Sums of Generalized Weighted Composition Operators from Weighted Bergman Spaces Induced by Doubling Weights into Bloch-Type Spaces

Abstract

The single generalized weighted composition operator $D_{u,\psi }^n$ on various spaces of analytic functions has been investigated for decades, i.e., $D_{u,\psi }^{n}f=u·(f^{\left(n\right)}\circ \psi )$, where $f\in H\left(\mathbb{D}\right)$. However, the study of the finite sum of generalized weighted composition operators with different orders, i.e., $P_{U,\psi }^{k}f=u_0·f\circ \psi +u_1·f^{\prime }\circ \psi +\cdots +u_k·f^{\left(k\right)}\circ \psi ,$ is far from complete. The boundedness, compactness and essential norm of sums of generalized weighted composition operators from weighted Bergman spaces with doubling weights into Bloch-type spaces are investigated. We show a rigidity property of $P_{U,\psi }^k$. Specifically, the boundedness and compactness of the sum $P_{U,\psi }^k$ is equivalent to those of each $D_{u_n,\psi }^n$, $0\le n\le k.$

1. Introduction

Let $\mathbb{D}$ denote the unit disk in the complex plane $\mathbb{C}$, with its boundary $\partial \mathbb{D}$ representing the unit circle. Furthermore, let $H\left(\mathbb{D}\right)$ be the collection of all functions that are analytic within $\mathbb{D}$. For any point a within $\mathbb{D}$, we define ${\sigma }_a$ as the automorphism of $\mathbb{D}$ that shifts the origin to a, given by ${\sigma }_a\left(w\right)={\displaystyle \frac{a-w}{1-\overline{a}w}}$. For any positive real number $\beta$, the Bloch-type space ${\mathcal{B}}^{\beta }$ is the set of all functions g in $H\left(\mathbb{D}\right)$ that meets the following requirement:

$${\parallel g\parallel }_{{\mathcal{B}}^{\beta }}=\left|g\left(0\right)\right|+\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2{)}^{\beta }\left|g^{\prime }\left(w\right)\right|<\infty .$$

When furnished with this norm ${\parallel ·\parallel }_{{\mathcal{B}}^{\beta }}$, ${\mathcal{B}}^{\beta }$ is a Banach space. We also define $\parallel g\parallel$ as ${sup}_{w\in \mathbb{D}}{(1-|w|}^2{)}^{\beta }\left|g^{\prime }\left(w\right)\right|$. Specifically, ${\mathcal{B}}^1$ is equivalent to $\mathcal{B}$, known as the classical Bloch space. For more information on the various operators that act on the Bloch space, the reader is directed to the literature referenced in [1,2,3,4,5,6].

Given a non-negative function $\omega$ that is integrable over the interval $[0,1]$, denoted by $\omega \in L^1\left([0,1]\right)$. The extension of this function to the unit disk $\mathbb{D}$, defined by $\omega \left(z\right)=\omega \left(\right|z\left|\right)$ for all $z\in \mathbb{D}$, is known as a radial weight. The set of doubling weights, denoted by $\widehat{\mathcal{D}}$, includes all radial weights $\omega$ that fulfill the specific criterion (as detailed in [7]):

$$\widehat{\omega }\left(r\right)\le C\widehat{\omega }\left({\displaystyle \frac{r+1}{2}}\right),0\le r<1,$$

where $C=C\left(\omega \right)$ is a constant with a value of at least 1. Throughout this paper, the function $\widehat{\omega }\left(z\right)$ is specified as the integral ${\int }_{\left|z\right|}^1\omega \left(t\right)dt$.

Let $0<p<\infty$ and $\omega \in \widehat{\mathcal{D}}$. The weighted Bergman space $A_{\omega }^p$, which arises from a doubling weight $\omega$, is defined as the collection of all functions $f\in H\left(\mathbb{D}\right)$ that satisfy the following condition:

$${\parallel f\parallel }_{A_{\omega }^p}^p={\int }_{\mathbb{D}}{\left|f\left(z\right)\right|}^p\omega \left(z\right)dA\left(z\right)<\infty .$$

In this context, $dA$ represents the normalized area measure over $\mathbb{D}$. If $\omega \left(z\right)$ takes the form ${(\alpha +1)(1-|z|}^2{)}^{\alpha }$ with $-1<\alpha <\infty$, we refer to $A_{\omega }^p$ as $A_{\alpha }^p$, which is recognized as the standard weighted Bergman space. Throughout this paper, we assume that $\widehat{\omega }\left(z\right)>0$ for all $z\in \mathbb{D}$. If this is not the case, then $A_{\omega }^p$ is equivalent to $H\left(\mathbb{D}\right)$.

We will denote the set of all analytic functions that map the unit disk $\mathbb{D}$ onto itself as $S\left(\mathbb{D}\right)$. Suppose we have a function $\psi$ that is a member of $S\left(\mathbb{D}\right)$, the composition operator $C_{\psi }$ is specified as:

$$C_{\psi }f=f\circ \psi ,\mathrm{for}\mathrm{every}f\in H\left(\mathbb{D}\right).$$

The main emphasis in the study of composition operators is to establish a connection between the operator-theoretic attributes of $C_{\psi }$ and the function-theoretic properties of $\psi$. For a comprehensive study of the various characteristics of composition operators, one should refer to the existing literature [8,9] and the references cited therein.

For every $n\in \mathbb{N}$, the n-th differentiation operator, indicated by $D^n$, is characterized by $D^{n}f=f^{\left(n\right)}$, with the understanding that $f\in H\left(\mathbb{D}\right)$ and that $f^{\left(0\right)}=f$. In particular, when $n=1$, we arrive at the standard derivative operator D, which is frequently unbounded across a variety of spaces consisting of analytic functions.

Given that $u\in H\left(\mathbb{D}\right)$ and $\psi \in S\left(\mathbb{D}\right)$, we introduce the generalized weighted composition operator $D_{u,\psi }^n$. This operator is an extension of the traditional weighted composition operator and is also known as the weighted differentiation composition operator, defined as:

$$D_{u,\psi }^{n}f=u·(f^{\left(n\right)}\circ \psi ),\mathrm{for}f\in H\left(\mathbb{D}\right).$$

In the particular case where $n=0$, the operator $D_{u,\psi }^n$ is identified as the conventional weighted composition operator, denoted by $u{C}_{\psi }$. The notion of the operator $D_{u,\psi }^n$ was introduced by Zhu, the first author of this manuscript, as noted in [10]. For additional understanding and findings concerning the generalized weighted composition operator on analytic function spaces, one should refer to the literature cited in [10,11,12,13,14,15,16].

