Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-

1.   Introduction

Let $\mathbb{D} = \{ z\in \mathbb{C}\; /\; |z| < 1\; \}$ be the unit disk in the complex space. $\mathcal{O}(\mathbb{D})$ denotes the space of functions that are holomorphic in $\mathbb{D}$ and $\mathcal{H}^\infty(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions on $\mathbb{D}$ with the norm $\left\| f\right\|_\infty = \sup_{z\in \mathbb{D}}\left| f(z)\right|.$ For a holomorphic self-mapping $\varphi$ of $\mathbb{D}$ $(\varphi(\mathbb{D})\subset \mathbb{D})$ and a holomorphic function $u$: $\mathbb{D}\longrightarrow \mathbb{C}$, the pair $(u, \varphi)$ induces the linear operator $W_{\varphi, \; u}$: $\mathcal{O}(\mathbb{D})\longrightarrow \mathcal{O}(\mathbb{D})$ defined by

$W_{\varphi, \; u}$ which is called weighted composition operator with symbols $u$ and $\varphi.$ Observe that $W_{\varphi, \; u}(f) = M_uC_\varphi(f)$, where $M_u(f) = u.f,$ is the multiplication operator with symbol $u$, and $C_\varphi(f) = f\circ \varphi$, is the composition operator with symbol $\varphi$.

If $u \equiv 1$, then $W_{\varphi, \; u} = C_\varphi$, and if $\varphi$ is the identity $(\varphi(z) = z)$, then $W_{\varphi, \; u} = M_u.$

During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in $\mathbb{C}$ or $\mathbb{C}^n$. We refer the readers to the monographs ^{[1,3,5,13,18,20,23]} for detailed information and the references therein.

For $a \in \mathbb{D}$ the Möbius transformation $\varphi_{a}(z)$ is defined by

For each $a \in \mathbb{D}$, the Green's function with logarithmic singularity at $a \in \mathbb{D}$ is denoted by

The pseudohyperbolic distance $\rho$: $\mathbb{D} \times \mathbb{D} \longrightarrow[0, 1)$ is defined by

We will denote by

It is easy to check that $\rho(a, z)$ satisfies the following inequalities:

For $0 < \alpha < \infty$, recall that an $f \in \mathcal{O}(\mathbb{D})$ is said to belong to the $\alpha$-Bloch space $\mathcal{B}^\alpha$ if

With the norm $\|f\| = | f(0)|+ {\mathcal{B}}_\alpha(f)$, $\mathcal{B}^\alpha$ is a Banach space. When $\alpha = 1$, $\mathcal{B}^1 = \mathcal{B}$ is the well-known Bloch space. For more information on Bloch spaces we refer the interested reader to ^[19]. Let $\mathcal{B}_0 ^\alpha$ be the space which consists of all $f \in \mathcal{B}$ satisfying

This space is called the little Bloch space. See ^[19] for more information on Bloch spaces.

Let $\alpha\geq 0$. The Bers-type space, denoted by ${\cal H}_\alpha^\infty(\mathbb{D})$, is a Banach space defined by

equipped with the norm

Note that, ${\cal H}_{\alpha}^\infty(\mathbb{D})$ is a Banach space with the norm $\| \|_{{\cal H}_{\alpha}^\infty(\mathbb{D})}.$

When $\alpha = 0$, ${\mathcal H}_0^\infty(\mathbb{D})$ is just the bounded analytic function space ${\mathcal H}^\infty(\mathbb{D})$. For more information about several studied on Bers-type spaces we refer to ^[3,20].

Let $K$: $[0, \; \infty)\longrightarrow (0, \; \infty)$ be right continuous and nondecreasing function. The authors Ahmed and Bakhit in ^[7] introduced the ${\mathcal N}_K(\mathbb{D})$ spaces as follows:

The analytic $\mathcal{N}_{K}(\mathbb{D})$-space is defined by

equipped with the norm

Remark 1.1. We make the following observations:

$(1)$ If $K(t) = t^p$, then $\mathcal{N}_{K}(\mathbb{D}) = \mathcal{N}_p(\mathbb{D}),$ since $g(z, a)\approx (1-|\varphi_a|^2)$.

$(2)$ If $K(t)\equiv 1$, then $\mathcal{N}_{1}(\mathbb{D}) = \mathcal{A}^2$ (the Bergman space), where for $0 < p < \infty$, the Bergman space ${\mathcal A}^p$ is the set of analytic functions $f$ in the unit disk $\mathbb{D}$ with

Remark 1.2. In the study of the space $\mathcal{N}_{K}(\mathbb{D})$, the authors in ^[7] assume that the following condition

is satisfied, so that the ${\cal N}_{K}(\mathbb{D})$ space is not trivial.

