Essential Norm of Products of Volterra-Type Operators and Composition Operators on Iterated Banach-Type Spaces

Abstract

Let $H\left(\mathbb{D}\right)$ be the set of analytic functions on the open unit disk $\mathbb{D}$. For $n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$, define the iterated weighted-type Banach space ${\mathcal{V}}_n:=\left\{f\in H\left(\mathbb{D}\right):{sup}_{z\in \mathbb{D}}{(1-|z|}^2\left)\right|f^{\left(n\right)}\left(z\right)|<\infty \right\}.$ In this work, we study the boundedness and the essential norm of products of Volterra-type operators and composition operators on iterated weighted-type Banach spaces.

1. Introduction

Let $H\left(\mathbb{D}\right)$ denote the set of analytic functions on the open unit disk $\mathbb{D}$, and let $S\left(\mathbb{D}\right)$ represent the set of analytic self-maps of $\mathbb{D}$. Given $\phi \in S\left(\mathbb{D}\right)$, the composition operator is defined by

$$C_{\phi }f=f\circ \phi ,\forall f\in H\left(\mathbb{D}\right).$$

This operator has been extensively studied. For a comprehensive treatment of the theory of composition operators on function spaces, we refer the reader to the books [1,2].

For $\psi \in H\left(\mathbb{D}\right)$, the Volterra-type operator is defined by

$$V_{\psi }f\left(z\right)={\int }_0^{z}f\left(w\right){\psi }^{\prime }\left(w\right)dw,$$

introduced by Pommerenke in [3]. In particular, he showed that $V_{\psi }$ is bounded on $H^2$ if and only if $\psi \in \mathrm{BMOA}$.

Volterra-type operators have garnered significant attention following the foundational work of Aleman and Siskakis [4,5] on Hardy and Bergman spaces. These operators have important theoretical applications in the analysis of integral equations and semigroup dynamics within complex function theory [5]. Although the term Volterra-type also appears in applied fields such as nonlinear image processing and machine learning, those contexts refer to Volterra series-nonlinear integral models distinct from the analytic operators considered here. Such series are used to model phenomena involving memory and higher-order interactions, as in nonlinear system identification and signal prediction [6] and in recurrent neural networks for speech recognition, natural language processing, and time-series forecasting [7]. In contrast, the Volterra-type operators studied in this work are firmly rooted in operator theory, where they are employed to characterize boundedness, compactness, and spectral properties on spaces of analytic functions.

Given $\psi \in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$, the product of Volterra and composition operators on $H\left(\mathbb{D}\right)$ was defined by Li and Stević in [8,9,10,11] as follows:

$$C_{\phi }{V}_{\psi }f\left(z\right)={\int }_0^{\phi \left(z\right)}f\left(w\right){\psi }^{\prime }\left(w\right)dw,\mathrm{and}{V}_{\psi }{C}_{\phi }f\left(z\right)={\int }_0^{z}f\left(\phi \left(w\right)\right){\psi }^{\prime }\left(w\right)dw.$$

(1)

Notably, if $\psi$ is the identity map $id$, the operator $V_{\psi }{C}_{\phi }$ becomes the linear operator:

$$I_{\phi }f\left(z\right):=V_{id}{C}_{\phi }f\left(z\right)={\int }_0^z(f\circ \phi )\left(w\right))dw.$$

For $n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$, the iterated weighted-type Banach space is defined as follows:

$${\mathcal{V}}_n:=\left\{f\in H\left(\mathbb{D}\right):\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|f^{\left(n\right)}\left(z\right)|<\infty \right\}.$$

We adopt the convention that $f^{\left(0\right)}=f$.

When $n=0$, 1, and 2, the corresponding function spaces are the growth space ${\mathcal{A}}^{-1}$, the Bloch space $\mathcal{B}$, and the Zygmund space $\mathcal{Z}$, respectively.

As shown in [12], for each $n\in {\mathbb{N}}_0$, the map

$$f\mapsto {\parallel f\parallel }_{s{\mathcal{V}}_n}:=\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|f^{\left(n\right)}\left(z\right)|$$

defines a seminorm on ${\mathcal{V}}_n$. The space becomes a Banach space under the norm

$${\parallel f\parallel }_{{\mathcal{V}}_n}:=\sum _{k=0}^{n-1}|f^{\left(k\right)}{\left(0\right)|+\parallel f\parallel }_{s{\mathcal{V}}_n}.$$

For $n\in \mathbb{N}$ and $k=1,\dots ,n-1$, we have $f\in {\mathcal{V}}_n$ if and only if $f^{\left(k\right)}\in {\mathcal{V}}_{n-k}$, and the norm satisfies

$${\parallel f\parallel }_{{\mathcal{V}}_n}=\sum _{j=0}^{k-1}|f^{\left(j\right)}\left(0\right)|+\parallel f^{\left(k\right)}{\parallel }_{{\mathcal{V}}_{n-k}}.$$

(2)

According to Theorem 1 in [13], ${\mathcal{V}}_n$ is the dual of the Hardy space $H^{1/n}$ for all $n\ge 2$. As shown in [14], the sequence of spaces ${\mathcal{V}}_n$ is nested, and the inclusion ${\mathcal{V}}_n\hookrightarrow {\mathcal{V}}_{n-1}$ is continuous with an operator norm not exceeding 1. Since the Zygmund space ${\mathcal{V}}_2$ is contained in the disk algebra, this holds for all ${\mathcal{V}}_n$ with $n\ge 2$.

Additionally, for $n\ge 3$, ${\mathcal{V}}_n$ is an algebra, and functions in ${\mathcal{V}}_n$ satisfy the following pointwise estimates:

$$\left|f\left(z\right)\right|\le \left\{\begin{array}{cc}{\displaystyle \frac{1}{1-{\left|z\right|}^2}}{\parallel f\parallel }_{{\mathcal{V}}_n},\hfill & \mathrm{if}n=0,\hfill \\ max\left\{1,{\displaystyle \frac{1}{2}}log{\displaystyle \frac{1+\left|z\right|}{1-\left|z\right|}}\right\}{\parallel f\parallel }_{{\mathcal{V}}_n},\hfill & \mathrm{if}n=1,\hfill \\ {\parallel f\parallel }_{{\mathcal{V}}_n},\hfill & \mathrm{if}n\ge 2,\hfill \end{array}\right.$$

(3)

as shown in [14].

The spaces ${\mathcal{V}}_n$ are a special case of the nth weighted-type spaces introduced by Stević [15]:

$${\mathcal{V}}_{\mu ,n}:=\left\{f\in {H\left(\mathbb{D}\right):\parallel f\parallel }_{s{\mathcal{V}}_{\mu ,n}}:=\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left|f^{\left(n\right)}\left(z\right)\right|<\infty \right\},$$

where $\mu \left(z\right)$ is a continuous positive function. The corresponding norm is

$${\parallel f\parallel }_{{\mathcal{V}}_{\mu ,n}}:=\sum _{k=0}^{n-1}|f^{\left(k\right)}{\left(0\right)|+\parallel f\parallel }_{s{\mathcal{V}}_{\mu ,n}}.$$

These spaces naturally appear in complex analysis and operator theory and include many well-known examples, such as the Hardy space $H^{\infty }$, the Bloch space, and the Zygmund space.

