A Linear Composition Operator on the Bloch Space

Abstract

Let $n\in {\mathbb{N}}_0$, $\psi$ be an analytic self-map on $\mathbb{D}$ and u be an analytic function on $\mathbb{D}$. The single operator $D_{u,\psi }^n$ acting on various spaces of analytic functions has been a subject of investigation for many years. It is defined as $(D_{u,\psi }^{n}f)(z)=u(z)f^{(n)}(\psi (z)),f\in H(\mathbb{D})$. However, the study of the operator $P_{\overrightarrow{u},\psi }^k$, which represents a finite sum of these operators with varying orders, remains incomplete. The boundedness, compactness and essential norm of the operator $P_{\overrightarrow{u},\psi }^k$ on the Bloch space are investigated in this paper, and several characterizations for these properties are provided.

1. Introduction

The goal of this paper is to study the boundedness, compactness and essential norm of a linear composition operator on the Bloch space. Let $\mathbb{D}$ be the unit disk within the complex plane $\mathbb{C}$, and let $H(\mathbb{D})$ denote the set of functions that are analytic on $\mathbb{D}$. We conventionally refer to the collection of all self-maps that are analytic on $\mathbb{D}$ as $S(\mathbb{D})$. For a given $u\in H(\mathbb{D})$ and $\psi \in S(\mathbb{D})$, the weighted composition operator $u{C}_{\psi }$ is defined by

$$(u{C}_{\psi }f)(z)=u(z)f(\psi (z)),f\in H(\mathbb{D}).$$

The operator $u{C}_{\psi }$ is the composition operator $C_{\psi }$ when $u=1$. The primary focus in the field of composition operators is to correlate the operator-theoretic characteristics of $C_{\psi }$ with the function-theoretic properties of $\psi$. For an in-depth examination of various properties of composition operators, one can referee [1,2] and the references therein.

Furthermore, when $u\in H(\mathbb{D})$, $\psi \in S(\mathbb{D})$ and $n\in {\mathbb{N}}_0$, the generalized weighted composition operator, also recognized as the weighted differentiation composition operator and denoted by $D_{u,\psi }^n$, is defined by

$$(D_{u,\psi }^{n}f)(z)=u(z)f^{(n)}(\psi (z)),f\in H(\mathbb{D}).$$

In the special case where $n=0$, $D_{u,\psi }^n$ simplifies to the weighted composition operator. The operator $D_{u,\psi }^n$ was introduced by Zhu in [3]. For a comprehensive understanding of the generalized weighted composition operator on some analytic function spaces, one can referee [3,4,5,6,7] and the references therein.

Let $n,k\in {\mathbb{N}}_0$, $\psi \in S(\mathbb{D})$, and $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$. After the publication of [8], Stević suggested his colleagues to study the following operator:

$$T_{\overrightarrow{u},\psi }^{k,n}f=\sum _{j=0}^k{u}_j·f^{(n+j)}\circ \psi =\sum _{j=0}^k{D}_{u_j,\psi }^{n+j}f,f\in H(\mathbb{D}),$$

along with some related operators on ${\mathbb{C}}^n$. Some results in this direction are presented in [9]. When $n=0$, we denote $T_{\overrightarrow{u},\psi }^{k,0}$ by $P_{\overrightarrow{u},\psi }^k$, which has been recently studied in [10], and a particular case was also investigated in [11]. In particular, when $k=0$, $P_{\overrightarrow{u},\psi }^k$ is just the weighted composition operator $u_0{C}_{\psi }$ and the $T_{\overrightarrow{u},\psi }^{0,n}$ is the operator $D_{u_0,\psi }^n$.

A function $f\in H(\mathbb{D})$ is said to be in the Bloch space, denoted by $\mathcal{B}$, if it satisfies

$${\parallel f\parallel }_{\beta }=\underset{z\in \mathbb{D}}{\sup }\left(1-{|z|}^2\right)|f^{\prime }(z)|<\infty .$$

It is a well-established fact that $\mathcal{B}$ is a Banach space when equipped with the norm ${\parallel f\parallel }_{\mathcal{B}}=|f(0)|+{\parallel f\parallel }_{\beta }$. We say that an $f\in H(\mathbb{D})$ belongs to the little Bloch space, denoted by ${\mathcal{B}}_0$, if ${\lim }_{|z|\to 1}|f^{\prime }(z)|(1-{|z|}^2)=0.$ It is well known that $C_{\psi }:\mathcal{B}\to \mathcal{B}$ is bounded by Schwarz–Pick Lemma for any $\psi \in S(\mathbb{D})$. The compactness for $C_{\psi }:\mathcal{B}\to \mathcal{B}$ was initially studied in [12], which showed that $C_{\psi }:\mathcal{B}\to \mathcal{B}$ is compact if and only if

$$\underset{s\to 1}{\lim }\underset{|\psi (z)|>s}{\sup }{\displaystyle \frac{1-{|z|}^2}{1-{|\psi (z)|}^2}}|{\psi }^{\prime }(z)|=0.$$

Tjani [13] proved that $C_{\psi }:\mathcal{B}\to \mathcal{B}$ is compact if and only if ${\lim }_{|a|\to 1}{\parallel C_{\psi }{\sigma }_a\parallel }_{\mathcal{B}}=0$, where ${\sigma }_a(z)={\displaystyle \frac{a-z}{1-\overline{a}z}}$. Additionally, Wulan et al. [14] presented a new characterization for the compactness of $C_{\psi }:\mathcal{B}\to \mathcal{B}$, showing that $C_{\psi }:\mathcal{B}\to \mathcal{B}$ is compact if and only if ${\lim }_{n\to \infty }{\parallel {\psi }^n\parallel }_{\mathcal{B}}=0.$ The essential norm of a bounded linear operator is a significant concept in functional analysis, as it provides a measure of the “size" of the operator beyond the behavior of compact operators. In the context of the composition operator acting on the Bloch space, several researchers have made contributions to understand its essential norm. Montes-Rodríguez and Zhao gave an exact essential norm of the composition operator $C_{\psi }:\mathcal{B}\to \mathcal{B}$ in [15] and [16], respectively. For further insights into composition operators $C_{\psi }:\mathcal{B}\to \mathcal{B}$, one may refer to the literature [2,12,15,16,17,18]. See [19,20,21,22,23,24] for the study of weighted composition operators $u{C}_{\psi }:\mathcal{B}\to \mathcal{B}$. Furthermore, the study of generalized weighted composition operators on the Bloch space has been explored in [6,7].

Motivated by the above-mentioned works, by combining the methods of [6,7,9], the aim of this article is to study the operator $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$. Several characterizations for the boundedness, compactness and essential norm for the operator $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ are provided. Our results generalized many results in the literature.

Throughout this paper, we say that $A\lesssim B$ if there exists a constant C such that $A\le CB$. The symbol $A\approx B$ means that $A\lesssim B\lesssim A$.

2. Boundedness of ${\mathit{P}}_{\overrightarrow{\mathit{u}},\mathit{\psi }}^{\mathit{k}}:\mathcal{B}\to \mathcal{B}$

Lemma 1 

([2]). Let n be a positive integer and $f\in \mathcal{B}$. Then the following inequalities hold.

(i) 

$|f(z)|\le {\parallel f\parallel }_{\mathcal{B}}\log {\displaystyle \frac{e}{1-{|z|}^2}}$, $z\in \mathbb{D}.$

(ii) 

$|f^{(n)}(z)|\lesssim {\displaystyle \frac{{\parallel f\parallel }_{\mathcal{B}}}{{(1-|z|}^2{)}^n}}$, $z\in \mathbb{D}.$

Throughout this paper, we always assume that $u_{k+1}=0$ for the simplicity of the notation.

Theorem 1. 

Let $k\in {\mathbb{N}}_0$, $\psi \in S(\mathbb{D})$ and $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$. Then, the following statements are equivalent:

(a) 

The operator $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded.

(b) 

${\sup }_{a\in \mathbb{D}}{\parallel P_{\overrightarrow{u},\psi }^k{f}_{t,a}\parallel }_{\mathcal{B}}<\infty ,$ for each $t=0,1,\cdots ,k+1.$ Here,

$$\begin{array}{c}\hfill f_{0,a}(z)={\left(\log {\displaystyle \frac{e}{1-\overline{a}z}}\right)}^2{(\log {\displaystyle \frac{e}{1-{|a|}^2}})}^{-1},z\in \mathbb{D},\end{array}$$

and

$$\begin{array}{c}\hfill f_{t,a}(z)={\displaystyle \frac{1-{|a|}^2}{1-\overline{a}z}}{\sigma }_a^t(z)={\displaystyle \frac{1-{|a|}^2}{1-\overline{a}z}}{\left({\displaystyle \frac{a-z}{1-\overline{a}z}}\right)}^t,z\in \mathbb{D},\mathit{for}t=1,2,\cdots ,k+1.\end{array}$$

(c) 

$$\begin{array}{ccc}& & M_0=\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)|\log {\displaystyle \frac{e}{1-{|\psi (z)|}^2}}<\infty ;\hfill \\ & & M_t=\underset{z\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_{t-1}(z){\psi }^{\prime }(z)+u_t^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^t}}<\infty ,\mathit{for}t=1,2,\cdots ,k+1.\hfill \end{array}$$

Proof. 