Lemma 1.1. ([8,Lemma 2.2]) Assume that the function $K$ satisfies $(1.1)$.For each $w\in \mathbb{D}$, let

for $z\in \mathbb{D}$. Then $h_\omega$ satisfies the following conditions:

(i) $h_w \in \mathcal{N}_{K}(\mathbb{D}).$

(ii) $\|h_w\|_{\mathcal{N}_{K}(\mathbb{D})}\lesssim 1.$

(iii) $\sup\limits _{\omega \in \mathbb{D}}\|h_\omega\|_{{\mathcal N}_K(\mathbb{D})} \leq 1$.

Several important properties of the $\mathcal{N}_{K}(\mathbb{D})$-spaces and $H_{\alpha}^{\infty}(\mathbb{D})$ spaces and also of weighted composition operators from $\mathcal{N}_{K}(\mathbb{D})$-spaces to the spaces $H_{\alpha}^{\infty}(\mathbb{D})$ and from $H_{\alpha}^{\infty}(\mathbb{D})$-spaces to $\mathcal{N}_{K}(\mathbb{D})$ have been characterized in ^[7,8,15].

We cite here main results from ^[15] for the readers' convenience.

Theorem 1.1. (^[15,22])Let $K$: $[0, \infty)\rightarrow [0, \infty)$, be a nondecreasing function and $\varphi$ be a holomorphic self-map of $\mathbb{D}$. For$\alpha \in (0, \infty)$ and $u\in {\mathcal{O}}(\mathbb{D})$. The weighted composition operator

$(1)$ is bounded if and only if

$(2)$ is compact if and only if

Remark 1.3. When $K(t) = t^p$, Theorem 1.1 coincides with [22,Thoerem 3,Corollary 2].

Theorem 1.2. (^[15,22]) Let $K$: $[0, \infty)\rightarrow [0, \infty)$, be a nondecreasing function and $\varphi$ be a holomorphic self-map of $\mathbb{D}$. For$\alpha \in (0, \infty)$ and $u\in {\mathcal{O}}(\mathbb{D})$.Then the following properties hold:

$(1)$ The weighted composition operator $W_{\varphi, \; u} = uC_\varphi$: ${\cal H}_{\alpha}^{\infty}(\mathbb{D}) \longrightarrow \mathcal{N}_{K}(\mathbb{D})$ is bounded.

$(2)$ $u$ and $\varphi$ satisfy

$(3)$ $u$ and $\varphi$ satisfy

Remark 1.4. When $K(t) = t^p$, Theorem 1.2 coincides with [22,Theorem 1].

Theorem 1.3. (^[15,22]) $K$: $[0, \infty)\rightarrow [0, \infty)$, be a nondecreasing function and $\varphi$ be a holomorphic self-map of $\mathbb{D}$. For$\alpha \in (0, \infty)$ and $u\in {\mathcal{O}}(\mathbb{D})$, then the following are equivalent:

(i) $W_{\varphi, \; u}$: ${\cal H}_\alpha^{\infty}(\mathbb{D})\longrightarrow \mathcal{N}_{K}(\mathbb{D})$ iscompact operator.

(ii) $u$ and $\varphi$ satisfy

(iii) $u$ and $\varphi$ satisfy

Remark 1.5. When $K(t) = t^p$, Theorem 1.3 coincides with [22,Corollary 1].

Lemma 1.2. ([7,Proposition 2.1])For each right continuous and nondecreasing function $K$: $[0, \infty) \rightarrow [0, \infty)$, the following inclusion holds:

Our goal here is to investigate the boundedness and compactness of the difference of two weighted composition operators acting from $\mathcal{N}_{K}(\mathbb{D})$ -spaces to ${\mathcal H}_{\alpha}^\infty(\mathbb{D})$-spaces and form ${\mathcal H}_{\alpha}^\infty(\mathbb{D})$-spaces to $\mathcal{N}_{K}(\mathbb{D})$-spaces. To this end we introduce analytic maps $\varphi, \psi$: $\mathbb{D} \longrightarrow \mathbb{D}$ and $u, v$: $\mathbb{D } \longrightarrow \mathbb{C}$ and look at the operator

2.   Main results

2.1.   Differences of weighted composition operators from $\mathcal{N}_{K}(\mathbb{D})$ into ${\cal H}_{\alpha}^\infty(\mathbb{D})$

In this section we study the boundedness and compactness of two differences weighted composition operators

In fact, the following results corresponds to the results obtained in ^{[2,4,6,9,10,11,12,16,21]}.