The Table 1 shows how different choices of the weight function $\mu \left(z\right)$ and the order n give rise to various classical and generalized function spaces.

Table 1. Examples of spaces ${\mathcal{V}}_{\mu ,n}$ for various weights $\mu$ and derivative orders n.

Space Name	Order n	Weight $\mathit{\mu }\left(\mathit{z}\right)$	Symbol
Set of bounded analytic functions on $\mathbb{D}$	0	1	$H^{\infty }$
$\alpha$-Growth space	0	${(1-|z|}^2{)}^{\alpha },\alpha >0$	${\mathcal{A}}^{-\alpha }$
Weighted Banach space	0	$\mu \left(z\right)$	$H_{\mu }^{\infty }$
Bloch space	1	$1-{\left|z\right|}^2$	$\mathcal{B}$
$\alpha$-Bloch space	1	${(1-|z|}^2{)}^{\alpha },\alpha >0$	${\mathcal{B}}_{\alpha }$
Logarithmic Bloch space	1	${(1-|z|}^2)log{\displaystyle \frac{2}{1-{\left|z\right|}^2}}$	${\mathcal{B}}_{log}$
Zygmund space	2	$1-{\left|z\right|}^2$	$\mathcal{Z}$
$\alpha$-Zygmund space	2	${(1-|z|}^2{)}^{\alpha },\alpha >0$	${\mathcal{Z}}_{\alpha }$

Table 1. Examples of spaces ${\mathcal{V}}_{\mu ,n}$ for various weights $\mu$ and derivative orders n.

Space Name	Order n	Weight $\mathit{\mu }\left(\mathit{z}\right)$	Symbol
Set of bounded analytic functions on $\mathbb{D}$	0	1	$H^{\infty }$
$\alpha$-Growth space	0	${(1-|z|}^2{)}^{\alpha },\alpha >0$	${\mathcal{A}}^{-\alpha }$
Weighted Banach space	0	$\mu \left(z\right)$	$H_{\mu }^{\infty }$
Bloch space	1	$1-{\left|z\right|}^2$	$\mathcal{B}$
$\alpha$-Bloch space	1	${(1-|z|}^2{)}^{\alpha },\alpha >0$	${\mathcal{B}}_{\alpha }$
Logarithmic Bloch space	1	${(1-|z|}^2)log{\displaystyle \frac{2}{1-{\left|z\right|}^2}}$	${\mathcal{B}}_{log}$
Zygmund space	2	$1-{\left|z\right|}^2$	$\mathcal{Z}$
$\alpha$-Zygmund space	2	${(1-|z|}^2{)}^{\alpha },\alpha >0$	${\mathcal{Z}}_{\alpha }$

Although iterated weighted-type Banach spaces are mostly studied in complex analysis, their focus on derivative bounds and smoothness has parallels with tools in numerical analysis. For example, weighted Sobolev spaces are widely used in finite element methods for controlling approximation errors [16,17].

Numerous works have explored the boundedness and compactness of Volterra-type and composition operators across weighted-type Banach spaces. For instance, Li and Stević [9] studied these operators from $H^{\infty }$ to the Bloch space ${\mathcal{V}}_1$, and further, from $H^{\infty }$ and ${\mathcal{V}}_1$ to ${\mathcal{V}}_2$ [10]. In [11], they focused on the boundedness of Bloch-type spaces, ${\mathcal{V}}_{\alpha ,1}$. Ye explored similar topics on the logarithmic Bloch space [18], and in [19], Ye and Lin analyzed essential norms in Zygmund-type spaces, ${\mathcal{V}}_{\alpha ,2}$.

Colonna and Hmidouch [14] extended these results by studying weighted composition operators on ${\mathcal{V}}_n$. Motivated by their work, we study products of Volterra-type and composition operators on ${\mathcal{V}}_n$. Specifically, in Section 2, we characterize the boundedness of such operators from ${\mathcal{V}}_n$ to ${\mathcal{V}}_{\mu ,n}$; in Section 3, we estimate their essential norms. Our objective is to advance operator theory on analytic function spaces further.

Recall that if $\mathcal{X}$ and $\mathcal{Y}$ are two Banach spaces and ${\parallel ·\parallel }_{\mathcal{X}\to \mathcal{Y}}$ denotes the operator norm, then the essential norm of a bounded linear operator $\mathcal{S}:\mathcal{X}\to \mathcal{Y}$ is its distance to the set of compact operators L mapping $\mathcal{X}$ to $\mathcal{Y}$; that is,

$${\parallel \mathcal{S}\parallel }_e={inf\{\parallel \mathcal{S}-L\parallel }_{\mathcal{X}\to \mathcal{Y}}:\mathrm{L}\mathrm{is}\mathrm{compact}\mathrm{operator}\}.$$

It is well known that ${\parallel \mathcal{S}\parallel }_e=0$ if and only if $\mathcal{S}:\mathcal{X}\to \mathcal{Y}$ is compact.

Throughout this paper, C denotes a positive constant whose value may vary between occurrences. The notation $A⪯B$ means $A\le cB$ for some $c>0$, and $A\asymp B$ means $c_{1}B\le A\le c_{2}B$ for positive constants $c_1,c_2$.

2. Boundedness of ${\mathit{V}}_{\mathit{\psi }}{\mathit{C}}_{\mathit{\phi }},{\mathit{C}}_{\mathit{\phi }}{\mathit{V}}_{\mathit{\psi }}:{\mathcal{V}}_{\mathit{n}}\to {\mathcal{V}}_{\mathit{\mu },\mathit{n}}$

In this section, we analyze the boundedness of $V_{\psi }{C}_{\phi }$ and $C_{\phi }{V}_{\psi }$ from ${\mathcal{V}}_n$ to ${\mathcal{V}}_{\mu ,n}$, and we derive an approximation of the operator norms.

We begin by recalling the following extension of Faá di Bruno’s formula:

Lemma 1

([15], Lemma 4). For $f,\psi \in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$, $z\in \mathbb{D}$,

$${\left(\psi (f\circ \phi )\right)}^{\left(n\right)}\left(z\right)=\sum _{k=0}^n{f}^{\left(k\right)}\left(\phi \left(z\right)\right)\sum _{l=k}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right){\psi }^{(n-l)}\left(z\right)B_{l,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(l-k+1)}\left(z\right)),$$

where

$$\begin{array}{c}\hfill B_{l,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(l-k+1)}\left(z\right))=\sum _{k_1,k_2,\dots ,k_l}{\displaystyle \frac{l!}{k_1!k_2!\dots k_l!}}\prod _{j=1}^l{\left({\displaystyle \frac{{\phi }^{\left(j\right)}\left(z\right)}{j!}}\right)}^{k_j},\end{array}$$

(4)

and the sum is taken over all nonnegative integers $k_1,k_2,\dots ,k_l$ such that $k=k_1+k_2+\dots +k_l$, and $k_1+2k_2+\dots +l{k}_l=l$.