$(a)\Rightarrow (b)$ Assume that $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded. For each $a\in \mathbb{D}$, it is easy to check that $f_{t,a}\in \mathcal{B}$ for all $t=0,1,\cdots ,k+1$. Moreover ${\sup }_{a\in \mathbb{D}}{\parallel f_{t,a}\parallel }_{\mathcal{B}}<\infty$ for all $t=0,1,\cdots ,k+1$. By the boundedness of $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$, we obtain

$$\underset{a\in \mathbb{D}}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{t,a}{\parallel }_{\mathcal{B}}\le \parallel P_{\overrightarrow{u},\psi }^k\parallel \underset{a\in \mathbb{D}}{\sup }\parallel f_{t,a}{\parallel }_{\mathcal{B}}\le C\parallel P_{\overrightarrow{u},\psi }^k\parallel <\infty ,$$

for all $t=0,1,\cdots ,k+1$, as desired.

$(b)\Rightarrow (c)$ Assume that $(b)$ holds. From the assumption we see that

$$\begin{array}{c}\hfill \underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{t,\psi (a)}\parallel }_{\mathcal{B}}<\infty ,\end{array}$$

(1)

for all $t=0,1,\cdots ,k+1$. For any $a\in \mathbb{D}$, it is easy to check that $f_{k+1,\psi (a)}\in \mathcal{B}$, $\parallel f_{k+1,\psi (a)}{\parallel }_{\mathcal{B}}<\infty$, $f_{k+1,\psi (a)}^{(i)}(\psi (a))=0$ for all $i=0,1,\cdots ,k$ and

$$\left|f_{k+1,\psi (a)}^{(k+1)}(\psi (a))\right|={\displaystyle \frac{(k+1)!}{{(1-|\psi (a)|}^2{)}^{k+1}}}.$$

Thus,

$$\begin{array}{ccc}& & \parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,\psi (a)}{\parallel }_{\mathcal{B}}\ge {(1-|a|}^2)|{(P_{\overrightarrow{u},\psi }^k{f}_{k+1,\psi (a)})}^{\prime }(a)|\hfill \\ & =& {(1-|a|}^2)|u_k(a){\psi }^{\prime }(a)||f_{k+1,\psi (a)}^{(k+1)}(\psi (a))|={\displaystyle \frac{{(1-|a|}^2)|u_k(a){\psi }^{\prime }(a)|(k+1)!}{{(1-|\psi (a)|}^2{)}^{k+1}}}.\hfill \end{array}$$

(2)

Therefore, by (1) we have

$$\begin{array}{c}\hfill M_{k+1}=\underset{a\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|a|}^2)|u_k(a){\psi }^{\prime }(a)|}{{(1-|\psi (a)|}^2{)}^{k+1}}}\le {\displaystyle \frac{1}{(k+1)!}}\underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,\psi (a)}\parallel }_{\mathcal{B}}<\infty .\end{array}$$

(3)

For any $a\in \mathbb{D}$, it is easy to check that $f_{k,\psi (a)}\in \mathcal{B},$$\parallel f_{k,\psi (a)}{\parallel }_{\mathcal{B}}<\infty$, $f_{k,\psi (a)}^{(i)}(\psi (a))=0\mathrm{for} \mathrm{all}i=0,1,\cdots ,k-1$ and

$$\begin{array}{c}\hfill \left|f_{k,\psi (a)}^{(k)}(\psi (a))\right|={\displaystyle \frac{k!}{{(1-|\psi (a)|}^2{)}^k}}.\end{array}$$

(4)

Using Lemma 1 and (4), we obtain

$$\begin{array}{ccc}& & \parallel P_{\overrightarrow{u},\psi }^k{f}_{k,\psi (a)}{\parallel }_{\mathcal{B}}\ge {(1-|a|}^2)|{(P_{\overrightarrow{u},\psi }^k{f}_{k,\psi (a)})}^{\prime }(a)|\hfill \\ & \ge & {(1-|a|}^2)|u_k^{\prime }(a)+u_{k-1}(a){\psi }^{\prime }(a)||f_{k,\psi (a)}^{(k)}{(\psi (a))|-(1-|a|}^2)|u_k(a){\psi }^{\prime }(a)||f_{k,\psi (a)}^{(k+1)}(\psi (a))|\hfill \\ & \gtrsim & {\displaystyle \frac{{(1-|a|}^2)|u_k^{\prime }(a)+u_{k-1}(a){\psi }^{\prime }(a)|k!}{{(1-|\psi (a)|}^2{)}^k}}-{\displaystyle \frac{\parallel f_{k,\psi (a)}{\parallel }_{\mathcal{B}}{(1-|a|}^2)|u_k(a){\psi }^{\prime }(a)|}{{(1-|\psi (a)|}^2{)}^{k+1}}}.\hfill \end{array}$$

(5)

Thus, using (1), (3) and (5), we have

$$\begin{array}{ccc}\hfill M_k& =& \underset{a\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|a|}^2)|u_k^{\prime }(a)+u_{k-1}(a){\psi }^{\prime }(a)|}{{(1-|\psi (a)|}^2{)}^k}}\hfill \\ & \lesssim & \underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k,\psi (a)}\parallel }_{\mathcal{B}}+\underset{a\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|a|}^2)|u_k(a){\psi }^{\prime }(a)|}{{(1-|\psi (a)|}^2{)}^{k+1}}}\hfill \\ & \lesssim & \underset{a\in \mathbb{D}}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{k,\psi (a)}{\parallel }_{\mathcal{B}}+\underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,\psi (a)}\parallel }_{\mathcal{B}}<\infty .\hfill \end{array}$$

(6)

This proves the case $t=k$. Now, we fix $1\le t\le k-1$ and assume that

$$\begin{array}{c}\hfill M_i<\infty ,\end{array}$$

(7)

for all $i=t+1,\cdots ,k$. We will prove that $M_t<\infty .$ For any $a\in \mathbb{D}$, it is easy to check that $f_{t,\psi (a)}\in \mathcal{B}$, $\parallel f_{t,\psi (a)}{\parallel }_{\mathcal{B}}<\infty$, $f_{t,\psi (a)}^{(s)}(\psi (a))=0\mathrm{for} \mathrm{all}s<t$ and

$$\begin{array}{c}\hfill \left|f_{t,\psi (a)}^{(t)}(\psi (a))\right|={\displaystyle \frac{t!}{{(1-|\psi (a)|}^2{)}^t}}.\end{array}$$

(8)

Using Lemma 1 and (8), we have

$$\begin{array}{ccc}& & \parallel P_{\overrightarrow{u},\psi }^k{f}_{t,\psi (a)}{\parallel }_{\mathcal{B}}\ge {(1-|a|}^2)|{(P_{\overrightarrow{u},\psi }^k{f}_{t,\psi (a)})}^{\prime }(a)|\hfill \\ & \ge & {(1-|a|}^2)\left|u_t^{\prime }(a)+u_{t-1}(a){\psi }^{\prime }(a)\right|\left|f_{t,\psi (a)}^{(t)}(\psi (a))\right|\hfill \\ & & -\sum _{i=t+1}^{k+1}{(1-|a|}^2)\left|u_i^{\prime }(a)+u_{i-1}(a){\psi }^{\prime }(a)\right|\left|f_{t,\psi (a)}^{(i)}(\psi (a))\right|\hfill \\ & \gtrsim & {\displaystyle \frac{{(1-|a|}^2)\left|u_t^{\prime }(a)+u_{t-1}(a){\psi }^{\prime }(a)\right|t!}{{(1-|\psi (a)|}^2{)}^t}}\hfill \\ & & -\sum _{i=t+1}^{k+1}{\displaystyle \frac{\parallel f_{t,\psi (a)}{\parallel }_{\mathcal{B}}{(1-|a|}^2)\left|u_i^{\prime }(a)+u_{i-1}(a){\psi }^{\prime }(a)\right|}{{(1-|\psi (a)|}^2{)}^i}}.\hfill \end{array}$$

(9)

Thus, by (1), (3), (6) and (9), we have

$$\begin{array}{ccc}& & M_t=\underset{a\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|a|}^2)|u_t^{\prime }(a)+u_{t-1}(a){\psi }^{\prime }(a)|}{{(1-|\psi (a)|}^2{)}^t}}\hfill \\ & \lesssim & \underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{t,\psi (a)}\parallel }_{\mathcal{B}}+\sum _{i=t+1}^{k+1}\underset{a\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|a|}^2)\left|u_i^{\prime }(a)+u_{i-1}(a){\psi }^{\prime }(a)\right|}{{(1-|\psi (a)|}^2{)}^i}}\hfill \\ & \lesssim & \sum _{i=t}^{k+1}\underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,\psi (a)}\parallel }_{\mathcal{B}}<\infty ,\hfill \end{array}$$

(10)

for every $t=1,2,\cdots ,k$.