Let k be a non-negative integer, $\psi \in S\left(\mathbb{D}\right)$, and let U represent the sequence $\{u_0,u_1,\dots ,u_k\}$ such that each $u_j$ is an element of $H\left(\mathbb{D}\right)$. In the work [17] by Wang, Wang, and Guo, the operator $P_{U,\psi }^k$ was introduced, which is defined by the sum:

$$P_{U,\psi }^{k}f=\sum _{j=0}^k{u}_j·(f^{\left(j\right)}\circ \psi )=\sum _{j=0}^k{D}_{u_j,\psi }^{j}f,\mathrm{for}f\in H\left(\mathbb{D}\right).$$

The researchers in [17] explored the boundedness and compactness characteristics of the operator $P_{U,\psi }^k$ within certain analytic function spaces. For an in-depth examination of this subject, the reader is encouraged to consult [17,18,19,20]. Nevertheless, the investigation into the finite sum of generalized weighted composition operators of varying orders remains largely unresolved.

The objective of this article is to study the boundedness and compactness of the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$, which refers to the sum of the generalized weighted composition operators from weighted Bergman spaces with doubling weights to Bloch-type spaces. We demonstrate a rigidity characteristic of $P_{U,\psi }^k$. Specifically, the boundedness and compactness of $P_{U,\psi }^k$ are equivalent to those of each $D_{u_n,\psi }^n$, for $0\le n\le k$. Furthermore, we provide a calculation for the essential norm of the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$.

It should be observed that the essential norm of a bounded linear operator $T:X\to Y$ is measured by its distance to the set of compact operators $\mathcal{K}$ that map from X to Y. This is expressed as:

$${\parallel T\parallel }_{e,X\to Y}=inf\left\{{\parallel T-\mathcal{K}\parallel }_{X\to Y}:\mathcal{K}\mathrm{is}\mathrm{compact}\right\},$$

recognizing that X and Y are Banach spaces, and ${\parallel ·\parallel }_{X\to Y}$ denotes the norm associated with the operator.

Throughout this paper, the symbol C denotes a positive constant that is contingent upon the context and may vary from one line to the next. We define $A\lesssim B$ to indicate the existence of a constant C for which $A\le CB$. The notation $A\approx B$ signifies that $A\lesssim B$ and $B\lesssim A$ simultaneously.

2. Boundedness of ${\mathit{P}}_{\mathit{U},\psi }^{\mathit{k}}:{\mathit{A}}_{\omega }^{\mathit{p}}\to {\mathcal{B}}^{\beta }$

In this section, we investigate the boundedness of the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$. We need some notations and the following lemma, which can be found in [21].

For every $\eta \in \mathbb{D}$, the associated Carleson square at $\eta$ is defined by the set:

$$S\left(\eta \right)=\left\{r{e}^{i\theta }:\left|\eta \right|\le r<1,|\mathrm{Arg}\eta -\theta |<{\displaystyle \frac{1-\left|\eta \right|}{2}}\right\}.$$

For a radial weight $\omega$, the integral over the Carleson square is given by

$$\omega \left(S\left(\eta \right)\right)={\int }_{S\left(\eta \right)}\omega \left(z\right)dA\left(z\right).$$

It is evident that $\omega \left(S\left(\eta \right)\right)\approx (1-|\eta \left|\right)\widehat{\omega }\left(\eta \right)$. For an in-depth understanding of the properties of doubling weights, the reader is directed to [7,22] and the associated literature.

Lemma 1.

Let $\omega \in \widehat{\mathcal{D}}$, $0<p<\infty$, $n\in {\mathbb{N}}_0$ and $f\in A_{\omega }^p$. Then, there exists a constant $C=C(\omega ,n,p)>0$, such that

$$|f^{\left(n\right)}\left(w\right)|\le C{\displaystyle \frac{{\parallel f\parallel }_{A_{\omega }^p}}{{\left(\omega \left(S\left(w\right)\right)\right)}^{1/p}{(1-|w|}^2{)}^n}},w\in \mathbb{D}.$$

(1)

From now on, we assume that $u_j=0$ when $j=-1$ and $u_j^{\prime }=0$ when $j=k+1$ for the simplicity of the notations. Set

$$\begin{array}{c}\hfill M_j=\underset{w\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)+u_j^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}}<\infty ,\mathrm{for}\mathrm{all}j=0,1,\cdots ,k+1.\end{array}$$

Theorem 1.

Let $\beta >0$, $k\in {\mathbb{N}}_0$, $1<p<\infty ,$$\omega \in \widehat{\mathcal{D}}$, $\psi \in S\left(\mathbb{D}\right)$ and U denote the sequence $\{u_0,u_1,···,u_k\}$ such that $u_j\in H\left(\mathbb{D}\right)$. Then, the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded if, and only if,

$$\sum _{i=0}^{k+1}{M}_i<\infty .$$

Furthermore, if $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded, then

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\approx \sum _{i=0}^{k+1}{M}_i.\end{array}$$

Proof. 

Suppose that ${\sum }_{i=0}^{k+1}{M}_i<\infty .$ Let $g\in A_{\omega }^p$. By Lemma 1, we have

$$\begin{array}{ccc}\hfill \parallel P_{U,\psi }^k{g\parallel }_{{\mathcal{B}}^{\beta }}& =& |P_{U,\psi }^{k}g\left(0\right)|+\parallel P_{U,\psi }^{k}g\parallel =|\sum _{j=0}^k{u}_j\left(0\right)g^{\left(j\right)}\left(\psi \left(0\right)\right)|\hfill \\ & & +\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2{)}^{\beta }\left|\sum _{j=0}^k\left(u_j^{\prime }\left(w\right)g^{\left(j\right)}\left(\psi \left(w\right)\right)+u_j\left(w\right){\psi }^{\prime }\left(z\right)g^{(j+1)}\left(\psi \left(w\right)\right)\right)\right|\hfill \\ & \le & {C\parallel g\parallel }_{A_{\omega }^p}\left(\sum _{j=0}^k{\displaystyle \frac{|u_j\left(0\right)|}{{\left(\omega \left(S\left(\psi \left(0\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(0\right)|}^2{)}^j}}+\sum _{j=0}^{k+1}{M}_j\right)\hfill \\ & \lesssim & {\parallel g\parallel }_{A_{\omega }^p}\left(C+\sum _{j=0}^{k+1}{M}_j\right)<\infty ,\hfill \end{array}$$

(2)

which implies that $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded.