Noting that ${\left(V_{\psi }{C}_{\phi }f\right)}^{\prime }\left(z\right)=f\left(\phi \left(z\right)\right){\psi }^{\prime }\left(z\right)$ and ${\left(C_{\phi }{V}_{\psi }f\right)}^{\prime }\left(z\right)=f\left(\phi \left(z\right)\right){(\psi \circ \phi )}^{\prime }\left(z\right))$. Then, for $n\in \mathbb{N}$,

$$\begin{array}{c}\hfill {\left(C_{\phi }{V}_{\psi }f\right)}^{\left(n\right)}\left(z\right)=\sum _{k=0}^{n-1}{f}^{\left(k\right)}\left(\phi \left(z\right)\right){\theta }_k\left(z\right),{\left(V_{\psi }{C}_{\phi }f\right)}^{\left(n\right)}\left(z\right)=\sum _{k=0}^{n-1}{f}^{\left(k\right)}\left(\phi \left(z\right)\right)N_k\left(z\right),\end{array}$$

(5)

where

$$\begin{array}{cc}\hfill N_k\left(z\right)& =\sum _{l=k}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{l}}\right){\psi }^{(n-l)}\left(z\right)B_{l,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(l-k+1)}\left(z\right)),\hfill \\ \hfill {\theta }_k\left(z\right)& =\sum _{l=k}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{l}}\right){\left((\psi \circ \phi )\right)}^{(n-l)}\left(z\right)B_{l,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(l-k+1)}\left(z\right))\hfill \\ \hfill & =\sum _{l=k}^{n-1}\left(\genfrac{}{}{0pt}{}{n-1}{l}\right)[\sum _{m=0}^{n-l}{\psi }^{\left(m\right)}\left(\phi \left(z\right)\right)\hfill \\ & \times B_{n-l,m}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(n-l-m+1)}\left(z\right))B_{l,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(l-k+1)}\left(z\right)){\displaystyle ]}\hfill \end{array}$$

for all $k=0,\dots ,n-1.$

Now we are ready to state the first main result of this section.

Theorem 1.

Fix $n\in \mathbb{N}$. Let $\psi \in H\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then, the operator $V_{\psi }{C}_{\phi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded if and only if

$$\begin{array}{ccc}& & \underset{z\in \mathbb{D}}{sup}\mu \left(z\right){\displaystyle [}\sum _{k=0}^{n-2}|N_k\left(z\right)|+|N_{n-1}\left(z\right)|log\frac{e}{1-{\left|\phi \left(z\right)\right|}^2}{\displaystyle ]}<\infty .\hfill \end{array}$$

Moreover, if $V_{\psi }{C}_{\phi }$ is bounded, then

$$\begin{array}{cc}\hfill \parallel V_{\psi }{C}_{\phi }\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)[{\sum }_{k=0}^{n-2}|N_k\left(z\right)|+|N_{n-1}\left(z\right)|log\frac{e}{1-{\left|\phi \left(z\right)\right|}^2}].\end{array}$$

Proof. 

Assume that $M:=\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)[{\sum }_{k=0}^{n-2}|N_k\left(z\right)|+|N_{n-1}\left(z\right)|log\frac{e}{1-{\left|\phi \left(z\right)\right|}^2}]$ is finite. Let $f\in {\mathcal{V}}_n$ with ${\parallel f\parallel }_{{\mathcal{V}}_n}=1$. Then, for $k=0,\dots ,n-2,$$f^{\left(k\right)}\in V_{n-k}$. Therefore, by (3) and (2), for $z\in \mathbb{D}$, we have

$$\begin{array}{c}\hfill |f^{\left(k\right)}\left(\phi \left(z\right)\right)|\le \parallel f^{\left(k\right)}{\parallel }_{V_{n-k}}\le 1.\end{array}$$

(6)

Moreover, $f^{(n-1)}\in {\mathcal{V}}_1$; thus, again by (3) and using (2), we have

$$\begin{array}{c}\hfill |f^{(n-1)}\left(\phi \left(z\right)\right)|\le \parallel f^{(n-1)}{\parallel }_{{\mathcal{V}}_1}log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\le {\parallel f\parallel }_{{\mathcal{V}}_n}log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}=log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}.\end{array}$$

(7)

Therefore, by (5)–(7), we obtain

$$\begin{array}{cc}\hfill \mu \left(z\right)|{\left(V_{\psi }{C}_{\phi }f\right)}^{\left(n\right)}\left(z\right)|& =\mu \left(z\right)\left|\sum _{k=0}^{n-1}{f}^{\left(k\right)}\left(\phi \left(z\right)\right)N_k\left(z\right)\right|\hfill \\ & \le \mu \left(z\right)\sum _{k=0}^{n-2}|f^{\left(k\right)}\left(\phi \left(z\right)\right)\left|\right|N_k\left(z\right)|+|f^{(n-1)}\left(\phi \left(z\right)\right)\left|\right|N_{n-1}\left(z\right)|\hfill \\ & \le \mu \left(z\right)\left[\sum _{k=0}^{n-2}|N_k\left(z\right)|+|N_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right].\hfill \end{array}$$

Taking the supremum over all z in $\mathbb{D}$, we obtain

$$\parallel V_{\psi }{C}_{\phi }{f\parallel }_{s{\mathcal{V}}_{\mu ,n}}\le \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[\sum _{k=0}^{n-2}|N_k\left(z\right)|+|N_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right].$$

(8)

Noting that $\left(V_{\psi }{C}_{\phi }f\right)\left(0\right)=0$, and for $j\in \{1,\dots ,n-1\}$, by (3) and using (2), we have

$$\begin{array}{cc}\hfill |{\left(V_{\psi }{C}_{\phi }f\right)}^{\left(j\right)}\left(0\right)|& =\left|\sum _{k=0}^{j-1}{f}^{\left(k\right)}\left(\phi \left(0\right)\right)\sum _{l=k}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{j-1}{l}}\right){\left(\psi \right)}^{(j-l)}\left(0\right)B_{l,k}({\phi }^{\prime }\left(0\right),\dots ,{\phi }^{(l-k+1)}\left(0\right))\right|\hfill \\ & \le \sum _{k=0}^{j-1}\left|\sum _{l=k}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{j-1}{l}}\right){\left(\psi \right)}^{(j-l)}\left(0\right)B_{l,k}({\phi }^{\prime }\left(0\right),\dots ,{\phi }^{(l-k+1)}\left(0\right))\right|\hfill \\ & ⪯\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{k=0}^{j-1}|N_k\left(z\right)|\hfill \\ & ⪯M.\hfill \end{array}$$

(9)

Summing over all $j\in \{0,\dots ,n-1\}$ gives

$$\begin{array}{c}\hfill \sum _{j=0}^{n-1}|{\left(V_{\psi }{C}_{\phi }f\right)}^{\left(j\right)}\left(0\right)|⪯M.\end{array}$$

(10)

Combining (8) and (10), we obtain

$$\parallel V_{\psi }{C}_{\phi }{f\parallel }_{{\mathcal{V}}_{\mu ,n}}⪯M,$$

which proves that $V_{\psi }{C}_{\phi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded. Moreover

$$\parallel V_{\psi }{C}_{\phi }\parallel ⪯M.$$

Conversely, assume $V_{\psi }{C}_{\phi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded. Firstly, we show that

$$\begin{array}{c}\hfill M_k:=\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)|N_k\left(z\right)|⪯\parallel V_{\psi }{C}_{\phi }\parallel \end{array}$$