Next, we prove that $M_0<\infty$. For any $a\in \mathbb{D}$, we see that $\parallel f_{0,\psi (a)}{\parallel }_{\mathcal{B}}<\infty$ and

$$\begin{array}{c}\hfill |f_{0,\psi (a)}(\psi (a))|=\log {\displaystyle \frac{e}{1-{|\psi (a)|}^2}}.\end{array}$$

(11)

Using Lemma 1 and (11), we have

$$\begin{array}{ccc}\hfill \parallel P_{\overrightarrow{u},\psi }^k{f}_{0,\psi (a)}{\parallel }_{\mathcal{B}}& \ge & {(1-|a|}^2)|{(P_{\overrightarrow{u},\psi }^k{f}_{0,\psi (a)})}^{\prime }(a)|\hfill \\ & \gtrsim & {(1-|a|}^2)|u_0^{\prime }(a)|\log {\displaystyle \frac{e}{1-{|\psi (a)|}^2}}\hfill \\ & & -\sum _{t=1}^{k+1}{\displaystyle \frac{\parallel f_{0,\psi (a)}{\parallel }_{\mathcal{B}}{(1-|a|}^2)|u_t^{\prime }(a)+u_{t-1}(a){\psi }^{\prime }(a)|}{{(1-|\psi (a)|}^2{)}^t}}.\hfill \end{array}$$

(12)

Thus, using (1), (3), (10) and (12), we have

$$\begin{array}{ccc}& & M_0=\underset{a\in \mathbb{D}}{\sup }{(1-|a|}^2)|u_0^{\prime }(a)|\log {\displaystyle \frac{e}{1-{|\psi (a)|}^2}}\hfill \\ & \lesssim & \underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{0,\psi (a)}\parallel }_{\mathcal{B}}+\sum _{t=1}^{k+1}\underset{a\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|a|}^2)|u_t^{\prime }(a)+u_{t-1}(a){\psi }^{\prime }(a)|}{{(1-|\psi (a)|}^2{)}^t}}\hfill \\ & \lesssim & \underset{a\in \mathbb{D}}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{0,\psi (a)}{\parallel }_{\mathcal{B}}+\sum _{t=1}^{k+1}\underset{a\in \mathbb{D}}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{t,\psi (a)}\parallel }_{\mathcal{B}}<\infty ,\hfill \end{array}$$

as desired.

$(c)\Rightarrow (a)$ Suppose that $(c)$ holds. Let $f\in \mathcal{B}$. By Lemma 1,

$$\begin{array}{ccc}\hfill \parallel P_{\overrightarrow{u},\psi }^k{f\parallel }_{\mathcal{B}}& =& |P_{\overrightarrow{u},\psi }^{k}f(0)|+\parallel P_{\overrightarrow{u},\psi }^k{f\parallel }_{\beta }=|\sum _{t=0}^k{u}_t(0)f^{(t)}(\psi (0))|\hfill \\ & & +\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|\sum _{t=0}^k(u_t^{\prime }(z)f^{(t)}(\psi (z))+u_t(z){\psi }^{\prime }(z)f^{(t+1)}(\psi (z)))|\hfill \\ & \le & \sum _{t=0}^k|u_t(0)||f^{(t)}(\psi (0))|+\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)||f(\psi (z))|\hfill \\ & & +\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)\sum _{t=1}^{k+1}|u_t^{\prime }(z)+u_{t-1}(z){\psi }^{\prime }(z)||f^{(t)}(\psi (z))|\hfill \\ & \lesssim & {\parallel f\parallel }_{\mathcal{B}}(|u_0(0)|\log {\displaystyle \frac{e}{1-{|\psi (0)|}^2}}+\sum _{t=1}^k{\displaystyle \frac{|u_t(0)|}{{(1-|\psi (0)|}^2{)}^t}}\hfill \\ & & +\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)|\log {\displaystyle \frac{e}{1-{|\psi (z)|}^2}}+\sum _{t=1}^{k+1}\underset{z\in \mathbb{D}}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_{t-1}(z){\psi }^{\prime }(z)+u_t^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^t}})\hfill \\ & \lesssim & {\parallel f\parallel }_{\mathcal{B}}(C+\sum _{t=0}^{k+1}{M}_t)<\infty .\hfill \end{array}$$

(13)

So $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded. Thus $(a)$ holds. □

Next, we give another characterization of the boundedness of $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$. For this purpose, we state some definitions and some lemmas which will be used.

Let $v:\mathbb{D}\to R_{+}$ be a radial weight, i.e., $v(z)=v(|z|)$ for all $z\in \mathbb{D}$. Set

$$H_v^{\infty }=\{f\in {H(\mathbb{D}):\parallel f\parallel }_v=\underset{z\in \mathbb{D}}{\sup }v(z)|f(z)|<\infty \}.$$

$H_v^{\infty }$ is called the weighted space. We denote $H_v^{\infty }$ by $H_{v_{\alpha }}^{\infty }$ when $v=v_{\alpha }(z)={(1-|z|}^2{)}^{\alpha }(0<\alpha <\infty )$. For a radial weight v, let

$$\tilde{v}=(\sup \{|f(z)|:f\in H_v^{\infty }{,\parallel f\parallel }_v\le 1{\})}^{-1},z\in \mathbb{D}.$$

By [20], we see that ${\tilde{v}}_{\alpha }(z)=v_{\alpha }(z)$ and ${\tilde{v}}_{\log }=v_{\log }$ when $v=v_{\log }(z)={\left(\log {\displaystyle \frac{e}{1-{|z|}^2}}\right)}^{-1}.$

Lemma 2 

([25]). Let v and w be radial, non-increasing weights tending to zero at the boundary of $\mathbb{D}$. Then $u{C}_{\psi }:H_v^{\infty }\to H_w^{\infty }$ is bounded if and only if

$$\underset{n\ge 0}{\sup }{\displaystyle \frac{\parallel u{\psi }^n{\parallel }_w}{\parallel {\eta }^n{\parallel }_v}}<\infty .$$

Moreover,

$$\begin{array}{c}\hfill \underset{n\ge 0}{\sup }{\displaystyle \frac{\parallel u{\psi }^n{\parallel }_w}{\parallel {\eta }^n{\parallel }_v}}\approx \underset{z\in \mathbb{D}}{\sup }{\displaystyle \frac{w(z)}{\tilde{v}(\psi (z))}}|u(z)|=\parallel u{C}_{\psi }{\parallel }_{H_v^{\infty }\to H_w^{\infty }}.\end{array}$$

Theorem 2. 

Let $k\in {\mathbb{N}}_0$, $\psi \in S(\mathbb{D})$ and $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$. Then $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded if and only if $u_0\in \mathcal{B}$,

$$\begin{array}{c}\hfill \underset{n\ge 1}{\sup }\log n{\parallel u_0^{\prime }{\psi }^n\parallel }_{v_1}<\infty ,\end{array}$$

and

$$\begin{array}{c}\hfill \underset{n\ge 1}{\sup }{n}^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1}<\infty ,\mathit{for}t=1,2,\cdots ,k+1.\end{array}$$

Proof. 

