallel }_{{\mathcal{Z}}_0\to B_{\mu }}{\parallel f-p_n\parallel }_{\mathcal{Z}}\to 0$$

as $n\to \mathrm{\infty }$. Since the operator $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is bounded, we have $\varphi D{C}_{\phi }({\mathcal{Z}}_0)\subseteq B_{\mu ,0}$, which implies the boundedness of$\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu ,0}$. □

Theorem 2.6 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.

• (i) $\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu ,0}$ is compact;

• (ii) $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu ,0}$ is compact;

• (iii)
  $$\underset{|z|\to 1}{lim}\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}=0$$

  (2.27)

and

$$\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}=0.$$

(2.28)

Proof of Theorem 2.6 (i) ⇒ (ii). This implication is trivial.

• (ii) ⇒ (iii). Assume that $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu ,0}$ is compact. Then $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu ,0}$ is bounded. From the proof of Theorem 2.5, we have

  $$\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|=0$$

  (2.29)

and

$$\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2|=0.$$

(2.30)

Hence, if ${\parallel \phi \parallel }_{\mathrm{\infty }}<1$, from (2.29) and (2.30), we obtain

$$\underset{|z|\to 1}{lim}\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}\le \frac{1}{1-{\parallel \phi \parallel }_{\mathrm{\infty }}^2}\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2|=0$$

and

$$\begin{array}{r}\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}\\ \le ln\frac{e}{1-{\parallel \phi \parallel }_{\mathrm{\infty }}^2}\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|=0,\end{array}$$

from which the result follows in this case.

Now assume that ${\parallel \phi \parallel }_{\mathrm{\infty }}=1$. Let ${(z_k)}_{k\in N}$ be a sequence such that $|\phi (z_k)|\to 1$ as $k\to \mathrm{\infty }$. Since $\varphi D{C}_{\phi }:{\mathcal{Z}}_0\to B_{\mu }$ is compact, by Theorem 2.4, we have

$$\underset{|\phi (z)|\to 1}{lim}\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}=0$$

(2.31)

and

$$\underset{|\phi (z)|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}=0.$$

(2.32)

From (2.30) and (2.31), it follows that for every $\epsilon >0$, there exists an $r\in (0,1)$ such that $\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}<\epsilon$, when $r<|\phi (z)|<1$, and there exists a $\sigma \in (0,1)$ such that $\mu (|z|)|\varphi (z){({\phi }^{\prime }(z))}^2|\le \epsilon (1-r^2)$, when $\sigma <|z|<1$. Therefore, when $\sigma <|z|<1$ and $r<|\phi (z)|<1$, we have

$$\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}<\epsilon .$$

(2.33)

On the other hand, if $\sigma <|z|<1$ and $|\phi (z)|\le r$, we obtain

$$\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}<\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-r^2}<\epsilon .$$

(2.34)

Inequality (2.33) together with (2.34) gives the (2.27). Similarly, (2.29) and(2.32) imply (2.28).

• (iii) ⇒ (i). Let $f\in \mathcal{Z}$. Then we have

  $$\begin{array}{r}\mu (|z|)|{(\varphi D{C}_{\phi }f)}^{\prime }(z)|\\ \le C[\mu (|z|)\frac{|\varphi (z){({\phi }^{\prime }(z))}^2|}{1-{|\phi (z)|}^2}+\mu (|z|)|\varphi (z){\phi }^{″}(z)+{\varphi }^{\prime }(z){\phi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}]{\parallel f\parallel }_{\mathcal{Z}}.\end{array}$$

Taking the supremum in this inequality over all $f\in \mathcal{Z}$ such that ${\parallel f\parallel }_{\mathcal{Z}}\le 1$, then letting $|z|\to 1$, and using (2.27) and (2.28), we obtain

$$\underset{|z|\to 1}{lim}\underset{{\parallel f\parallel }_{\mathcal{Z}}\le 1}{sup}\mu (|z|)|{(\varphi D{C}_{\phi }f)}^{\prime }(z)|=0.$$

From Lemma 2.2 it follows that the operator $\varphi D{C}_{\phi }:\mathcal{Z}\to B_{\mu ,0}$ is compact. □

Similarly to the proofs of Theorems 2.3-2.6, we can get the following results; weomit the proof.

Theorem 2.7 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.

• (i) $\varphi C_{\phi }D:\mathcal{Z}\to B_{\mu }$ is bounded;

• (ii) $\varphi C_{\phi }D:{\mathcal{Z}}_0\to B_{\mu }$ is bounded;

• (iii)
  $$\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)\frac{|\varphi (z){\phi }^{\prime }(z)|}{1-{|\phi (z)|}^2}<\mathrm{\infty }$$

and

$$\underset{z\in \mathrm{\Delta }}{sup}\mu (|z|)|{\varphi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}<\mathrm{\infty }.$$

Theorem 2.8 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.

• (i) $\varphi C_{\phi }D:\mathcal{Z}\to B_{\mu }$ is compact;

• (ii) $\varphi C_{\phi }D:{\mathcal{Z}}_0\to B_{\mu }$ is compact;

• (iii) $\varphi C_{\phi }D:\mathcal{Z}\to B_{\mu }$ is bounded,

  $$\underset{|\phi (z)|\to 1}{lim}\mu (|z|)\frac{|\varphi (z){\phi }^{\prime }(z)|}{1-{|\phi (z)|}^2}=0$$

and

$$\underset{|\phi (z)|\to 1}{lim}\mu (|z|)|{\varphi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}=0.$$

Theorem 2.9 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then$\varphi C_{\phi }D:{\mathcal{Z}}_0\to B_{\mu ,0}$is bounded if and only if$\varphi C_{\phi }D:{\mathcal{Z}}_0\to B_{\mu }$is bounded and

$$\underset{|z|\to 1}{lim}\mu (|z|)|\varphi (z){\phi }^{\prime }(z)|=0\mathit{\text{and}}\underset{|z|\to 1}{lim}\mu (|z|)|{\varphi }^{\prime }(z)|=0.$$

Theorem 2.10 Let φ be an analytic self-map of Δ, and let ϕ be an analytic function in Δ. Suppose that μ is normal. Then the following statements are equivalent.

• (i) $\varphi C_{\phi }D:\mathcal{Z}\to B_{\mu ,0}$ is compact;

• (ii) $\varphi C_{\phi }D:{\mathcal{Z}}_0\to B_{\mu ,0}$ is compact;

• (iii)
  $$\underset{|z|\to 1}{lim}\mu (|z|)\frac{|\varphi (z){\phi }^{\prime }(z)|}{1-{|\phi (z)|}^2}=0$$

and

$$\underset{|z|\to 1}{lim}\mu (|z|)|{\varphi }^{\prime }(z)|ln\frac{e}{1-{|\phi (z)|}^2}=0.$$