(11)

for any $k=0,\dots ,n-1$ arguing by induction on n. Since $1\in {\mathcal{V}}_n$ and $V_{\psi }{C}_{\phi }$ is bounded, we obtain

$$M_0=\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)|{\left(V_{\psi }{C}_{\phi }1\right)}^{\left(n\right)}\left(z\right)|\le \parallel V_{\psi }{C}_{\phi }{\parallel \parallel 1\parallel }_{{\mathcal{V}}_n}=\parallel V_{\psi }{C}_{\phi }\parallel ,$$

which confirms the basic case of the induction. Next, assume for $j\in \{0,\dots ,k-1\}$

$$\begin{array}{c}\hfill M_j⪯\parallel V_{\psi }{C}_{\phi }\parallel .\end{array}$$

(12)

Consider $p_k\left(z\right)=z^k$ as a test function and note that ${\displaystyle \frac{d^j{p}_k\left(z\right)}{d{z}^j}}=0$ for $j>k$. Thus, we have

$$\begin{array}{cc}\hfill {\left(V_{\psi }{C}_{\phi }{p}_k\right)}^{\left(n\right)}\left(z\right)& =\sum _{j=0}^{n-1}{p}_k^{\left(j\right)}\left(\phi \left(z\right)\right)N_j\left(z\right))\hfill \\ & =\sum _{j=0}^k{p}_k^{\left(j\right)}\left(\phi \left(z\right)\right)N_j\left(z\right))\hfill \\ & ={\left(\phi \left(z\right)\right)}^k{N}_0\left(z\right)+\sum _{j=1}^{k-1}k\dots (k-j+1){\left(\phi \left(z\right)\right)}^{k-j}{N}_j\left(z\right))+k!N_k\left(z\right)).\hfill \end{array}$$

Therefore, by (12) and since $p_k\in {\mathcal{V}}_n$ and the operator is bounded,

$$\begin{array}{cc}\hfill M_k& ={\displaystyle \frac{1}{k!}}\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)|{\left(V_{\psi }{C}_{\phi }{p}_k\right)}^{\left(n\right)}\left(z\right)-{\left(\phi \left(z\right)\right)}^k{N}_0\left(z\right)\hfill \\ & -\sum _{j=1}^{k-1}(k-1)\dots (k-j+1){\left(\phi \left(z\right)\right)}^{k-j}{N}_j\left(z\right))|\hfill \\ & \le {\displaystyle \frac{1}{k!}}\left(\parallel V_{\psi }{C}_{\phi }{p}_k{\parallel }_{{\mathcal{V}}_n}+M_0+\sum _{j=1}^{k-1}k\dots (k-j+1){\left(\phi \left(z\right)\right)}^{k-j}{M}_j\right)\hfill \\ & ⪯\parallel V_{\psi }{C}_{\phi }\parallel .\hfill \end{array}$$

Hence, $M_k⪯\parallel V_{\psi }{C}_{\phi }\parallel$ for $k\in \{0,\dots ,n-1\}$.

Next, we prove that the quantity M is finite. Fix $a\in \mathbb{D}$ such that $\left|a\right|>1/2$, and consider the function $\left\{f_a\right\}$ as in ([14], Lemma 3.3)

$$\begin{array}{c}\hfill f_a\left(z\right)={(log{\displaystyle \frac{e}{1-{\left|a\right|}^2}})}^{-1}{\displaystyle \frac{{(-1)}^{n-1}{(1-\overline{a}z)}^{n-1}}{{\overline{a}}^{n-1}(n-1)!}}\left({\alpha }_n+{\beta }_{n}log(1-\overline{a}z)+{log}^2(1-\overline{a}z)\right)\end{array}$$

(13)

where

$$\begin{array}{cc}\hfill {\beta }_n& =-2\left(1+{(-1)}^n\sum _{k=0}^{n-2}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k}}\right){\displaystyle \frac{{(-1)}^k}{n-1-k}}\right)\mathrm{and}\hfill \\ \hfill {\alpha }_n& =1+2{(-1)}^n\sum _{k=0}^{n-2}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k}}\right){\displaystyle \frac{{(-1)}^k}{n-k-1}}\sum _{j=1}^{n-k-2}{\displaystyle \frac{1}{j}}-{\beta }_n{(-1)}^n\sum _{k=0}^{n-2}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k}}\right){\displaystyle \frac{{(-1)}^k}{n-1-k}}\hfill \end{array}$$

and recalling that $S:=\underset{\left|a\right|>1/2}{sup}{\parallel f_a\parallel }_{{\mathcal{V}}_n}$ is finite.

Fix $w\in \mathbb{D}$ and assume first $\left|\phi \right(w\left)\right|>1/2.$ Since $f_{\phi \left(w\right)}\in {\mathcal{V}}_n$ and by the boundedness of $V_{\psi }{C}_{\phi }$ from ${\mathcal{V}}_n$ to ${\mathcal{V}}_{\mu ,n}$, then

$$\begin{array}{ccc}\hfill \parallel V_{\psi }{C}_{\phi }\parallel S& \ge & \mu \left(w\right)|{\left(V_{\psi }{C}_{\phi }{f}_{\phi \left(w\right)}\right)}^{\left(n\right)}\left(w\right)|\hfill \\ & =& \mu \left(w\right)|\sum _{k=0}^{n-2}{f}^{\left(k\right)}\left(\phi \left(w\right)\right)N_k\left(w\right)+f_{\phi \left(w\right)}^{(n-1)}{N}_{n-1}\left(w\right)|\hfill \\ & \ge & \mu \left(w\right)|f_{\phi \left(w\right)}^{(n-1)}\left(\phi \left(w\right)\right)\left|\right|N_{n-1}\left(w\right)|-\mu \left(w\right)\sum _{k=0}^{n-2}|f_{\phi \left(w\right)}^{\left(k\right)}\left(\phi \left(w\right)\right)\left|\right|N_k\left(w\right)|.\hfill \end{array}$$

By (13), $f_{\phi \left(w\right)}^{(n-1)}\left(\phi \left(w\right)\right)=log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}$. Recalling that for $k\in \{0,\dots ,n-2\}$, by (2),

$$|f_{\phi \left(w\right)}^{\left(k\right)}\left(\phi \left(w\right)\right)|\le \parallel f_{\phi \left(w\right)}^{\left(k\right)}{\parallel }_{{\mathcal{V}}_{n-k}}\le {\parallel f_{\phi \left(w\right)}\parallel }_{{\mathcal{V}}_n}\le S,$$

we obtain

$$\begin{array}{cc}\hfill \mu \left(z\right)log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}\left|N_{n-1}\left(w\right)\right|& \le \parallel V_{\psi }{C}_{\phi }\parallel S+S\sum _{k=0}^{n-2}{M}_k⪯\parallel V_{\psi }{C}_{\phi }\parallel .\hfill \end{array}$$

(14)

Next, assume $\left|\phi \right(w\left)\right|\le 1/2$. Then

$$\begin{array}{c}\hfill \mu \left(w\right)log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}|N_{n-1}\left(w\right)|\le {\displaystyle \frac{4}{3}}\mu \left(w\right)|N_{n-1}\left(w\right)|\le {\displaystyle \frac{4}{3}}{M}_{n-1}⪯\parallel V_{\psi }{C}_{\phi }\parallel .\end{array}$$