We have proved that $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded if and only if $(c)$ in Theorem 1 holds. By [26], we see that $M_0<\infty$ is equivalent to $u_0^{\prime }{C}_{\psi }:H_{v_{\log }}^{\infty }\to H_{v_1}^{\infty }$ is bounded. By Lemma 2, this is equivalent to

$$\begin{array}{c}\hfill \underset{n\ge 1}{\sup }{\displaystyle \frac{\parallel u_0^{\prime }{\psi }^{n-1}{\parallel }_{v_1}}{\parallel {\eta }^{n-1}{\parallel }_{v_{\log }}}}<\infty .\end{array}$$

(14)

By [26], $M_t<\infty$ is equivalent to $(u_{t-1}{\psi }^{\prime }+u_t^{\prime })C_{\psi }:H_{v_t}^{\infty }\to H_{v_1}^{\infty }$ is bounded, for $t=1,2,\cdots ,k+1$. By Lemma 2, this is equivalent to

$$\begin{array}{c}\hfill \underset{n\ge 1}{\sup }{\displaystyle \frac{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}{\parallel }_{v_1}}{\parallel {\eta }^{n-1}{\parallel }_{v_t}}}<\infty ,\mathrm{for}t=1,2,\cdots ,k+1.\end{array}$$

(15)

By [20],

$$\underset{n\to \infty }{\lim }{n}^{\alpha }\parallel {\eta }^{n-1}{\parallel }_{v_{\alpha }}={({\displaystyle \frac{2\alpha }{e}})}^{\alpha }\mathrm{and}\underset{n\to \infty }{\lim }\log n{\parallel {\eta }^n\parallel }_{v_{\log }}=1,$$

when $\alpha >0$. Therefore, $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded if and only if

$$\begin{array}{ccc}& & \max \{\parallel u_0{\parallel }_{\mathcal{B}},\underset{n\ge 1}{\sup }\log n{\parallel u_0^{\prime }{\psi }^n\parallel }_{v_1}\}\hfill \\ & =& \max \{\parallel u_0{\parallel }_{\mathcal{B}},\underset{n\ge 2}{\sup }\log (n-1){\parallel u_0^{\prime }{\psi }^{n-1}\parallel }_{v_1}\}\approx \underset{n\ge 1}{\sup }{\displaystyle \frac{\parallel u_0^{\prime }{\psi }^{n-1}{\parallel }_{v_1}}{\parallel {\eta }^{n-1}{\parallel }_{v_{\log }}}}<\infty ,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{n\ge 1}{\sup }{n}^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1}\approx \underset{n\ge 1}{\sup }{\displaystyle \frac{n^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1}}{n^t{\parallel {\eta }^{n-1}\parallel }_{v_t}}}<\infty ,\mathrm{for}t=1,2,\cdots ,k+1.\end{array}$$

The proof is complete. □

3. Essential Norm of ${\mathit{P}}_{\overrightarrow{\mathit{u}},\mathit{\psi }}^{\mathit{k}}:\mathcal{B}\to \mathcal{B}$

For $r\in (0,1)$, let $K_{r}f(z)=f(rz)$. Then $K_r$ is compact on $\mathcal{B}$ or ${\mathcal{B}}_0$ with $\parallel K_r{\parallel }_{\mathcal{B}}\le 1.$ The following lemma whose proof follows from Proposition 2.1 in [26], Lemma 4.2 in [16] and Cauchy’s integral formula.

Lemma 3. 

There is a sequence $\left\{r_k\right\}$, with $0<r_k<1$ tending to 1, such that the compact operator $L_n={\displaystyle \frac{1}{n}}{\sum }_{k=1}^n{K}_{r_k}$ acting on ${\mathcal{B}}_0$ satisfies:

(i) 

For any $t\in [0,1)$, $n\in \mathbb{N}$, ${\lim }_{n\to \infty }{\sup }_{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }_{|z|\le t}|{((I-L_n)f)}^{(n)}(z)|=0$.

(ii) 

$$\underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }\underset{|z|>s}{\sup }|(I-L_n)f(z)|{(\log {\displaystyle \frac{1}{1-{|z|}^2}})}^{-1}\le 1,$$

for s sufficiently close to 1, and

(iii) 

${\lim }_{n\to \infty }{\sup }_{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }_{|z|\le s}|(I-L_n)f(z)|=0$, for the above s.

(iv) 

${\lim \sup }_{n\to \infty }\parallel I-L_n\parallel \le 1$.

Furthermore, the same is true for the sequence of biadjoints $L_n^{**}$ (the same form as $L_n$ on ${\mathcal{B}}_0$) on $\mathcal{B}$.

Lemma 4. 

Let $k\in {\mathbb{N}}_0$, $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$, $\psi \in S(\mathbb{D})$ with ${\parallel \psi \parallel }_{\infty }<1$. If $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded, then $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is compact.

Proof. 

By the assumption that $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded and Theorem 1, we obtain

$$T_0:=\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)|<\infty ,$$

and

$$T_t:=\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_t^{\prime }(z)+u_{t-1}(z){\psi }^{\prime }(z)|<\infty ,\mathrm{for}t=1,2,\cdots ,k+1.$$

Let ${\left\{f_n\right\}}_{n\in \mathbb{N}}$ be bounded in $\mathcal{B}$ such that $f_n\to 0$ uniformly on compact subsets of $\mathbb{D}$ as $n\to \infty$. Cauchy’s estimate implies that $f_n^{(t)}\to 0$ uniformly on compact subsets of $\mathbb{D}$ as $n\to \infty$ for $t=0,1,\cdots ,k+1$. Thus, $|P_{\overrightarrow{u},\psi }^k{f}_n(0)|\to 0$ as $n\to \infty$ and for the compact subset $K=\{\psi (z):|\psi (z)|\le \parallel \psi {\parallel }_{\infty }\}\subseteq \mathbb{D}$, we have

$$\begin{array}{c}\hfill \underset{z\in \mathbb{D}}{\sup }|f_n^{(t)}(\psi (z))|\to 0\mathrm{as}n\to \infty .\end{array}$$

(16)

Using (16) and Lemma 1, we have

$$\begin{array}{ccc}& & \underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|\sum _{t=0}^k(u_t^{\prime }(z)f_n^{(t)}(\psi (z))+u_t(z){\psi }^{\prime }(z)f_n^{(t+1)}(\psi (z)))|\hfill \\ & \le & \underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)||f_n(\psi (z))|+\sum _{t=1}^{k+1}\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_t^{\prime }(z)+u_{t-1}(z){\psi }^{\prime }(z)||f_n^{(t)}(\psi (z))|\hfill \\ & \le & T_0\underset{z\in \mathbb{D}}{\sup }|f_n(\psi (z))|+\sum _{t=1}^{k+1}{T}_t\underset{z\in \mathbb{D}}{\sup }|f_n^{(t)}(\psi (z))|,\hfill \end{array}$$

which implies that $\parallel P_{\overrightarrow{u},\psi }^k{f}_n{\parallel }_{\mathcal{B}}\to 0$ as $n\to \infty .$ Hence, $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is compact. The proof is complete. □

Next, we state and prove the main results in this section. For simplicity, let

$$\begin{array}{c}\hfill A_i=\underset{|a|\to 1}{\lim }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,a}\parallel }_{\mathcal{B}},\mathrm{for}i=0,1,\cdots ,k+1;\end{array}$$

$$\begin{array}{c}\hfill B_0=\underset{r\to 1}{\lim }\underset{|\psi (z)|>r}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)|\log {\displaystyle \frac{e}{1-{|\psi (z)|}^2}};\end{array}$$

$$\begin{array}{ccc}& & B_t=\underset{r\to 1}{\lim }\underset{|\psi (z)|>r}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_{t-1}(z){\psi }^{\prime }(z)+u_t^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^t}},\mathrm{for}t=1,2,\cdots ,k+1.\hfill \end{array}$$

Theorem 3. 

Let $k\in {\mathbb{N}}_0$, $\psi \in S(\mathbb{D})$ with ${\parallel \psi \parallel }_{\infty }=1$ and $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$. If $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded, then

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\approx \underset{0\le i\le k+1}{\max }{A}_i\approx \underset{0\le i\le k+1}{\max }{B}_i.\end{array}$$

Proof. 