Assume that $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded. For $b\in \mathbb{D}$, take

$$\begin{array}{c}\hfill f_{j,b}={\displaystyle \frac{1-{\left|b\right|}^2}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}(1-\overline{b}z)}}{\sigma }_b^j\left(z\right),\mathrm{for}\mathrm{all}j=0,1,\cdots ,k+1.\end{array}$$

(3)

After a calculation, it is observed that $f_{j,b}$ is an element of $A_{\omega }^p$. Additionally, the norm of $f_{j,b}$ in $A_{\omega }^p$, denoted as $\parallel f_{j,b}{\parallel }_{A_{\omega }^p}$, is bounded above by a constant C for every j ranging from 0 to $k+1$. Given the bounded nature of the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$, we deduce that

$$\underset{b\in \mathbb{D}}{sup}\parallel P_{U,\psi }^k{f}_{j,b}{\parallel }_{{\mathcal{B}}^{\beta }}\le \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\underset{b\in \mathbb{D}}{sup}\parallel f_{j,b}{\parallel }_{A_{\omega }^p}\le C{\parallel P_{U,\psi }^k\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}<\infty ,$$

for all $j=0,1,\cdots ,k+1$. Since $f_{k+1,\psi \left(a\right)}^{\left(i\right)}\left(\psi \left(b\right)\right)=0$ for all $i=0,1,\cdots ,k$ and

$$|f_{k+1,\psi \left(b\right)}^{(k+1)}\left(\psi \left(b\right)\right)|={\displaystyle \frac{(k+1)!}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^{k+1}}},$$

we have

$$\begin{array}{ccc}& & {(1-|b|}^2{)}^{\beta }\left|{\left(P_{U,\psi }^k{f}_{k+1,\psi \left(b\right)}\right)}^{\prime }\left(b\right)\right|\hfill \\ & =& {(1-|b|}^2{)}^{\beta }|u_0^{\prime }\left(b\right)f_{k+1,\psi \left(b\right)}\left(\psi \left(b\right)\right)+u_k\left(b\right){\psi }^{\prime }\left(b\right)f_{k+1,\psi \left(b\right)}^{(k+1)}\left(\psi \left(b\right)\right)\hfill \\ & & +\sum _{j=1}^k\left(u_j^{\prime }\left(b\right)+u_{j-1}\left(b\right){\psi }^{\prime }\left(b\right)\right)f_{k+1,\psi \left(b\right)}^{\left(j\right)}\left(\psi \left(b\right)\right)|\hfill \\ & =& {(1-|b|}^2{)}^{\beta }|u_k\left(b\right){\psi }^{\prime }\left(b\right)\left|\right|f_{k+1,\psi \left(b\right)}^{(k+1)}\left(\psi \left(b\right)\right)|\hfill \\ & =& {\displaystyle \frac{{(1-|b|}^2{)}^{\beta }\left|u_k\left(b\right){\psi }^{\prime }\left(b\right)\right|(k+1)!}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^{k+1}}},\hfill \end{array}$$

(4)

which implies that

$$\begin{array}{c}\hfill M_{k+1}\lesssim \underset{b\in \mathbb{D}}{sup}\parallel P_{U,\psi }^k{f}_{k+1,\psi \left(b\right)}{\parallel }_{{\mathcal{B}}^{\beta }}\lesssim {\parallel P_{U,\psi }^k\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}.\end{array}$$

(5)

Since

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge \parallel P_{U,\psi }^k{f}_{k,\psi \left(b\right)}{\parallel }_{{\mathcal{B}}^{\beta }}\ge {(1-|b|}^2{)}^{\beta }\left|{\left(P_{U,\psi }^k{f}_{k,\psi \left(b\right)}\right)}^{\prime }\left(b\right)\right|\hfill \\ & \ge & {(1-|b|}^2{)}^{\beta }|u_k^{\prime }\left(b\right)+u_{k-1}\left(b\right){\psi }^{\prime }\left(b\right)|\left|f_{k,\psi \left(b\right)}^{\left(k\right)}{\left(\psi \left(b\right)\right)|-(1-\left|b\right|}^2{)}^{\beta }\right|u_k\left(b\right){\psi }^{\prime }\left(b\right)\left|\right|f_{k,\psi \left(b\right)}^{(k+1)}\left(\psi \left(b\right)\right)|\hfill \\ & \ge & {\displaystyle \frac{{(1-|b|}^2{)}^{\beta }|u_k^{\prime }\left(b\right)+u_{k-1}\left(b\right){\psi }^{\prime }\left(b\right)|k!}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^k}}-{\displaystyle \frac{C\parallel f_{k,\psi \left(b\right)}{\parallel }_{A_{\omega }^p}{(1-|b|}^2{)}^{\beta }\left|u_k\left(b\right){\psi }^{\prime }\left(b\right)\right|}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^{k+1}}},\hfill \end{array}$$

(6)

we have

$$\begin{array}{ccc}\hfill M_k& \lesssim & \underset{b\in \mathbb{D}}{sup}{\parallel P_{U,\psi }^k{f}_{k,\psi \left(b\right)}\parallel }_{{\mathcal{B}}^{\beta }}+C\underset{b\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|b|}^2{)}^{\beta }\left|u_k\left(b\right){\psi }^{\prime }\left(b\right)\right|}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^{k+1}}}\hfill \\ & \lesssim & \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}.\hfill \end{array}$$

(7)

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

$$\begin{array}{c}\hfill M_i\lesssim {\parallel P_{U,\psi }^k\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }},i=j+1,\cdots ,k.\end{array}$$

(8)

We next prove

$$M_j\lesssim {\parallel P_{U,\psi }^k\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}.$$

After a calculation, we obtain $f_{j,\psi \left(b\right)}^{\left(s\right)}\left(\psi \left(b\right)\right)=0$ for all $s<j$ and

$$\begin{array}{c}\hfill |f_{j,\psi \left(b\right)}^{\left(j\right)}\left(\psi \left(b\right)\right)|={\displaystyle \frac{j!}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^j}}.\end{array}$$

(9)