(15)

Combining (14) and (15) and taking the supremum over all w in $\mathbb{D}$, we obtain

$$\begin{array}{c}\hfill \underset{w\in \mathbb{D}}{sup}\mu \left(w\right)log{\displaystyle \frac{e}{1-{\left|\phi \left(w\right)\right|}^2}}|N_{n-1}\left(w\right)|⪯ \parallel V_{\psi }{C}_{\phi }\parallel .\end{array}$$

(16)

It follows, by (11) and (16),

$$M⪯ \parallel V_{\psi }{C}_{\phi }\parallel .$$

which completes our proof. □

Note that if $\phi$ is the identity map $id$, then $V_{\psi }=V_{\psi }{C}_{id}$, and $N_k\left(z\right)=\left(\genfrac{}{}{0pt}{}{n-1}{k}\right){\psi }^{(n-k)}\left(z\right)$ for $k=0,\dots n-1$. Moreover, if $\psi$ is the identity map $id$, then $I_{\phi }=V_{id}{C}_{\phi }$, $N_k\left(z\right)=B_{n-1,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(n-k)}\left(z\right))$ for $k=0,\dots n-2$, while $N_{n-1}\left(z\right)={\left({\phi }^{\prime }\left(z\right)\right)}^{n-1}$. Hence, by Theorem 1, we deduce the following:

Corollary 1.

Fix $n\in \mathbb{N}$ and let $\psi \in H\left(\mathbb{D}\right)$. Then, the operator $V_{\psi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded if and only if

$$\begin{array}{c}\hfill \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[|{\psi }^{\prime }\left(z\right)|log{\displaystyle \frac{e}{1-{\left|z\right|}^2}}+\sum _{k=2}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k}}\right)|{\psi }^{\left(k\right)}\left(z\right)|\right]<\infty .\end{array}$$

Moreover, if $V_{\psi }$ is bounded, then

$$\parallel V_{\psi }\parallel \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[|{\psi }^{\prime }\left(z\right)|log\frac{e}{1-{\left|z\right|}^2}+\sum _{k=2}^n\left(\genfrac{}{}{0pt}{}{n-1}{k}\right)|{\psi }^{\left(k\right)}\left(z\right)|\right].$$

Corollary 2.

Fix $n\in \mathbb{N}$, and let $\phi \in S\left(\mathbb{D}\right)$. Then, the operator $I_{\phi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded if and only if

$$\begin{array}{c}\hfill \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[\sum _{k=0}^{n-2}|B_{n-1,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(n-k)}\left(z\right))|+|{\phi }^{\prime }{\left(z\right)|}^{n-1}log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right]<\infty .\end{array}$$

Moreover, if $I_{\phi }$ is bounded, then

$$\begin{array}{cc}\hfill \parallel I_{\phi }\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[{\sum }_{k=0}^{n-2}|B_{n-1,k}({\phi }^{\prime }\left(z\right),\dots ,{\phi }^{(n-k)}\left(z\right))|+|{\phi }^{\prime }{\left(z\right)|}^{n-1}log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right].\end{array}$$

We now state the second main result of this section.

Theorem 2.

Fix $n\in \mathbb{N}$. Let $\psi \in H\left(\mathbb{D}\right)$, and $\phi \in S\left(\mathbb{D}\right)$. Then, the operator $C_{\phi }{V}_{\psi }:{\mathcal{V}}_n\to :{\mathcal{V}}_{\mu ,n}$ is bounded if and only if

$$\begin{array}{c}\hfill \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[\sum _{k=0}^{n-2}|{\theta }_k\left(z\right)|+|{\theta }_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{|\phi \left(z\right)|}^2}}\right]<\infty .\end{array}$$

Moreover, if $C_{\phi }{V}_{\psi }$ is bounded, then

$$\begin{array}{cc}\hfill \parallel C_{\phi }{V}_{\psi }\parallel & \asymp \underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[{\sum }_{k=0}^{n-2}|{\theta }_k\left(z\right)|+|{\theta }_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right].\end{array}$$

Proof. 

Assume that $\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[{\sum }_{k=0}^{n-2}|{\theta }_k\left(z\right)|+|{\theta }_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right]$ is finite. Let $f\in {\mathcal{V}}_n$ with ${\parallel f\parallel }_{{\mathcal{V}}_n}=1$. Following the argument used in the proof of Theorem 1, we obtain

$$\begin{array}{c}\hfill |f^{\left(k\right)}\left(\phi \left(z\right)\right)|\le \parallel f^{\left(k\right)}{\parallel }_{V_{n-k}}\le 1,\mathrm{for}k=0,\dots ,n-2.\end{array}$$

(17)

Moreover,

$$\begin{array}{c}\hfill |f^{(n-1)}{\left(\phi \left(z\right)\right)|\le \parallel f\parallel }_{{\mathcal{V}}_n}log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}=log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}.\end{array}$$

(18)

Therefore, by (5), (17) and (18), we have

$$\begin{array}{cc}\hfill \parallel C_{\phi }{V}_{\psi }{f\parallel }_{{\mathcal{V}}_{\mu ,n}}& =\sum _{j=0}^{n-1}|{\left(C_{\phi }{V}_{\psi }f\right)}^{\left(j\right)}\left(0\right)|+\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left|{\left(C_{\phi }{V}_{\psi }f\right)}^{\left(n\right)}\left(z\right)\right|\hfill \\ & =\sum _{j=0}^{n-1}\left|\sum _{k=0}^{j-1}{f}^{\left(k\right)}\left(\phi \left(0\right)\right)\sum _{l=k}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{j-1}{l}}\right){(\psi \circ \phi )}^{(j-l)}\left(0\right)B_{l,k}({\phi }^{\prime }\left(0\right),\dots ,{\phi }^{(l-k+1)}\left(0\right))\right|\hfill \\ & +\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left|\sum _{k=0}^{n-1}{f}^{\left(k\right)}\left(\phi \left(z\right)\right){\theta }_k\left(z\right)\right|\hfill \\ & \le \sum _{j=0}^{n-1}\sum _{k=0}^{j-1}\left|\sum _{l=k}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{j-1}{l}}\right){(\psi \circ \phi )}^{(j-l)}\left(0\right)B_{l,k}({\phi }^{\prime }\left(0\right),\dots ,{\phi }^{(l-k+1)}\left(0\right))\right|\hfill \\ & +\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{k=0}^{n-2}|f^{\left(k\right)}\left(\phi \left(z\right)\right)\left|\right|{\theta }_k\left(z\right)|+|f^{(n-1)}\left(\phi \left(z\right)\right)\left|\right|{\theta }_{n-1}\left(z\right)|\hfill \\ & ⪯\sum _{j=0}^{n-1}\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{k=0}^{j-1}|{\theta }_k\left(z\right)|+\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[\sum _{k=0}^{n-2}|{\theta }_k\left(z\right)|+|{\theta }_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right]\hfill \\ & ⪯\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[\sum _{k=0}^{n-2}|{\theta }_k\left(z\right)|+|{\theta }_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right]\hfill \end{array}$$

which proves that $C_{\phi }{V}_{\psi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded. Moreover,

$$\parallel C_{\phi }{V}_{\psi }\parallel ⪯\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[\sum _{k=0}^{n-2}|{\theta }_k\left(z\right)|+|{\theta }_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right].$$

Conversely, assume $C_{\phi }{V}_{\psi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded. To prove that

$$\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left[\sum _{k=0}^{n-2}|{\theta }_k\left(z\right)|+|{\theta }_{n-1}\left(z\right)|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\right]$$

is finite, one can use the same argument as in the proof of Theorem 1. The details are omitted since the proof is straightforward. □

3. Essential Norms

We begin the section by providing Lemma 4.1 of [14] and a useful result proved by Tjani in [20] that will be used to estimate the essential norm of integral-type operators on ${\mathcal{V}}_n$.