We first prove that

$$\underset{0\le i\le k+1}{\max }{A}_i\le {\parallel P_{\overrightarrow{u},\psi }^k\parallel }_{e,\mathcal{B}\to \mathcal{B}}.$$

Let $a\in \mathbb{D}\setminus \left\{0\right\}$. It is easy to see that $f_{i,a}(i=0,1,\cdots ,k+1)\in \mathcal{B}$ and $\left\{f_{i,a}\right\}$ converges to 0 uniformly on compact subsets of $\mathbb{D}$ as $|a|\to 1$. Since $f_{i,a}\in {\mathcal{B}}_0$ for $i=0,1,\cdots ,k+1$, this ensures that $\left\{f_{i,a}\right\}$ tends to 0 weakly in $\mathcal{B}$, and thus for any compact operator $K:\mathcal{B}\to \mathcal{B}$, we have

$$\underset{|a|\to 1}{\lim }{\parallel K{f}_{i,a}\parallel }_{\mathcal{B}}=0,\mathrm{for}\mathrm{all}i=0,1,\cdots ,k+1.$$

Hence, for each $i=0,1,\cdots ,k+1$,

$$\begin{array}{ccc}\hfill \parallel P_{\overrightarrow{u},\psi }^k{-K\parallel }_{\mathcal{B}\to \mathcal{B}}& \gtrsim & \underset{|a|\to 1}{\lim \sup }{\parallel (P_{\overrightarrow{u},\psi }^k-K)f_{i,a}\parallel }_{\mathcal{B}}\gtrsim A_i,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}=\underset{K}{\inf }{\parallel P_{\overrightarrow{u},\psi }^k-K\parallel }_{\mathcal{B}\to \mathcal{B}}\gtrsim \underset{0\le i\le k+1}{\max }{A}_i.\end{array}$$

Next, let ${\left\{z_n\right\}}_{n\in \mathbb{N}}$ be a sequence in $\mathbb{D}$ with $|\psi (z_n)|\to 1$ as $n\to \infty$ such that

$$\begin{array}{c}\hfill \underset{r\to 1}{\lim }\underset{|\psi (z)|>r}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_k(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^{k+1}}}=\underset{n\to \infty }{\lim }{\displaystyle \frac{(1-|z_n{|}^2)|u_k(z_n){\psi }^{\prime }(z_n)|}{(1-|\psi (z_n){{|}^2)}^{k+1}}}.\end{array}$$

(17)

For each n, define

$$f_{k+1,n}(z)={\displaystyle \frac{1-|\psi (z_n){|}^2}{1-\overline{\psi (z_n)}z}}{\sigma }_{\psi (z_n)}^{k+1}(z),z\in \mathbb{D}.$$

Then $f_{k+1,n}\in \mathcal{B}$ and $\parallel f_{k+1,n}{\parallel }_{\mathcal{B}}\lesssim 1,f_{k+1,n}^{(i)}(\psi (z_n))=0$ for all $i=0,1,\cdots ,k$ and

$$\begin{array}{c}\hfill |f_{k+1,n}^{(k+1)}(\psi (z_n))|={\displaystyle \frac{(k+1)!}{(1-|\psi (z_n){{|}^2)}^{k+1}}}.\end{array}$$

(18)

Similarly, ${\left\{f_{k+1,n}\right\}}_{n\in \mathbb{N}}$ is bounded in $\mathcal{B}$ and $f_{k+1,n}\to 0$ weakly in $\mathcal{B}$. Thus, for any compact operator $K:\mathcal{B}\to \mathcal{B}$, ${\lim }_{n\to \infty }{\parallel K{f}_{k+1,n}\parallel }_{\mathcal{B}}=0$. Further, we obtain

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{-K\parallel }_{\mathcal{B}\to \mathcal{B}}\ge \underset{n\to \infty }{\lim \sup }\parallel (P_{\overrightarrow{u},\psi }^k-K)f_{k+1,n}{\parallel }_{\mathcal{B}}\ge \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,n}\parallel }_{\mathcal{B}},\end{array}$$

and hence

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}=\underset{K}{\inf }\parallel P_{\overrightarrow{u},\psi }^k{-K\parallel }_{\mathcal{B}\to \mathcal{B}}\ge \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,n}\parallel }_{\mathcal{B}}.\end{array}$$

(19)

Using (18) and (19) we obtain

$$\begin{array}{ccc}& & \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\ge \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,n}\parallel }_{\mathcal{B}}\hfill \\ & \ge & \underset{n\to \infty }{\lim \sup }(1-|z_n{|}^2)\left|\sum _{t=0}^k\right(u_t^{\prime }(z_n)f_{k+1,n}^{(t)}(\psi (z_n))+u_t(z_n){\psi }^{\prime }(z_n)f_{k+1,n}^{(t+1)}(\psi (z_n))\left)\right|\hfill \\ & =& \underset{n\to \infty }{\lim \sup }{\displaystyle \frac{(k+1)!(1-|z_n{|}^2)|u_k(z_n){\psi }^{\prime }(z_n)|}{(1-|\psi (z_n){{|}^2)}^{k+1}}}.\hfill \end{array}$$

(20)

Since $|\psi (z_n)|\to 1$ as $n\to \infty$, it follows from (17) and (20) that

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\gtrsim B_{k+1}.\end{array}$$

(21)

Similarly to the proof of Theorem 1 and the above statements, we can obtain

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\gtrsim B_t\mathrm{for}\mathrm{all}t=0,1,\cdots ,k.\end{array}$$

(22)

Finally, we prove that

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\lesssim \underset{0\le i\le k+1}{\max }{A}_i\mathrm{and}{\parallel P_{\overrightarrow{u},\psi }^k\parallel }_{e,\mathcal{B}\to \mathcal{B}}\lesssim \underset{0\le i\le k+1}{\max }{B}_i.\end{array}$$

Let $\left\{L_n\right\}$ be given in Lemma 3. Since each $L_n^{**}$ is compact on $\mathcal{B}$, $P_{\overrightarrow{u},\psi }^k{L}_n^{**}$ is also compact on $\mathcal{B}$ and we obtain

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\le \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k-P_{\overrightarrow{u},\psi }^k{L}_n^{**}\parallel }_{\mathcal{B}\to \mathcal{B}}.\end{array}$$

(23)

Therefore, we only need to prove that

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k-P_{\overrightarrow{u},\psi }^k{L}_n^{**}\parallel }_{\mathcal{B}\to \mathcal{B}}\lesssim \underset{0\le i\le k+1}{\max }{A}_i\end{array}$$

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k-P_{\overrightarrow{u},\psi }^k{L}_n^{**}\parallel }_{\mathcal{B}\to \mathcal{B}}\lesssim \underset{0\le i\le k+1}{\max }{B}_i.\end{array}$$

For any $f\in \mathcal{B}$ such that ${\parallel f\parallel }_{\mathcal{B}}\le 1$, we consider

$$\begin{array}{ccc}& & \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\le \underset{n\to \infty }{\lim \sup }\parallel P_{\overrightarrow{u},\psi }^k-P_{\overrightarrow{u},\psi }^k{L}_n^{**}\parallel \hfill \\ & =& \underset{n\to \infty }{\lim \sup }\parallel P_{\overrightarrow{u},\psi }^k(I-L_n^{**})\parallel \le \underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k(I-L_n^{**})f\parallel }_{\mathcal{B}}\hfill \\ & =& \underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }|P_{\overrightarrow{u},\psi }^k(I-L_n^{**})f(0)|+\underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k(I-L_n^{**})f\parallel }_{\beta }\hfill \\ & =& \underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }\left|\sum _{j=0}^k{u}_j(0){\left((I-L_n^{**})f\right)}^{(j)}(\psi (0))\right|\hfill \\ & & +\underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }\parallel \sum _{j=0}^k{u}_j{\left((I-L_n^{**})f\right)}^{(j)}\circ \psi {\parallel }_{\beta }.\hfill \end{array}$$

(24)

Lemma 3 guarantees that

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }\sum _{j=0}^k|u_j(0)||{\left((I-L_n^{**})f\right)}^{(j)}(\psi (0))|=0.\end{array}$$

(25)

Now we consider

$$\begin{array}{c}\hfill J=\underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }\parallel \sum _{j=0}^k{u}_j{\left((I-L_n^{**})f\right)}^{(j)}\circ \psi {\parallel }_{\beta }.\end{array}$$

Let $f\in \mathcal{B}$ with ${\parallel f\parallel }_{\mathcal{B}}\le 1$. Then

$$\begin{array}{ccc}& & J=\underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }\parallel \sum _{j=0}^k{u}_j{\left((I-L_n^{**})f\right)}^{(j)}\circ \psi {\parallel }_{\beta }\hfill \\ & =& \underset{n\to \infty }{\lim \sup }\underset{z\in \mathbb{D}}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)\left((I-L_n^{**})f\right)(\psi (z))+u_k(z){\psi }^{\prime }(z){\left((I-L_n^{**})f\right)}^{(k+1)}(\psi (z))\hfill \\ & & +\sum _{j=1}^k\left(u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)\right){\left((I-L_n^{**})f\right)}^{(j)}(\psi (z))|\hfill \\ & \le & \underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|\le r_N}{\sup }{(1-|z|}^2)|\left((I-L_n^{**})f\right)(\psi (z))||u_0^{\prime }(z)|\hfill \\ & & +\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|\left((I-L_n^{**})f\right)(\psi (z))||u_0^{\prime }(z)|\hfill \\ & & +\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|\le r_N}{\sup }{(1-|z|}^2)\sum _{j=1}^k\left|{\left((I-L_n^{**})f\right)}^{(j)}(\psi (z))\right||u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|\hfill \\ & & +\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\sum _{j=1}^k\left|{\left((I-L_n^{**})f\right)}^{(j)}(\psi (z))\right||u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|\hfill \\ & & +\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|\le r_N}{\sup }{(1-|z|}^2)\left|{\left((I-L_n^{**})f\right)}^{(k+1)}(\psi (z))\right||u_k(z){\psi }^{\prime }(z)|\hfill \\ & & +\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left|{\left((I-L_n^{**})f\right)}^{(k+1)}(\psi (z))\right||u_k(z){\psi }^{\prime }(z)|\hfill \\ & =& Q_1+Q_2+Q_3+Q_4+Q_5+Q_6,\hfill \end{array}$$