Using (9) and Lemma 1, we have

$$\begin{array}{ccc}\hfill \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}& \ge & \parallel P_{U,\psi }^k{f}_{j,\psi \left(b\right)}{\parallel }_{{\mathcal{B}}^{\beta }}\ge {(1-|b|}^2{)}^{\beta }\left|{\left(P_{U,\psi }^k{f}_{j,\psi \left(b\right)}\right)}^{\prime }\left(b\right)\right|\hfill \\ & \ge & {(1-|b|}^2{)}^{\beta }|u_j^{\prime }\left(b\right)+u_{j-1}\left(b\right){\psi }^{\prime }\left(b\right)|\left|f_{j,\psi \left(b\right)}^{\left(j\right)}\left(\psi \left(b\right)\right)\right|\hfill \\ & & -\sum _{i=j+1}^{k+1}{(1-|b|}^2{)}^{\beta }|u_i^{\prime }\left(b\right)+u_{i-1}\left(b\right){\psi }^{\prime }\left(b\right)||f_{j,\psi \left(b\right)}^{\left(i\right)}\left(\psi \left(b\right)\right)|\hfill \\ & \ge & {\displaystyle \frac{{(1-|b|}^2{)}^{\beta }|u_j^{\prime }\left(b\right)+u_{j-1}\left(b\right){\psi }^{\prime }\left(b\right)|j!}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^j}}\hfill \\ & -& \sum _{i=j+1}^{k+1}{\displaystyle \frac{C\parallel f_{j,\psi \left(b\right)}{\parallel }_{A_{\omega }^p}{(1-|b|}^2{)}^{\beta }|u_i^{\prime }\left(b\right)+u_{i-1}\left(b\right){\psi }^{\prime }\left(b\right)|}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^i}}.\hfill \end{array}$$

(10)

Thus, by (5), (7) and (10), we have

$$\begin{array}{ccc}\hfill M_j& \lesssim & \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}+C\sum _{i=j+1}^{k+1}\underset{b\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|b|}^2{)}^{\beta }|u_i^{\prime }\left(b\right)+u_{i-1}\left(b\right){\psi }^{\prime }\left(b\right)|}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^i}}\hfill \\ & \lesssim & \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}.\hfill \end{array}$$

(11)

Therefore, $M_j<\infty ,\mathrm{for}\mathrm{all}j=1,\cdots ,k+1$.

Finally, we show that $M_0<\infty$. It is easy to see that

$$\begin{array}{c}\hfill |f_{0,\psi \left(b\right)}\left(\psi \left(b\right)\right)|={\displaystyle \frac{1}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}}}.\end{array}$$

(12)

Using Lemma 1 and (12), we have

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge {\parallel P_{U,\psi }^k{f}_{0,\psi \left(b\right)}\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & {(1-|b|}^2{)}^{\beta }|u_0^{\prime }\left(b\right)f_{0,\psi \left(b\right)}\left(\psi \left(b\right)\right)+\sum _{j=1}^{k+1}\left(u_j^{\prime }\left(b\right)+u_{j-1}\left(b\right){\psi }^{\prime }\left(b\right)\right)f_{0,\psi \left(b\right)}^{\left(j\right)}\left(\psi \left(b\right)\right)|\hfill \\ & \ge & {\displaystyle \frac{{(1-|b|}^2{)}^{\beta }\left|u_0^{\prime }\left(b\right)\right|}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}}}-\sum _{j=1}^{k+1}{\displaystyle \frac{C\parallel f_{0,\psi \left(b\right)}{\parallel }_{A_{\omega }^p}{(1-|b|}^2{)}^{\beta }|u_j^{\prime }\left(b\right)+u_{j-1}\left(b\right){\psi }^{\prime }\left(b\right)|}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^j}}.\hfill \end{array}$$

(13)

Using (5), (7), (11) and (13), we have

$$\begin{array}{ccc}\hfill M_0& \lesssim & \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}+\sum _{j=1}^{k+1}\underset{b\in \mathbb{D}}{sup}{\displaystyle \frac{{(1-|b|}^2{)}^{\beta }|u_j^{\prime }\left(b\right)+u_{j-1}\left(b\right){\psi }^{\prime }\left(b\right)|}{{\left(\omega \left(S\left(\psi \left(b\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(b\right)|}^2{)}^j}}\hfill \\ & \lesssim & \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }},\hfill \end{array}$$

(14)

as desired. From the above proof, we obtain

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\approx \sum _{i=0}^{k+1}{M}_i,\end{array}$$

as desired. The proof of the theorem is now complete. □

According to the proof of Theorem 1, we easily obtain the following result.

Corollary 1.

Let $\beta >0$, $k\in {\mathbb{N}}_0$, $1<p<\infty$, $\omega \in \widehat{\mathcal{D}}$, $\psi \in S\left(\mathbb{D}\right)$ and U denote the sequence $\{u_0,u_1,···,u_k\}$ such that $u_j\in H\left(\mathbb{D}\right)$. Then, $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded if, and only if, each $D_{u_n,\psi }^n:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded for every $0\le n\le k$.

3. Essential Norm and Compactness of ${\mathit{P}}_{\mathit{U},\psi }^{\mathit{k}}:{\mathit{A}}_{\omega }^{\mathit{p}}\to {\mathcal{B}}^{\beta }$

In this part of the work, we provide a calculation for the essential norm of $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$. To accomplish this goal, we initially introduce a series of lemmas that will be employed later in the verification of the main results in this section.

Given that $\omega \in \widehat{\mathcal{D}}$ and $p\in (1,\infty )$, it is deduced from Theorem 7 in [22], that there is an isomorphism ${\left(A_{\omega }^p\right)}^{*}\simeq A_{\omega }^{{\displaystyle \frac{p}{p-1}}}$. Applying Lemma 2.1 in [23], we obtain the following lemma.

Lemma 2.

Suppose $0<\beta <\infty$, $1<p<\infty$, $\omega \in \widehat{\mathcal{D}}$ such that $K:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded. Then, K is compact if, and only if, $\parallel K{h}_k{\parallel }_{{\mathcal{B}}^{\beta }}\to 0$ as $k\to \infty$ whenever $\left\{h_k\right\}$ is bounded in $A_{\omega }^p$ and uniformly converges to 0 on any compact subset of $\mathbb{D}$ as $k\to \infty$.

Lemma 3.

Let $0<\beta <\infty$, $k\in {\mathbb{N}}_0$, $1<p<\infty$, $\omega \in \widehat{\mathcal{D}}$, U denote the sequence $\{u_0,u_1,···,u_k\}$ such that $u_j\in H\left(\mathbb{D}\right)$ and $\psi \in S\left(\mathbb{D}\right)$ with ${\parallel \psi \parallel }_{\infty }<1$ such that $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded. Then, $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is compact.

Proof. 