Lemma 2

([14], Lemma 4.1). For each nonnegative integer $n\ge 2$, if $\left\{f_k\right\}$ is a bounded sequence in ${\mathcal{V}}_n$ converging to zero uniformly on compact subsets of $\mathbb{D}$, then

$$\underset{k\to \infty }{lim}\parallel f_k{\parallel }_{\infty }=\underset{k\to \infty }{lim}\parallel f_k^{\prime }{\parallel }_{\infty }=\dots =\underset{k\to \infty }{lim}{\parallel f_k^{(n-2)}\parallel }_{\infty }=0.$$

Lemma 3

([20], Lemma 2.10). Let $T:\mathcal{X}\to \mathcal{Y}$ be a bounded linear operator, where $\mathcal{X}$ and $\mathcal{Y}$ are Banach spaces of analytic functions on $\mathbb{D}$. Suppose

(i) 

the point evaluation functionals on $\mathcal{Y}$ are continuous;

(ii) 

the closed unit ball of $\mathcal{X}$ is a compact subset of $\mathcal{X}$ in the topology of uniform convergence on compact sets;

(iii) 

T is continuous if $\mathcal{X}$ has the supremum norm and $\mathcal{Y}$ is given the topology of uniform convergence on compact sets.

Under these conditions, T is a compact operator if and only if for any bounded sequence $\left\{f_m\right\}$ in $\mathcal{X}$ converging to 0 uniformly on compact subsets of $\mathbb{D}$, the sequence $\left\{T{f}_m\right\}$ converges to 0 in norm.

The following result is crucial for estimating the essential norm of integral-type operator $V_{\psi }{C}_{\phi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$.

Lemma 4.

Fix $n\in {\mathbb{N}}_0$ and let $0\le r<1$. For $f\in {\mathcal{V}}_n$, the dilation function $T_r$ in ${\mathcal{V}}_n$ is defined by $T_{r}f\left(z\right):=f\left(rz\right)$ for all $z\in \mathbb{D}$. Then

(a) 

$$\begin{array}{c}\hfill \parallel T_r\parallel = 1,\end{array}$$

(19)

(b) 

for each $s\in (0,1)$ and $\epsilon >0$, there exists $r\in (0,1)$ such that

$$\begin{array}{c}\hfill \underset{{\parallel f\parallel }_{{\mathcal{V}}_n}=1}{sup}\underset{\left|z\right|<1}{sup}|((I-T_r)f){)}^{\left(j\right)}\left(z\right)|<\epsilon ,\mathit{for}\mathit{all}j=0,\dots ,n-2,\end{array}$$

(20)

$$\begin{array}{c}\hfill \underset{{\parallel f\parallel }_{{\mathcal{V}}_n}=1}{sup}\underset{\left|z\right|\le s}{sup}|((I-T_r)f){)}^{\left(j\right)}\left(z\right)|<\epsilon ,\mathit{for}j=n-1,n,\end{array}$$

(21)

(c) 

$T_r$ is compact on ${\mathcal{V}}_n$

Proof. 

(a) Let $r\in (0,1)$ and f in ${\mathcal{V}}_n$. Then

$$\begin{array}{cc}\hfill \parallel T_r{f\parallel }_{{\mathcal{V}}_n}& =\sum _{k=0}^{n-1}{r}^k|f^{\left(k\right)}\left(0\right)|+r^n\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2)|f^{\left(n\right)}\left(rz\right)|\hfill \\ \hfill & =\sum _{k=0}^{n-1}|f^{\left(k\right)}\left(0\right)|+\underset{\left|w\right|<r}{sup}{(1-|w|}^2\left)\right||f^{\left(n\right)}\left(w\right)|\hfill \\ \hfill & \le \sum _{k=0}^{n-1}|f^{\left(k\right)}\left(0\right)|+\underset{\left|w\right|<1}{sup}{(1-|w|}^2\left)\right|f^{\left(n\right)}\left(w\right)|\hfill \\ \hfill & ={\parallel f\parallel }_{{\mathcal{V}}_n}.\hfill \end{array}$$

(22)

Thus, $T_r$ is bounded, and observing that

$$1=\parallel T_r{1\parallel }_{{\mathcal{V}}_n}\le \parallel T_r{\parallel \parallel 1\parallel }_{{\mathcal{V}}_n}=\parallel T_r\parallel ,$$

(23)

combining (22) and (23), we obtain $\parallel T_r\parallel =1.$

(b)

Let $f\in {\mathcal{V}}_n$ with ${\parallel f\parallel }_{{\mathcal{V}}_n}\le 1$. Fix $s\in (0,1)$ and let $\left\{r_m\right\}$ be a sequence in $(0,1)$ converging to 1 as $m\to \infty$. By the continuity of f on the closed unit disk, for $z\in \mathbb{D}$,

$$\underset{m\to \infty }{lim}\left((I-T_{r_m})f\right)\left(z\right)=\underset{m\to \infty }{lim}(f\left(z\right)-f\left(r_{m}z\right))=0,$$

and by part (a)

$$\parallel (I-T_{r_m}){f\parallel }_{{\mathcal{V}}_n}\le {\parallel f\parallel }_{{\mathcal{V}}_n}+\parallel T_{r_m}{f\parallel }_{{\mathcal{V}}_n}\le 2{\parallel f\parallel }_{{\mathcal{V}}_n}\le 2.$$

Thus, ${\displaystyle \frac{1}{2}}(I-T_{r_m})f$ is in the unit ball of ${\mathcal{V}}_n$. Note that by (3), ${\mathcal{V}}_n$ satisfies the second hypothesis in Lemma 3; therefore, the sequence $\left\{(I-T_{r_m})f\right\}$ has subsequence $\left\{(I-T_{r_{m_j}})f\right\}$ converging uniformly to 0 on every compact subset of $\mathbb{D}$. By Montel’s theorem, the functions ${\left((I-T_{r_m})f\right)}^{\left(j\right)}$, for $j\in \{1,\dots ,n\},$ converge uniformly to 0 on every compact subset of $\mathbb{D}$. Since every sequence in ${\mathcal{V}}_n$ converging uniformly on compact subsets of $\mathbb{D}$ converges uniformly on $\overline{\mathbb{D}}$ for $n\ge 2$, the subsequence ${\left((I-T_{r_m})f\right)}^{\left(j\right)}$, for $j\in \{0,\dots ,n-2\},$ converges uniformly to 0 on $\overline{\mathbb{D}}$. Thus, for every $\epsilon >0$ there exists $r\in (0,1)$ such that

$$\begin{array}{cc}\hfill \underset{z\in \mathbb{D}}{sup}\left|{\left((I-T_r)f\right)}^{\left(j\right)}\left(z\right)\right|& <\epsilon \mathrm{for}j\in \{0,\dots ,n-2\},\hfill \\ \hfill \underset{\left|z\right|\le s}{sup}\left|{\left((I-T_r)f\right)}^j\left(z\right)\right|& <\epsilon \mathrm{for}j=n-1,n.\hfill \end{array}$$

The conclusion follows after taking the supremum over all functions in the unit ball of ${\mathcal{V}}_n$.