(26)

where $N\in \mathbb{N}$ is large enough such that $r_n\ge {\displaystyle \frac{1}{2}}$ for all $n\ge N$,

$$Q_1:=\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|\le r_N}{\sup }{(1-|z|}^2)|\left((I-L_n^{**})f\right)(\psi (z))||u_0^{\prime }(z)|,$$

$$Q_2:=\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|\left((I-L_n^{**})f\right)(\psi (z))||u_0^{\prime }(z)|,$$

$$\begin{array}{c}\hfill Q_3:=\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|\le r_N}{\sup }{(1-|z|}^2)\sum _{j=1}^k\left|{\left((I-L_n^{**})f\right)}^{(j)}(\psi (z))\right||u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|,\end{array}$$

and

$$\begin{array}{c}\hfill Q_4:=\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\sum _{j=1}^k\left|{\left((I-L_n^{**})f\right)}^{(j)}(\psi (z))\right||u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|,\end{array}$$

$$Q_5:=\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|\le r_N}{\sup }{(1-|z|}^2)\left|{\left((I-L_n^{**})f\right)}^{(k+1)}(\psi (z))\right||u_k(z){\psi }^{\prime }(z)|$$

and

$$Q_6:=\underset{n\to \infty }{\lim \sup }\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left|{\left((I-L_n^{**})f\right)}^{(k+1)}(\psi (z))\right||u_k(z){\psi }^{\prime }(z)|.$$

Since $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded, from Theorem 1, we obtain that $T_t<\infty (t=0,1,\cdots ,k+1)$, where $T_t$ is defined in Lemma 4. Since $f_{r_n}^{(t)}\to f^{(t)}(t=0,1,\cdots ,k+1)$ uniformly on compact subsets of $\mathbb{D}$ as $n\to \infty$, by Lemma 3 we have

$$\begin{array}{c}\hfill Q_1\le T_0\underset{n\to \infty }{\lim \sup }\underset{|w|\le r_N}{\sup }|\left((I-L_n^{**})f\right)(w)|=0,\end{array}$$

(27)

$$\begin{array}{c}\hfill Q_3\le \sum _{j=1}^k{T}_j\underset{n\to \infty }{\lim \sup }\underset{|w|\le r_N}{\sup }|{\left((I-L_n^{**})f\right)}^{(j)}(w)|=0,\end{array}$$

(28)

and

$$\begin{array}{c}\hfill Q_5\le T_{k+1}\underset{n\to \infty }{\lim \sup }\underset{|w|\le r_N}{\sup }|{\left((I-L_n^{**})f\right)}^{(k+1)}(w)|=0.\end{array}$$

(29)

Next we consider $Q_6$. We have $Q_6={\lim \sup }_{n\to \infty }{Q}_{61},$ where

$$\begin{array}{c}\hfill Q_{61}:=\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left|{\left((I-L_n^{**})f\right)}^{(k+1)}(\psi (z))\right||u_k(z){\psi }^{\prime }(z)|.\end{array}$$

Using the fact that ${\parallel f\parallel }_{\mathcal{B}}\le 1$, Lemmas 1 and 3, we have

$$\begin{array}{ccc}\hfill Q_{61}& =& \underset{|\psi (z)|>r_N}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_k(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^{k+1}}}{(1-|\psi (z)|}^2{)}^{k+1}\left|{\left((I-L_n^{**})f\right)}^{(k+1)}(\psi (z))\right|\hfill \\ & \lesssim & \parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}\underset{|\psi (z)|>r_N}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_k(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^{k+1}}}\hfill \\ & \lesssim & {\displaystyle \frac{\parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}}{(k+1)!}}\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|u_k(z){\psi }^{\prime }(z)||f_{k+1,\psi (z)}^{(k+1)}(\psi (z))|\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& \lesssim & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k+1,\psi (z)}\right)}^{\prime }(z)|\lesssim \underset{|a|>r_N}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,a}\parallel }_{\mathcal{B}}.\hfill \end{array}$$

(31)

Taking limit as $N\to \infty$ we obtain

$$\begin{array}{c}\hfill Q_6=\underset{n\to \infty }{\lim \sup }{Q}_{61}\lesssim \underset{|a|\to 1}{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,a}\parallel }_{\mathcal{B}}=A_{k+1}.\end{array}$$

(32)

From (30),

$$\begin{array}{c}\hfill Q_6=\underset{n\to \infty }{\lim \sup }{Q}_{61}\lesssim \underset{r\to 1}{\lim }\underset{|\psi (z)|>r}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_k(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^{k+1}}}=B_{k+1}.\end{array}$$

(33)

Next we consider $Q_4$. We have $Q_4={\lim \sup }_{n\to \infty }{Q}_{41},$ where

$$\begin{array}{c}\hfill Q_{41}:=\sum _{j=1}^k\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left|{\left((I-L_n^{**})f\right)}^{(j)}(\psi (z))\right||u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|.\end{array}$$

Using the fact that ${\parallel f\parallel }_{\mathcal{B}}\le 1$, Lemma 1, Lemma 3 and (31), we have

$$\begin{array}{ccc}& & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left|{\left((I-L_n^{**})f\right)}^{(k)}(\psi (z))\right||u_k^{\prime }(z)+u_{k-1}(z){\psi }^{\prime }(z)|\hfill \\ & \lesssim & {\displaystyle \frac{\parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}}{k!}}\underset{|\psi (z)|>r_N}{\sup }{\displaystyle \frac{{k!(1-|z|}^2)|u_k^{\prime }(z)+u_{k-1}(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^k}}\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k,\psi (z)}\right)}^{\prime }(z)-u_k(z){\psi }^{\prime }(z)f_{k,\psi (z)}^{(k+1)}(\psi (z))|\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k,\psi (z)}\right)}^{\prime }(z)\right|+|u_k(z){\psi }^{\prime }(z)||f_{k,\psi (z)}^{(k+1)}(\psi (z))|\right)\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k,\psi (z)}\right)}^{\prime }(z)\right|+{\parallel f_{k,\psi (z)}\parallel }_{\mathcal{B}}{\displaystyle \frac{|u_k(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^{k+1}}}\right)\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }\underset{|a|>r_N}{\sup }{(1-|z|}^2)\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k,a}\right)}^{\prime }(z)\right|+\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k+1,a}\right)}^{\prime }(z)\right|\right)\hfill \\ & \lesssim & \underset{|a|>r_N}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{k,a}{\parallel }_{\mathcal{B}}+\underset{|a|>r_N}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{k+1,a}\parallel }_{\mathcal{B}}.\hfill \end{array}$$

(34)

Further, fix $1\le j\le k-1$ and assume that

$$\begin{array}{ccc}& & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|{\left((I-L_n^{**})f\right)}^{(i)}(\psi (z))||u_i^{\prime }(z)+u_{i-1}(z){\psi }^{\prime }(z)|\hfill \\ & \lesssim & \parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}\underset{|\psi (z)|>r_N}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_i^{\prime }(z)+u_{i-1}(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^i}}\hfill \\ & \lesssim & \underset{|a|>r_N}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,a}{\parallel }_{\mathcal{B}}+\sum _{t=i+1}^{k+1}\underset{|a|>r_N}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{t,a}\parallel }_{\mathcal{B}}\hfill \end{array}$$