Assume that $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded. By Theorem 1, we obtain

$${\Phi }_j:=\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|<\infty ,\mathrm{for}\mathrm{all}j=0,1,\cdots ,k+1.$$

Let ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ be a bounded sequence in $A_{\omega }^p$ such that $h_n\to 0$ uniformly on the compact subsets of $\mathbb{D}$ as $n\to \infty$. Cauchy’s estimates imply that $h_n^{\left(j\right)}\to 0,j=0,1,\cdots ,k+1$, uniformly on the compact subsets of $\mathbb{D}$ as $n\to \infty$. Therefore, by the fact that $\Omega =\{z:|z|\le \parallel \psi {\parallel }_{\infty }\}$ is a compact subset of $\mathbb{D}$, we have

$$\begin{array}{c}\hfill \underset{w\in \mathbb{D}}{sup}\left|h_n^{\left(j\right)}\left(\psi \left(w\right)\right)\right|\to 0\mathrm{as}n\to \infty .\end{array}$$

(15)

For $i=0,1,\cdots ,k+1$, since ${\left\{h_n^{\left(i\right)}\right\}}_{n\in \mathbb{N}}$ converges to zero uniformly on compact subsets of $\mathbb{D}$, we obtain that $|P_{U,\psi }^k{h}_n\left(0\right)|\to 0$ as $n\to \infty$. Using (15), we have

$$\begin{array}{ccc}\hfill \parallel P_{U,\psi }^k{h}_n{\parallel }_{{\mathcal{B}}^{\beta }}& =& |P_{U,\psi }^k{h}_n\left(0\right)|+\sum _{j=0}^{k+1}\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)\left|\right|h_n^{\left(j\right)}\left(\psi \left(w\right)\right)|\hfill \\ & \le & |P_{U,\psi }^k{h}_n\left(0\right)|+\sum _{j=0}^{k+1}{\Phi }_j\underset{w\in \mathbb{D}}{sup}\left|h_n^{\left(j\right)}\left(\psi \left(w\right)\right)\right|\to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

This proves that $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is compact. □

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

$$\begin{array}{ccc}& & B_j=\underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_{j-1}\left(w\right){\psi }^{\prime }\left(z\right)+u_j^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}},\mathrm{for}j=0,1,\cdots ,k+1;\hfill \end{array}$$

and

$$\begin{array}{ccc}& & B_{j,n}={\displaystyle \frac{(1-|w_n{|}^2{)}^{\beta }|u_{j-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)+u_j^{\prime }\left(w_n\right)|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^j}},\mathrm{for}j=0,1,\cdots ,k+1;\hfill \end{array}$$

where ${\left\{w_n\right\}}_{n\in \mathbb{N}}$ is a sequence in $\mathbb{D}$.

Theorem 2.

Let $k\in {\mathbb{N}}_0$, $0<\beta <\infty$, $1<p<\infty$, $\omega \in \widehat{\mathcal{D}}$, $\psi \in S\left(\mathbb{D}\right)$ and U denote the sequence $\{u_0,u_1,···,u_k\}$ such that $u_j\in H\left(\mathbb{D}\right)$. If $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded, then

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\approx \underset{0\le i\le k+1}{max}{B}_i\approx \sum _{i=0}^{k+1}{B}_i.\end{array}$$

Proof. 

When the norm of $\psi$ is strictly less than one, i.e., ${\parallel \psi \parallel }_{\infty }<1$, it is straightforward to demonstrate that the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is compact, as established by Lemma 3. In such a case, the asymptotic relations are trivially satisfied.

We now shift our focus to the case where the norm of $\psi$ equals one, i.e., ${\parallel \psi \parallel }_{\infty }=1$. Let $\left\{h_i\right\}$ be a sequence of functions within $A_{\omega }^p$ that converges uniformly to zero on all compact subsets as i tends to infinity. Given any compact operator $\mathcal{K}:A_{\omega }^p\to {\mathcal{B}}^{\beta }$, it follows from Lemma 2 that the limit of the norm of $\mathcal{K}{h}_i$ in ${\mathcal{B}}^{\beta }$ as i approaches infinity is zero; that is, ${lim}_{i\to \infty }{\parallel \mathcal{K}{h}_i\parallel }_{{\mathcal{B}}^{\beta }}=0$. Consequently,

$$\begin{array}{ccc}\hfill \parallel P_{U,\psi }^k{-\mathcal{K}\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}& \gtrsim & \underset{i\to \infty }{lim\; sup}{\parallel (P_{U,\psi }^k-\mathcal{K})h_i\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & \underset{i\to \infty }{lim\; sup}\parallel P_{U,\psi }^k{h}_i{\parallel }_{{\mathcal{B}}^{\beta }}-\underset{i\to \infty }{lim\; sup}{\parallel \mathcal{K}{h}_i\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & =& \underset{i\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{h}_i\parallel }_{{\mathcal{B}}^{\beta }}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}=\underset{\mathcal{K}}{inf}\parallel P_{U,\psi }^k{-\mathcal{K}\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\gtrsim \underset{i\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{h}_i\parallel }_{{\mathcal{B}}^{\beta }}.\end{array}$$

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

$$\begin{array}{c}\hfill B_{k+1}=\underset{n\to \infty }{lim\; sup}{B}_{k+1,n}.\end{array}$$

(16)

For each n, $f_{k+1,\psi \left(w_n\right)}^{\left(i\right)}\left(\psi \left(w_n\right)\right)=0$ for all $i=0,1,\cdots ,k$ and

$$\begin{array}{c}\hfill \left|f_{k+1,\psi \left(w_n\right)}^{(k+1)}\left(\psi \left(w_n\right)\right)\right|={\displaystyle \frac{(k+1)!}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^{k+1}}}.\end{array}$$

(17)

Here, $f_{j,b}$ is defined in (3). It is obvious that $f_{k+1,\psi \left(w_n\right)}\in A_{\omega }^p$ for any $n\in \mathbb{N}$ and $f_{k+1,\psi \left(w_n\right)}\to 0$ uniformly on each compact subset of $\mathbb{D}$ as $n\to \infty$. Hence,

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{f}_{k+1,\psi \left(w_n\right)}\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}(1-|w_n{{|}^2)}^{\beta }\left|\sum _{j=0}^k\left(u_j^{\prime }\left(w_n\right)f_{k+1,\psi \left(w_n\right)}^{\left(j\right)}\left(\psi \left(w_n\right)\right)+u_j\left(w_n\right){\psi }^{\prime }\left(w_n\right)f_{k+1,\psi \left(w_n\right)}^{(j+1)}\left(\psi \left(w_n\right)\right)\right)\right|\hfill \\ & =& \underset{n\to \infty }{lim\; sup}(k+1)!B_{k+1,n}\gtrsim B_{k+1}.\hfill \end{array}$$