(c)

Fix $0<r<1$. To show that $T_r$ is compact on ${\mathcal{V}}_n$, by Lemma 3, it suffices to show that each bounded sequence $\left\{f_k\right\}$ on ${\mathcal{V}}_n$ converging uniformly to 0 on compact subsets of $\mathbb{D}$, ${lim}_{k\to \infty }{\parallel T_r{f}_k\parallel }_{{\mathcal{V}}_n}=0$. Let $\left\{f_k\right\}$ be a bounded sequence on ${\mathcal{V}}_n$ converging uniformly to 0 on compact subsets of $\mathbb{D}$. We have

$$\begin{array}{cc}\hfill \parallel T_r{f}_k{\parallel }_{s{\mathcal{V}}_n}& =\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|{\left(T_r{f}_k\right)}^{\left(n\right)}\left(z\right)|\hfill \\ \hfill & =r^n\underset{z\in \mathbb{D}}{sup}{(1-|z|}^2\left)\right|f_k^{\left(n\right)}\left(rz\right)|\hfill \\ \hfill & \le r^n\underset{\left|w\right|<r}{sup}{(1-|w|}^2\left)\right|f_k^{\left(n\right)}\left(w\right)|\hfill \\ & \le \underset{\left|w\right|<r}{sup}\left|f_k^{\left(n\right)}\left(w\right)\right|.\hfill \end{array}$$

Since $f_k^{\left(n\right)}$ converges uniformly to zero on the disk with radius r, then $\parallel T_r{f}_k{\parallel }_{s{\mathcal{V}}_n}\to 0$ as $k\to \infty$.

Since {0} is compact of $\mathbb{D}$, then ${\left(T_r{f}_k\right)}^{\left(j\right)}\left(0\right)$ converges to 0, for $j\in \{0,\dots ,n-1\},$. Therefore, $\parallel T_r{f}_k{\parallel }_{{\mathcal{V}}_n}\to 0$ as $k\to \infty$. Thus, $T_r$ is compact on ${\mathcal{V}}_n$. □

Now, we are ready to state the main result of this section.

Theorem 3.

Fix $n\in \mathbb{N}$ and let $\psi \in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. If $V_{\psi }{C}_{\phi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded, then

$$\parallel V_{\psi }{C}_{\phi }{\parallel }_e\asymp \underset{s\to 1}{lim}\underset{s<\left|\phi \right(z\left)\right|<1}{sup}\mu \left(z\right)log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\left|N_{n-1}\left(z\right)\right|.$$

Proof. 

To prove the upper estimate, fix $s\in (0,1)$ and $\epsilon >0$, and choose $r\in (0,1)$ as in Lemma 4. It follows, for $j\in \{0,\dots ,n-1\}$, by (9)

$$\begin{array}{cc}\hfill & \underset{{\parallel f\parallel }_{{\mathcal{V}}_n}=1}{sup}|({\left(V_{\psi }{C}_{\phi }(I-T_r)f\right)}^{\left(j\right)}\left(0\right)|\hfill \\ & =\underset{{\parallel f\parallel }_{{\mathcal{V}}_n}=1}{sup}\left|\sum _{k=0}^{j-1}{\left((I-T_r)f\right)}^{\left(k\right)}\left(\phi \left(0\right)\right)\sum _{l=k}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{j-1}{l}}\right){\left(\psi \right)}^{(j-1-l)}\left(0\right)B_{l,k}({\phi }^{\prime }\left(0\right),\dots ,{\phi }^{(l-k+1)}\left(0\right))\right|\hfill \\ & \le \epsilon \sum _{k=0}^{j-1}\left|\sum _{l=k}^{j-1}\left({\displaystyle \genfrac{}{}{0pt}{}{j-1}{l}}\right){\left(\psi \right)}^{(j-1-l)}\left(0\right)B_{l,k}({\phi }^{\prime }\left(0\right),\dots ,{\phi }^{(l-k+1)}\left(0\right))\right|\hfill \\ & ⪯M\epsilon .\hfill \end{array}$$

(24)

Since $T_r$ is compact and $V_{\psi }{C}_{\phi }$ is bounded, $V_{\psi }{C}_{\phi }{T}_r$ is also compact. Thus, by (19), (20), (21) and (24),

$$\begin{array}{cc}\hfill \parallel V_{\psi }{C}_{\phi }{\parallel }_e& \le \parallel V_{\psi }{C}_{\phi }-V_{\psi }{C}_{\phi }{T}_r\parallel \hfill \\ \hfill & =\underset{{\parallel f\parallel }_{{\mathcal{V}}_n}=1}{sup}{\parallel \left(V_{\psi }{C}_{\phi }(I-T_r)\right)f\parallel }_{{\mathcal{V}}_n}\hfill \\ \hfill & =\underset{{\parallel f\parallel }_{{\mathcal{V}}_n}=1}{sup}(\sum _{j=0}^{n-1}|{\left(V_{\psi }{C}_{\phi }(I-T_r)f\right)}^{\left(j\right)}\left(0\right)|\hfill \\ \hfill & +\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\left|\sum _{k=0}^{n-2}{\left((I-T_r)f\right)}^{\left(k\right)}\left(\phi \left(z\right)\right)N_k\left(z\right)|+{\left((I-T_r)f\right)}^{(n-1)}\left(\phi \left(z\right)\right)N_{n-1}\left(z\right)\right|\hfill \\ & \le \epsilon \sum _{j=0}^{n-1}M+\underset{{\parallel f\parallel }_{{\mathcal{V}}_n}=1}{sup}(\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{k=0}^{n-2}|(I-T_r){f)}^{\left(k\right)}(\phi \left(z\right)\left|\right|N_k\left(z\right)|\hfill \\ & +\underset{\left|z\right|\le s}{sup}\mu \left(z\right)(I-T_r){f)}^{(n-1)}(\phi \left(z\right)\left|\right|N_{n-1}\left(z\right)|\hfill \\ & +\underset{s<\left|\phi \right(z\left)\right|<1}{sup}\mu \left(z\right)|(I-T_r){f)}^{(n-1)}(\phi \left(z\right)\left|\right|N_{n-1}\left(z\right)\left|\right))\hfill \\ & \le \epsilon \left((nM+\underset{z\in \mathbb{D}}{sup}\mu \left(z\right)\sum _{k=0}^{n-2}|N_k\left(z\right)|+\underset{|z|\le s}{sup}\mu \left(z\right)|N_{n-1}\left(z\right)|\right)\hfill \\ & +2\underset{s<\left|\phi \right(z\left)\right|<1}{sup}\mu \left(z\right)log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\left|N_{n-1}\left(z\right)\right|.\hfill \end{array}$$

Since $\epsilon$ is arbitrary, passing to the limit as $s\to 1$, we obtain

$$\parallel V_{\psi }{C}_{\phi }{\parallel }_e⪯\underset{s\to 1}{lim}\underset{s<\left|\phi \right(z\left)\right|<1}{sup}\mu \left(z\right)log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}\left)\right|N_{n-1}\left(z\right)|.$$

To prove the lower estimate, let $\left\{w_m\right\}$ be a sequence in $\mathbb{D}$ such that $\left|\phi \right(w_m\left)\right|\to 1$. Then, the sequence $f_m:=f_{\phi \left(w_m\right)}$ defined in the proof of Theorem 3 converges to 0 uniformly on compact subsets. Moreover, $G:=\underset{m\in \mathbb{N}}{sup}{\parallel f_m\parallel }_{{\mathcal{V}}_n}<\infty$.