(35)

for each $i=j+1,\cdots ,k$. Now we establish (35) for $i=j$. Using the fact that ${\parallel f\parallel }_{\mathcal{B}}\le 1$, Lemmas 1 and 3, (31) and (35), we have

$$\begin{array}{ccc}& & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|{\left((I-L_n^{**})f\right)}^{(j)}(\psi (z))||u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|\hfill \\ & \lesssim & {\displaystyle \frac{\parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}}{j!}}\underset{|\psi (z)|>r_N}{\sup }{\displaystyle \frac{{j!(1-|z|}^2)|u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^j}}\hfill \\ & =& {\displaystyle \frac{\parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}}{j!}}\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|{\left(P_{\overrightarrow{u},\psi }^k{f}_{j,\psi (z)}\right)}^{\prime }(z)\hfill \\ & & -\sum _{i=j+1}^k\left(u_i^{\prime }(z)+u_{i-1}(z){\psi }^{\prime }(z)\right)f_{j,\psi (z)}^{(i)}(\psi (z))-u_k(z){\psi }^{\prime }(z)f_{j,\psi (z)}^{(k+1)}(\psi (z))|\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{j,\psi (z)}\right)}^{\prime }(z)\right|+\sum _{i=j+1}^{k+1}|u_i^{\prime }(z)+u_{i-1}(z){\psi }^{\prime }(z)||f_{j,\psi (z)}^{(i)}(\psi (z))|\right)\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{j,\psi (z)}\right)}^{\prime }(z)\right|+\sum _{i=j+1}^{k+1}{\parallel f_{j,\psi (z)}\parallel }_{\mathcal{B}}{\displaystyle \frac{|u_i^{\prime }(z)+u_{i-1}(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^i}}\right)\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }\underset{|a|>r_N}{\sup }{(1-|z|}^2)(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{j,a}\right)}^{\prime }(z)\right|\hfill \\ & & +\sum _{i=j+1}^k\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{i,a}\right)}^{\prime }(z)\right|+\sum _{t=i+1}^{k+1}\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{t,a}\right)}^{\prime }(z)\right|\right)+\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k+1,a}\right)}^{\prime }(z)\right|)\hfill \\ & \lesssim & \underset{|a|>r_N}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{j,a}{\parallel }_{\mathcal{B}}+\sum _{i=j+1}^{k+1}\underset{|a|>r_N}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,a}\parallel }_{\mathcal{B}}\hfill \end{array}$$

(36)

for every $j=1,2,\cdots ,k.$ After a calculation, using (36), we have

$$\begin{array}{ccc}\hfill Q_{41}& \lesssim & \sum _{j=1}^k\parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}\underset{|\psi (z)|>r_N}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^j}}\hfill \end{array}$$

(37)

$$\begin{array}{ccc}& \lesssim & \sum _{j=1}^k\underset{|a|>r_N}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{j,a}{\parallel }_{\mathcal{B}}+\sum _{j=1}^k\sum _{i=j+1}^{k+1}\underset{|a|>r_N}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,a}\parallel }_{\mathcal{B}}.\hfill \end{array}$$

(38)

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

$$\begin{array}{c}\hfill Q_4=\underset{n\to \infty }{\lim \sup }{Q}_{41}\lesssim \underset{|a|\to 1}{\lim \sup }\sum _{i=1}^{k+1}{\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,a}\parallel }_{\mathcal{B}}=\sum _{i=1}^{k+1}{A}_i\lesssim \underset{1\le i\le k+1}{\max }{A}_i.\end{array}$$

(39)

From (37), we see that

$$\begin{array}{c}\hfill Q_4=\underset{n\to \infty }{\lim \sup }{Q}_{41}\lesssim \underset{r\to 1}{\lim }\underset{|\psi (z)|>r}{\sup }\sum _{j=1}^k{\displaystyle \frac{{(1-|z|}^2)|u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^j}}\lesssim \underset{1\le i\le k}{\max }{B}_i.\end{array}$$

(40)

We have $Q_2={\lim \sup }_{n\to \infty }{Q}_{21},$ where

$$\begin{array}{c}\hfill Q_{21}:=\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|\left((I-L_n^{**})f\right)(\psi (z))||u_0^{\prime }(z)|.\end{array}$$

Using the fact that ${\parallel f\parallel }_{\mathcal{B}}\le 1$, Lemma 1, Lemma 3, (31) and (36), we have

$$\begin{array}{ccc}\hfill Q_{21}& \le & \parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)|\log {\displaystyle \frac{e}{1-{|\psi (z)|}^2}}\hfill \\ & =& \parallel I-L_n^{**}{\parallel }_{\mathcal{B}\to \mathcal{B}}{\parallel f\parallel }_{\mathcal{B}}\underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)|{\left(P_{\overrightarrow{u},\psi }^k{f}_{0,\psi (z)}\right)}^{\prime }(z)\hfill \\ & & -\sum _{j=1}^k\left(u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)\right)f_{0,\psi (z)}^{(j)}(\psi (z))-u_k(z){\psi }^{\prime }(z)f_{0,\psi (z)}^{(k+1)}(\psi (z))|\hfill \\ & \le & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{0,\psi (z)}\right)}^{\prime }(z)\right|+\sum _{j=1}^{k+1}|u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)||f_{0,\psi (z)}^{(j)}(\psi (z))|\right)\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }{(1-|z|}^2)\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{0,\psi (z)}\right)}^{\prime }(z)\right|+\sum _{j=1}^{k+1}{\parallel f_{0,\psi (z)}\parallel }_{\mathcal{B}}{\displaystyle \frac{|u_j^{\prime }(z)+u_{j-1}(z){\psi }^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^j}}\right)\hfill \\ & \lesssim & \underset{|\psi (z)|>r_N}{\sup }\underset{|a|>r_N}{\sup }{(1-|z|}^2)(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{0,a}\right)}^{\prime }(z)\right|+\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{k+1,a}\right)}^{\prime }(z)\right|\hfill \\ & & +\sum _{j=1}^k\left(\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{j,a}\right)}^{\prime }(z)\right|+\sum _{i=j+1}^{k+1}\left|{\left(P_{\overrightarrow{u},\psi }^k{f}_{i,a}\right)}^{\prime }(z)\right|\right))\hfill \\ & \lesssim & \underset{|a|>r_N}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{0,a}{\parallel }_{\mathcal{B}}+\sum _{j=1}^{k+1}\underset{|a|>r_N}{\sup }\parallel P_{\overrightarrow{u},\psi }^k{f}_{j,a}{\parallel }_{\mathcal{B}}+\sum _{j=1}^k\sum _{i=j+1}^{k+1}\underset{|a|>r_N}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,a}\parallel }_{\mathcal{B}}.\hfill \end{array}$$

(41)

Taking limit as $N\to \infty$ we obtain

$$\begin{array}{c}\hfill Q_2=\underset{n\to \infty }{\lim \sup }{Q}_{21}\lesssim \underset{|a|\to 1}{\lim \sup }\sum _{i=0}^{k+1}{\parallel P_{\overrightarrow{u},\psi }^k{f}_{i,a}\parallel }_{\mathcal{B}}=\sum _{i=0}^{k+1}{A}_i\lesssim \underset{1\le i\le k+1}{\max }{A}_i.\end{array}$$

(42)

From (41), we see that

$$\begin{array}{c}\hfill Q_2=\underset{n\to \infty }{\lim \sup }{Q}_{21}\lesssim \underset{r\to 1}{\lim }\underset{|\psi (z)|>r}{\sup }{(1-|z|}^2)|u_0^{\prime }(z)|\log {\displaystyle \frac{e}{1-{|\psi (z)|}^2}}=B_0.\end{array}$$

(43)

Hence by (24)–(29), (32), (39) and (42) we obtain

$$\begin{array}{ccc}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}& \le & \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k-P_{\overrightarrow{u},\psi }^k{L}_n^{**}\parallel }_{\mathcal{B}\to \mathcal{B}}\hfill \\ & \le & \underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }{\parallel P_{\overrightarrow{u},\psi }^k(I-L_n^{**})f\parallel }_{\mathcal{B}}\hfill \\ & =& \underset{n\to \infty }{\lim \sup }\underset{{\parallel f\parallel }_{\mathcal{B}}\le 1}{\sup }\parallel \sum _{j=0}^k{u}_j{\left((I-L_n^{**})f\right)}^{(j)}\circ \psi {\parallel }_{\beta }\lesssim \underset{0\le i\le k+1}{\max }{A}_i.\hfill \end{array}$$

Similarly, by (24)–(29), (33), (40) and (43) we obtain

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\le \underset{n\to \infty }{\lim \sup }{\parallel P_{\overrightarrow{u},\psi }^k-P_{\overrightarrow{u},\psi }^k{L}_n^{**}\parallel }_{\mathcal{B}\to \mathcal{B}}\lesssim \underset{0\le i\le k+1}{\max }{B}_i.\end{array}$$

This completes the proof. □

From Theorem 3, we immediately obtain the following characterization for the compactness of $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$.

Corollary 1. 