(18)

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

$$\begin{array}{c}\hfill B_k=\underset{n\to \infty }{lim\; sup}{B}_{k,n}.\end{array}$$

(19)

For each n, $f_{k,\psi \left(w_n\right)}^{\left(i\right)}\left(\psi \left(w_n\right)\right)=0$ for every $i=0,1,\cdots ,k-1$ and

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

(20)

Here, $f_{j,b}$ is defined in (3). Similarly, we obtain

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{f}_{k,\psi \left(w_n\right)}\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}(1-|w_n{{|}^2)}^{\beta }\left|\sum _{j=0}^k\left(u_j^{\prime }\left(w_n\right)f_{k,\psi \left(w_n\right)}^{\left(j\right)}\left(\psi \left(w_n\right)\right)+u_j\left(w_n\right){\psi }^{\prime }\left(w_n\right)f_{k,\psi \left(w_n\right)}^{(j+1)}\left(\psi \left(w_n\right)\right)\right)\right|\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}(1-|w_n{{|}^2)}^{\beta }|u_k^{\prime }\left(w_n\right)+u_{k-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)|\left|f_{k,\psi \left(w_n\right)}^{\left(k\right)}\left(\psi \left(w_n\right)\right)\right|\hfill \\ & & -\underset{n\to \infty }{lim\; sup}(1-|w_n{{|}^2)}^{\beta }\left|u_k\left(w_n\right){\psi }^{\prime }\left(w_n\right)\right|\left|f_{k,\psi \left(w_n\right)}^{(k+1)}\left(\psi \left(w_n\right)\right)\right|\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{k!(1-|w_n{{|}^2)}^{\beta }|u_k^{\prime }\left(w_n\right)+u_{k-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^k}}\hfill \\ & & -\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{C\parallel f_{k,\psi \left(w_n\right)}{\parallel }_{A_{\omega }^p}(1-|w_n{|}^2{)}^{\beta }\left|u_k\left(w_n\right){\psi }^{\prime }\left(w_n\right)\right|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^{k+1}}},\hfill \end{array}$$

(21)

which implies that

$$\begin{array}{c}\hfill C\parallel f_{k,\psi \left(w_n\right)}{\parallel }_{A_{\omega }^p}{B}_{k+1}+{\parallel P_{U,\psi }^k\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge \underset{n\to \infty }{lim\; sup}k!B_{k,n}.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge B_k.\end{array}$$

(22)

Now, we fix $j\in [1,k-1]$ and suppose that

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\gtrsim B_i,\end{array}$$

(23)

for every $i=j+1,\cdots ,k$. We will prove that (23) holds for $i=j$. For this purpose, let ${\left\{w_n\right\}}_{n\in \mathbb{N}}\subset \mathbb{D}$ with $\left|\psi \right(w_n\left)\right|\to 1$ as $n\to \infty$ for which

$$\begin{array}{c}\hfill B_j=\underset{n\to \infty }{lim\; sup}{B}_{j,n}.\end{array}$$

(24)

We see that $f_{j,\psi \left(w_n\right)}^{\left(i\right)}\left(\psi \left(w_n\right)\right)=0$ for each n and $i=0,1,\cdots ,j-1$. Moreover,

$$\begin{array}{c}\hfill |f_{j,\psi \left(w_n\right)}^{\left(j\right)}\left(\psi \left(w_n\right)\right)|={\displaystyle \frac{j!}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^j}},\end{array}$$

(25)

${\left\{f_{j,\psi \left(w_n\right)}\right\}}_{n\in \mathbb{N}}$ is bounded in $A_{\omega }^p$ and converges to zero uniformly on compact subsets of $\mathbb{D}$. Similarly,

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{f}_{j,\psi \left(w_n\right)}\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{j!(1-|w_n{|}^2{)}^{\beta }|u_j^{\prime }\left(w_n\right)+u_{j-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)|\left|\psi {\left(w_n\right)}^n\right|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^j}}\hfill \\ & & -C\sum _{i=j+1}^{k+1}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\parallel f_{j,\psi \left(w_n\right)}{\parallel }_{A_{\omega }^p}(1-|w_n{|}^2{)}^{\beta }|u_i^{\prime }\left(w_n\right)+u_{i-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^i}}.\hfill \end{array}$$

(26)

Since $\left|\psi \right(w_n\left)\right|\to 1$ as $n\to \infty$, from (19), (24) and (26), we obtain

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\gtrsim \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{f}_{j,\psi \left(w_n\right)}\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{j!(1-|w_n{{|}^2)}^{\beta }|u_j^{\prime }\left(w_n\right)+u_{j-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^j}}\hfill \\ & & -C\sum _{i=j+1}^{k+1}{\parallel f_{j,\psi \left(w_n\right)}\parallel }_{A_{\omega }^p}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{(1-|w_n{|}^2{)}^{\beta }|u_i^{\prime }\left(w_n\right)+u_{i-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^i}}\hfill \\ & \gtrsim & \underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{j!(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}}\hfill \\ & & -\sum _{i=j+1}^{k+1}\underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_i^{\prime }\left(w\right)+u_{i-1}\left(w\right){\psi }^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^i}}.\hfill \end{array}$$

Thus, applying (23),

$$\begin{array}{c}\hfill \underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}}\lesssim {\parallel P_{U,\psi }^k\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}.\end{array}$$

Therefore, for all $j=1,2,\cdots ,k$,

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\gtrsim \underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}}=B_j.\end{array}$$

(27)

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

$$\begin{array}{c}\hfill B_0=\underset{n\to \infty }{lim\; sup}{B}_{0,n}.\end{array}$$

(28)

It is easy to check that ${\left\{f_{0,\psi \left(w_n\right)}\right\}}_{n\in \mathbb{N}}$ is bounded in $A_{\omega }^p$ and converges to zero uniformly on compact subsets of $\mathbb{D}$. Moreover,

$$\begin{array}{c}\hfill |f_{0,\psi \left(w_n\right)}\left(\psi \left(w_n\right)\right)|={\displaystyle \frac{1}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}}}.\end{array}$$

(29)