Let $W:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ be a compact operator. Then, by Lemma 3, $\underset{m\to \infty }{lim}{\parallel W{f}_m\parallel }_{{\mathcal{V}}_{\mu ,n}}=0$. Hence, by Lemma 2 and (13),

$$\begin{array}{cc}\hfill G\parallel V_{\psi }{C}_{\phi }-W\parallel & \ge \underset{m\to \infty }{lim\; sup}{\parallel (V_{\psi }{C}_{\phi }-W)f_m\parallel }_{{\mathcal{V}}_{\mu ,n}}\hfill \\ & \ge \underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)\left|{\left(V_{\psi }{C}_{\phi }{f}_m\right)}^{\left(k\right)}\left(w_m\right)\right|\hfill \\ & =\underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)|\sum _{k=0}^{n-2}{f}_m^{\left(k\right)}\left(\phi \left(w_m\right)\right)N_k\left(w_m\right)+f_m^{(n-1)}\left(\phi \left(w_m\right)\right)N_{n-1}\left(w_m\right)|\hfill \\ & =\underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)log{\displaystyle \frac{e}{1-|\phi \left(w_n\right){|}^2}}\left|N_{n-1}\left(w_m\right)\right|.\hfill \end{array}$$

Taking the infimum over all compact operators $W:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$, we obtain

$$\underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}|N_{n-1}\left(w_m\right)|⪯\parallel V_{\psi }{C}_{\phi }{\parallel }_e.$$

□

By concentrating on the component operators $V_{\psi }$ and $I_{\phi }$, and using the identities $V_{\psi }=V_{\psi }{C}_{id}$ and $I_{\phi }=V_{id}{C}_{\phi }$ along with Theorem 3, we obtain the following results:

Corollary 3.

Fix $n\in \mathbb{N}$ and let $\psi \in H\left(\mathbb{D}\right)$. If $V_{\psi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded, then

$$\parallel V_{\psi }{\parallel }_e\asymp \underset{s\to 1}{lim}\underset{s<\left|z\right|<1}{sup}\mu \left(z\right)\left|{\psi }^{\prime }\left(z\right)\right|log{\displaystyle \frac{e}{1-{\left|z\right|}^2}}.$$

Corollary 4.

Fix $n\in \mathbb{N}$ and let $\phi \in S\left(\mathbb{D}\right)$. If $I_{\psi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded, then

$$\parallel I_{\phi }{\parallel }_e\asymp \underset{s\to 1}{lim}\underset{s<\left|\phi \right(z\left)\right|<1}{sup}\mu \left(z\right){\left|{\phi }^{\prime }\left(z\right)\right|}^{n-1}log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}.$$

We are now ready to state the second main result of this section.

Theorem 4.

Fix $n\in \mathbb{N}$ and let $\psi \in H\left(\mathbb{D}\right)$ and $\phi \in S\left(\mathbb{D}\right)$. If $C_{\phi }{V}_{\psi }:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ is bounded, then

$$\parallel C_{\phi }{V}_{\psi }{\parallel }_e\asymp \underset{s\to 1}{lim}\underset{s<\left|\phi \right(z\left)\right|<1}{sup}\mu \left(z\right)\left|{\theta }_{n-1}\left(z\right)\right|log{\displaystyle \frac{e}{1-{\left|\phi \left(z\right)\right|}^2}}.$$

Proof. 

To prove the lower estimate, let $\left\{w_m\right\}$ be a sequence in $\mathbb{D}$ such that $\left|\phi \right(w_m\left)\right|\to 1$. Then, the sequence $f_m:=f_{\phi \left(w_m\right)}$ defined in the proof of Theorem 3 converges to 0 uniformly on compact subsets. Moreover, $G:=\underset{m\in \mathbb{N}}{sup}{\parallel f_m\parallel }_{{\mathcal{V}}_n}<\infty$.

Let $T:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$ be a compact operator. Then, by Lemma 3, $\underset{m\to \infty }{lim}{\parallel T{f}_m\parallel }_{{\mathcal{V}}_{\mu ,n}}=0$. Hence, by Lemma 2 and (13),

$$\begin{array}{cc}\hfill G\parallel C_{\phi }{V}_{\psi }-T\parallel & \ge \underset{m\to \infty }{lim\; sup}{\parallel (C_{\phi }{V}_{\psi }-T)f_m\parallel }_{{\mathcal{V}}_{\mu ,n}}\hfill \\ & \ge \underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)\left|{\left(C_{\phi }{V}_{\psi }{f}_m\right)}^{\left(k\right)}\left(w_m\right)\right|\hfill \\ & =\underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)|\sum _{k=0}^{n-2}{f}_m^{\left(k\right)}\left(\phi \left(w_m\right)\right){\theta }_k\left(w_m\right)+f_m^{(n-1)}\left(\phi \left(w_m\right)\right){\theta }_{n-1}\left(w_m\right)|\hfill \\ & =\underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)log{\displaystyle \frac{e}{1-|\phi \left(w_n\right){|}^2}}\left|{\theta }_{n-1}\left(w_m\right)\right|.\hfill \end{array}$$

Taking the infimum over all compact operators $T:{\mathcal{V}}_n\to {\mathcal{V}}_{\mu ,n}$, we obtain

$$\underset{m\to \infty }{lim\; sup}\mu \left(w_m\right)log{\displaystyle \frac{e}{1-|\phi \left(w_m\right){|}^2}}|{\theta }_{n-1}\left(w_m\right)|⪯\parallel C_{\phi }{V}_{\psi }{\parallel }_e.$$

To prove the upper estimate, fix $s\in (0,1)$ and $\epsilon >0$, and choose $r\in (0,1)$ as in Lemma 4. By considering $C_{\phi }{V}_{\psi }$ instead of $V_{\psi }{C}_{\phi }$ and applying the same approach used in the proof of the upper estimate in Theorem 3, then the result follows directly. □

4. Conclusions

In this paper, we studied products of Volterra-type and composition operators on the iterated weighted-type Banach spaces ${\mathcal{V}}_n$. We gave clear conditions that describe when these operators are bounded from ${\mathcal{V}}_n$ to the weighted space ${\mathcal{V}}_{\mu ,n}$. We also estimated their essential norms, which help us to understand how close these operators are to being compact.

These results help improve the understanding of how certain operators behave in spaces of analytic functions. They show how Volterra-type and composition operators work together in more general settings that involve weights and higher-order derivatives. This work can also be a starting point for future studies on similar types of operators or on other function spaces.