Let $k\in {\mathbb{N}}_0$, $\psi \in S(\mathbb{D})$ and $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$. If $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded, then the following statements are equivalent.

(a) 

The operator $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is compact.

(b) 

${\lim }_{|\psi (a)|\to 1}{\parallel P_{\overrightarrow{u},\psi }^k{f}_{t,\psi (a)}\parallel }_{\mathcal{B}}=0,$ for $t=0,1,\cdots ,k+1.$

(c) 

$$\begin{array}{c}\hfill \underset{r\to 1}{\lim }\underset{|\psi (z)|>r}{\sup }{\displaystyle \frac{{(1-|z|}^2)|u_{t-1}(z){\psi }^{\prime }(z)+u_t^{\prime }(z)|}{{(1-|\psi (z)|}^2{)}^t}}=0,\mathit{for}t=0,1,2,\cdots ,k+1.\end{array}$$

(44)

By [25,26], we have the following lemma.

Lemma 5. 

Let v and w be radial, non-increasing weights tending to zero at the boundary of $\mathbb{D}$. Suppose $u{C}_{\psi }:H_v^{\infty }\to H_w^{\infty }$ is bounded. Then,

$$\begin{array}{c}\hfill \parallel u{C}_{\psi }{\parallel }_{e,H_v^{\infty }\to H_w^{\infty }}=\underset{s\to 1}{\lim }\underset{|\psi (z)|>s}{\sup }{\displaystyle \frac{w(z)}{\tilde{v}(\psi (z))}}|u(z)|=\underset{n\to \infty }{\lim \sup }{\displaystyle \frac{\parallel u{\psi }^n{\parallel }_w}{\parallel {\eta }^n{\parallel }_v}}.\end{array}$$

Using Lemma 5, we give another characterization for the essential norm of $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$.

Theorem 4. 

Let $k\in {\mathbb{N}}_0$, $\psi \in S(\mathbb{D})$ and $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$. If $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded, then

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\approx \underset{0\le i\le k+1}{\max }{D}_i,\end{array}$$

where $D_0={\lim \sup }_{n\to \infty }\log n{\parallel u_0^{\prime }{\psi }^n\parallel }_{v_1}$ and

$$\begin{array}{c}\hfill D_t=\underset{n\to \infty }{\lim \sup }{n}^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1},\mathit{for}t=1,2,\cdots ,k+1.\end{array}$$

Proof. 

According to Lemma 2, we known that the boundedness of $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is equivalent to the boundedness of $u_0^{\prime }{C}_{\psi }:H_{v_{\log }}^{\infty }\to H_{v_1}^{\infty }$ and $(u_{t-1}{\psi }^{\prime }+u_t^{\prime })C_{\psi }:H_{v_t}^{\infty }\to H_{v_1}^{\infty }$ for all $t=1,2,\cdots ,k+1$.

The upper estimate. By [20],

$$\underset{n\to \infty }{\lim }{n}^{\alpha }\parallel {\eta }^{n-1}{\parallel }_{v_{\alpha }}={({\displaystyle \frac{2\alpha }{e}})}^{\alpha }\mathrm{and}\underset{n\to \infty }{\lim }\log n{\parallel {\eta }^n\parallel }_{v_{\log }}=1,$$

when $\alpha >0$. By Lemma 5, we obtain

$$\begin{array}{ccc}\hfill \parallel u_0^{\prime }{C}_{\psi }{\parallel }_{e,H_{v_{\log }}^{\infty }\to H_{v_1}^{\infty }}& =& \underset{n\to \infty }{\lim \sup }{\displaystyle \frac{\parallel u_0^{\prime }{\psi }^{n-1}{\parallel }_{v_1}}{\parallel {\eta }^{n-1}{\parallel }_{v_{\log }}}}=\underset{n\to \infty }{\lim \sup }{\displaystyle \frac{\log (n-1)\parallel u_0^{\prime }{\psi }^{n-1}{\parallel }_{v_1}}{\log (n-1)\parallel {\eta }^{n-1}{\parallel }_{v_{\log }}}}\hfill \\ & \approx & \underset{n\to \infty }{\lim \sup }\log (n-1)\parallel u_0^{\prime }{\psi }^{n-1}{\parallel }_{v_1}=\underset{n\to \infty }{\lim \sup }\log n{\parallel u_0^{\prime }{\psi }^n\parallel }_{v_1},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime })C_{\psi }{\parallel }_{e,H_{v_t}^{\infty }\to H_{v_1}^{\infty }}& =& \underset{n\to \infty }{\lim \sup }{\displaystyle \frac{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}{\parallel }_{v_1}}{\parallel {\eta }^{n-1}{\parallel }_{v_t}}}\hfill \\ & =& \underset{n\to \infty }{\lim \sup }{\displaystyle \frac{n^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1}}{n^t{\parallel {\eta }^{n-1}\parallel }_{v_t}}}\hfill \\ & \approx & \underset{n\to \infty }{\lim \sup }{n}^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1},t=1,2,\cdots ,k+1.\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}& \lesssim & \parallel u_0^{\prime }{C}_{\psi }{\parallel }_{e,H_{v_{\log }}^{\infty }\to H_{v_1}^{\infty }}+\sum _{t=1}^{k+1}{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime })C_{\psi }\parallel }_{e,H_{v_t}^{\infty }\to H_{v_1}^{\infty }}\hfill \\ & \approx & \sum _{t=0}^{k+1}{D}_t\lesssim \underset{0\le t\le k+1}{\max }{D}_t.\hfill \end{array}$$

The lower estimate. By Theorem 3 and Lemma 5, we have

$$\begin{array}{ccc}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}& \gtrsim & B_0={\parallel u_0^{\prime }{C}_{\psi }\parallel }_{e,H_{v_{\log }}^{\infty }\to H_{v_1}^{\infty }}=\underset{n\to \infty }{\lim \sup }{\displaystyle \frac{\parallel u_0^{\prime }{\psi }^{n-1}{\parallel }_{v_1}}{\parallel {\eta }^{n-1}{\parallel }_{v_{\log }}}}\hfill \\ & \approx & \underset{n\to \infty }{\lim \sup }\log (n-1){\parallel u_0^{\prime }{\psi }^{n-1}\parallel }_{v_1}\hfill \\ & =& \underset{n\to \infty }{\lim \sup }\log n{\parallel u_0^{\prime }{\psi }^n\parallel }_{v_1},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}& \gtrsim & B_t={\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime })C_{\psi }\parallel }_{e,H_{v_t}^{\infty }\to H_{v_1}^{\infty }}=\underset{n\to \infty }{\lim \sup }{\displaystyle \frac{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}{\parallel }_{v_1}}{\parallel {\eta }^{n-1}{\parallel }_{v_t}}}\hfill \\ & \approx & \underset{n\to \infty }{\lim \sup }{n}^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1},\mathrm{for}t=1,2,\cdots ,k+1.\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill \parallel P_{\overrightarrow{u},\psi }^k{\parallel }_{e,\mathcal{B}\to \mathcal{B}}\gtrsim \underset{0\le i\le k+1}{\max }{D}_i.\end{array}$$

This completes the proof of this theorem. □

From Theorem 4, we immediately obtain the following corollary.

Corollary 2. 

Let $k\in {\mathbb{N}}_0$, $\psi \in S(\mathbb{D})$ and $u_j\in H(\mathbb{D}),j=0,1,\dots ,k$. If $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is bounded, then $P_{\overrightarrow{u},\psi }^k:\mathcal{B}\to \mathcal{B}$ is compact if and only if ${\lim \sup }_{n\to \infty }\log n{\parallel u_0^{\prime }{\psi }^n\parallel }_{v_1}=0$ and

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim \sup }{n}^t{\parallel (u_{t-1}{\psi }^{\prime }+u_t^{\prime }){\psi }^{n-1}\parallel }_{v_1}=0,\mathrm{for}t=1,2,\cdots ,k+1.\end{array}$$

4. Conclusions

In this paper, we investigate the boundedness, compactness and essential norm of the operator $P_{\overrightarrow{u},\psi }^k$ and give several characterizations for these properties on the Bloch space. Our approaches are inspired by [6,7] for the studying of generalized weighted composition operators on Bloch-type spaces, and [9] for the studying of polynomial differentiation composition operators from $H^p$ spaces to weighted-type spaces. We combine the methods of these three articles. And our proof is more detailed than [10]. Finally, our results generalized many results in the literature. For example, by Theorems 3 and 4, we can obtain the characterization of the essential norm of weighted composition operators $u{C}_{\psi }$ on the Bloch space (see [20,22]) and generalized weighted composition operator $D_{u,\psi }^n$ on the Bloch space (see [7]).