So, by using Lemma 1 and (29), we obtain

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{f}_{0,\psi \left(w_n\right)}\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}(1-|w_n{{|}^2)}^{\beta }\left|\sum _{j=0}^k\left(u_j^{\prime }\left(w_n\right)f_{0,\psi \left(w_n\right)}^{\left(j\right)}\left(\psi \left(w_n\right)\right)+u_j\left(w_n\right){\psi }^{\prime }\left(w_n\right)f_{0,\psi \left(w_n\right)}^{(j+1)}\left(\psi \left(w_n\right)\right)\right)\right|\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}(1-|w_n{|}^2{)}^{\beta }|u_0^{\prime }\left(w_n\right)\left|\right|f_{0,\psi \left(w_n\right)}\left(\psi \left(w_n\right)\right)|\hfill \\ & & -\sum _{j=1}^{k+1}\underset{n\to \infty }{lim\; sup}(1-|w_n{{|}^2)}^{\beta }|u_j^{\prime }\left(w_n\right)+u_{j-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)||f_{0,\psi \left(w_n\right)}^{\left(j\right)}\left(\psi \left(w_n\right)\right)|\hfill \\ & \ge & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{(1-|w_n{|}^2{)}^{\beta }\left|u_0^{\prime }\left(w_n\right)\right|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}}}\hfill \\ & & -C\parallel f_{0,\psi \left(w_n\right)}{\parallel }_{A_{\omega }^p}\sum _{j=1}^{k+1}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{(1-|w_n{|}^2{)}^{\beta }|u_j^{\prime }\left(w_n\right)+u_{j-1}\left(w_n\right){\psi }^{\prime }\left(w_n\right)|}{{\left(\omega \left(S\left(\psi \left(w_n\right)\right)\right)\right)}^{1/p}(1-|\psi \left(w_n\right){{|}^2)}^j}}.\hfill \end{array}$$

(30)

Since $\left|\psi \right(w_n\left)\right|\to 1$ as $n\to \infty$, from (19), (24), (28) and (30) we obtain

$$\begin{array}{ccc}& & \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\ge \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k{f}_{0,\psi \left(w_n\right)}\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & \ge & \underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }\left|u_0^{\prime }\left(w\right)\right|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}}}-C\sum _{j=1}^{k+1}\underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(z\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}},\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\gtrsim B_0.\end{array}$$

(31)

Therefore, (27) and (31) imply that

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\gtrsim \underset{0\le i\le k+1}{max}{B}_i\approx \sum _{i=0}^{k+1}{B}_i.\end{array}$$

Finally, we prove that

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\lesssim \underset{0\le i\le k+1}{max}{B}_i\approx \sum _{i=0}^{k+1}{B}_i.\end{array}$$

For $\theta \in [0,1)$, set

$$\left({\mathcal{K}}_{\theta }f\right)\left(w\right)=f_{\theta }\left(w\right)=f\left(\theta w\right),f\in H\left(\mathbb{D}\right).$$

It is evident that $f_{\theta }$ uniformly converges to f on all compact subsets of $\mathbb{D}$ as $\theta \to 1$. Additionally, ${\mathcal{K}}_{\theta }:A_{\omega }^p\to A_{\omega }^p$ is compact, and $\parallel {\mathcal{K}}_{\theta }{\parallel }_{A_{\omega }^p\to A_{\omega }^p}\le 1$. Consider a sequence $\left\{{\theta }_n\right\}\subset (0,1)$ that approaches 1 as $n\to \infty$. For each $n\in \mathbb{N}$, the operator $P_{U,\psi }^k{\mathcal{K}}_{{\theta }_n}:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is compact. By the definition of the essential norm, we obtain

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\le \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k-P_{U,\psi }^k{\mathcal{K}}_{{\theta }_n}\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}.\end{array}$$

(32)

Therefore, we only need to prove that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k-P_{U,\psi }^k{\mathcal{K}}_{{\theta }_n}\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\lesssim \underset{0\le i\le k+1}{max}{B}_i.\end{array}$$

Let $f\in A_{\omega }^p$ such that ${\parallel f\parallel }_{A_{\omega }^p}\le 1$. We consider

$$\begin{array}{ccc}& & \parallel (P_{U,\psi }^k-P_{U,\psi }^k{\mathcal{K}}_{{\theta }_n}){f\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & =& \left|\sum _{j=0}^k\left(u_j\left(0\right)f^{\left(j\right)}\left(\psi \left(0\right)\right)-u_j\left(0\right)f_{{\theta }_n}^{\left(j\right)}\left(\psi \left(0\right)\right)\right)\right|+\parallel \sum _{j=0}^k{u}_j(f^{\left(j\right)}-f_{{\theta }_n}^{\left(j\right)})\circ \psi \parallel .\hfill \end{array}$$

(33)

It is clear that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}\left|\sum _{j=0}^k\left(u_j\left(0\right)f^{\left(j\right)}\left(\psi \left(0\right)\right)-u_j\left(0\right)f_{{\theta }_n}^{\left(j\right)}\left(\psi \left(0\right)\right)\right)\right|=0.\end{array}$$

(34)

Consider

$$\begin{array}{ccc}& & \underset{n\to \infty }{lim\; sup}\parallel \sum _{j=0}^k{u}_j(f^{\left(j\right)}-f_{{\theta }_n}^{\left(j\right)})\circ \psi \parallel \hfill \\ & =& \underset{n\to \infty }{lim\; sup}\underset{w\in \mathbb{D}}{sup}{(1-|w|}^2{)}^{\beta }\left|\sum _{j=0}^{k+1}\left(u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)\right)\left(f^{\left(j\right)}-f_{{\theta }_n}^{\left(j\right)}\right)\left(\psi \left(w\right)\right)\right|\hfill \\ & \lesssim & Q_3+Q_4.\hfill \end{array}$$

(35)

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

$$\begin{array}{c}\hfill Q_3:=\underset{n\to \infty }{lim\; sup}\underset{\left|\psi \left(w\right)\right|\le {\theta }_n}{sup}{(1-|w|}^2{)}^{\beta }\sum _{j=0}^{k+1}\left|\left(f^{\left(j\right)}-f_{{\theta }_n}^{\left(j\right)}\right)\left(\psi \left(w\right)\right)\right||u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|,\end{array}$$

$$\begin{array}{c}\hfill Q_4:=\underset{n\to \infty }{lim\; sup}\underset{\left|\psi \left(w\right)\right|>{\theta }_n}{sup}{(1-|w|}^2{)}^{\beta }\sum _{j=0}^{k+1}\left|\left(f^{\left(j\right)}-f_{{\theta }_n}^{\left(j\right)}\right)\left(\psi \left(w\right)\right)\right||u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|.\end{array}$$

Since $f_{{\theta }_n}^{\left(i\right)}\to f^{\left(i\right)}(i=0,1,\cdots ,k+1)$ uniformly on compact subsets of $\mathbb{D}$ as $n\to \infty$, we have

$$\begin{array}{c}\hfill Q_3\le \sum _{j=0}^{k+1}{\Phi }_j\underset{n\to \infty }{lim\; sup}\underset{\left|w\right|\le {\theta }_n}{sup}|f^{\left(j\right)}\left(w\right)-f_{{\theta }_n}^{\left(j\right)}\left(w\right)|=0,\end{array}$$

(36)

where

$${\Phi }_j:=\underset{z\in \mathbb{D}}{sup}{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|<\infty ,\mathrm{for}\mathrm{all}j=0,1,2,\cdots ,k+1.$$

For $Q_4$, we obtain $Q_4\le {lim\; sup}_{n\to \infty }{Q}_{41},$ where

$$\begin{array}{c}\hfill Q_{41}:=\sum _{j=0}^{k+1}\underset{\left|\psi \left(w\right)\right|>{\theta }_n}{sup}{(1-|w|}^2{)}^{\beta }\left|\left(f^{\left(j\right)}-f_{{\theta }_n}^{\left(j\right)}\right)\left(\psi \left(w\right)\right)\right||u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|.\end{array}$$

Using the fact that ${\parallel f\parallel }_{A_{\omega }^p}\le 1$ and Lemma 1, we obtain

$$\begin{array}{ccc}& & \underset{\left|\psi \left(w\right)\right|>{\theta }_n}{sup}{(1-|w|}^2{)}^{\beta }\left|\left(f^{\left(k\right)}-f_{{\theta }_n}^{\left(k\right)}\right)\left(\psi \left(w\right)\right)\right||u_k^{\prime }\left(w\right)+u_{k-1}\left(w\right){\psi }^{\prime }\left(w\right)|\hfill \\ & \le & {\displaystyle \frac{C\parallel f-f_{{\theta }_n}{\parallel }_{A_{\omega }^p}}{k!}}\underset{\left|\psi \left(w\right)\right|>{\theta }_n}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_k^{\prime }\left(w\right)+u_{k-1}\left(w\right){\psi }^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^k}}.\hfill \end{array}$$

(37)

After a calculation, we have

$$\begin{array}{ccc}\hfill Q_{41}& \le & \sum _{j=0}^{k+1}C{\parallel f-f_{{\theta }_n}\parallel }_{A_{\omega }^p}\underset{\left|\psi \left(w\right)\right|>{\theta }_n}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(z\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}}.\hfill \end{array}$$

(38)

From (38), we see that

$$\begin{array}{ccc}\hfill Q_4=\underset{n\to \infty }{lim\; sup}{Q}_{41}& \lesssim & \underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}\sum _{j=1}^k{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_j^{\prime }\left(w\right)+u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}}\hfill \\ & \lesssim & \sum _{j=0}^{k+1}{B}_i\lesssim \underset{0\le i\le k+1}{max}{B}_i.\hfill \end{array}$$

(39)

Hence, by (33)–(36) and (39), we have

$$\begin{array}{ccc}& & \underset{n\to \infty }{lim\; sup}{\parallel P_{U,\psi }^k-P_{U,\psi }^k{\mathcal{K}}_{{\theta }_n}\parallel }_{A_{\omega }^p\to {\mathcal{B}}^{\beta }}\hfill \\ & =& \underset{n\to \infty }{lim\; sup}\underset{{\parallel f\parallel }_{A_{\omega }^p}\le 1}{sup}{\parallel (P_{U,\psi }^k-P_{U,\psi }^k{\mathcal{K}}_{{\theta }_n})f\parallel }_{{\mathcal{B}}^{\beta }}\hfill \\ & =& \underset{n\to \infty }{lim\; sup}\underset{{\parallel f\parallel }_{A_{\omega }^p}\le 1}{sup}\parallel \sum _{j=0}^k{u}_j(f^{\left(j\right)}-f_{{\theta }_n}^{\left(j\right)})\circ \psi \parallel \hfill \\ & \lesssim & \sum _{j=0}^{k+1}{B}_i\lesssim \underset{0\le i\le k+1}{max}{B}_i.\hfill \end{array}$$

(40)

Hence, by (32) and (40), we have

$$\begin{array}{c}\hfill \parallel P_{U,\psi }^k{\parallel }_{e,A_{\omega }^p\to {\mathcal{B}}^{\beta }}\lesssim \sum _{j=0}^{k+1}{B}_i\lesssim \underset{0\le i\le k+1}{max}{B}_i.\end{array}$$

We have thus proved the theorem. □

From the proof of Theorem 2, we easily obtain the following corollary.

Corollary 2.

Let $\beta >0$, $k\in {\mathbb{N}}_0$, $1<p<\infty ,$$\omega \in \widehat{\mathcal{D}}$, $\psi \in S\left(\mathbb{D}\right)$ and U denote the sequence $\{u_0,u_1,···,u_k\}$ such that $u_j\in H\left(\mathbb{D}\right)$. If $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is bounded, then the following conditions are equivalent:

(i) $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is compact.

(ii)

$$\begin{array}{c}\hfill \underset{r\to 1}{lim}\underset{\left|\psi \right(w\left)\right|>r}{sup}{\displaystyle \frac{{(1-|w|}^2{)}^{\beta }|u_{j-1}\left(w\right){\psi }^{\prime }\left(w\right)+u_j^{\prime }\left(w\right)|}{{\left(\omega \left(S\left(\psi \left(w\right)\right)\right)\right)}^{1/p}{(1-|\psi \left(w\right)|}^2{)}^j}}=0,\mathrm{for}j=0,1,\cdots ,k+1.\end{array}$$

(iii) $D_{u_n,\psi }^n:A_{\omega }^p\to {\mathcal{B}}^{\beta }$ is compact for every $0\le n\le k$.

4. Conclusions

In this manuscript, we delve into the analysis of the boundedness, compactness and essential norm associated with the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$, offering a variety of characterizations for these attributes. Our methodologies draw inspiration from the studies in [14,15], which focus on generalized weighted composition operators, and [21], which examines product-type operators from $A_{\omega }^p$ spaces to Bloch-type spaces. We integrate the techniques presented in these three scholarly works. Our proof provides a more exhaustive explanation compared to that found in [17]. Ultimately, our findings extend numerous results found in the existing literature, as referenced in [21]. Additionally, we illustrate a rigidity property of $P_{U,\psi }^k$, highlighting that its boundedness and compactness are directly equivalent to those of each $D_{u_n,\psi }^n$, for $0\le n\le k$. Moreover, we furnish a computation for the essential norm of the operator $P_{U,\psi }^k:A_{\omega }^p\to {\mathcal{B}}^{\beta }$.