We are now ready to prove a necessary condition and a sufficient condition for the boundedness of $T_{\varphi, \psi}$: $\mathcal{N}_{K}(\mathbb{D})\longrightarrow {\cal H}_{\alpha}^\infty(\mathbb{D})$.

For that purpose, consider the following three conditions:

In order to prove the main results of this paper, the following auxiliary lemma is needed.

Lemma 2.1. ([17,Lemma 2.3]) For $f\in {\cal H}_\alpha^\infty(\mathbb{D})$ and $z, w\in \mathbb{D},$

Theorem 2.1. Let $K$: $[0, \; \infty)\longrightarrow [0, \; \infty)$ be a nondecreasing function, $\varphi$ and $\psi$ are holomorphic self-maps from $\mathbb{D}$ to $\mathbb{D}$. For $u, v \in \mathcal{O}(\mathbb{D})$ and $\alpha > 0.$ Then the following statements are equivalent:

$(1)$ $T_{\varphi, \psi}$: $\mathcal{N}_{K}(\mathbb{D})\longrightarrow {\cal H}_{\alpha}^{\infty}(\mathbb{D})$ is bounded.

$(2)$ $\varphi, \psi$ and $u, v$ satisfy the conditions $(2.1)$ and $(2.3)$.

$(3)$ $\varphi, \psi$ and $u, v$ satisfy the conditions $(2.2)$ and $(2.3).$

Proof. $(3)\Rightarrow (1).$ Assume that the functions $\varphi, \psi$ and $u, v$ satisfy the conditions $(2.2)$ and $(2.3)$, and we need to prove that $T_{\varphi, \psi}$ is bounded. In fact, let $f \in \mathcal{N}_{K}(\mathbb{D})$, then we have

Taking in to account that ${\cal N}_{K}(\mathbb{D})\subset {\cal H}_1^\infty(\mathbb{D})$ ([7,Proposition 2.1]), it follows from conditions $(2.2)$ and $(2.3)$ that

where $C$ is a positive constant. Therefore $T_{\varphi, \psi}$ is bounded form $\mathcal{N}_{K}(\mathbb{D})$ to ${\cal H}_{\alpha}^{\infty}(\mathbb{D})$ as required.

$(2)\Rightarrow (3).$ Observe that

which implies that $(2.3)$ holds.

Finally we show the implication $(1)\Rightarrow (2).$ Assume that $T_{\varphi, \; \psi}$ is bounded from $\mathcal{N}_{K}(\mathbb{D})$ to ${\cal H}_{\alpha}^{\infty}(\mathbb{D})$ and prove that $(2.1)$ and $(2.3)$ are hold. Since $T_{\varphi, \psi}$ is bounded, we have for all $f \in \mathcal{N}_{K}(\mathbb{D})$

For each $z \in \mathbb{D}$, set

be the function test in Lemma 1.1.

By taking into account Lemma 1.1, we have $h_w \in \mathcal{N}_{K}$ and $\|h_w\|_{\mathcal{N}_{K}(\mathbb{D})} \lesssim 1.$

Fix $\omega \in \mathbb{D}$, and consider the function $g_\omega$ defined by

for $z\in \mathbb{D}.$ We have

Thus $g_\omega \in {\mathcal N}_K(\mathbb{D})$. Note that

From the boundeness of

it then follows that

Hence the condition $(2.1)$ holds. On the other hand we have

where

and

In view of Lemma 2.1 and the condition $(2.1)$ we deduce that $|B(\omega)| < \infty$ for all $w \in \mathbb{D}$, which implies that $|A(\omega)| < \infty$ for all $w \in \mathbb{D}$. Thus, the condition $(2.3)$ is proved.

Remark 2.1. the statement $(1)$ of Theorem 1.1 follows easily for the simple case $v\equiv 0$ of Theorem 2.1.

Corollary 2.1. Let $K$: $[0, \; \infty)\longrightarrow [0, \; \infty)$ be a nondecreasing function, $\varphi$ and $\psi$ are holomorphic self-maps from $\mathbb{D}$ to $\mathbb{D}$. For $u \in \mathcal{O}(\mathbb{D})$ and $\alpha > 0,$ then, $uC_\varphi-uC_\psi$: ${\mathcal N}_K(\mathbb{D})\longrightarrow {\mathcal H}_\alpha^\infty(\mathbb{D})$ is bounded if and only if the following two conditions hold:

and

Proof. Assume that $T_{\varphi, \; \psi}$ is bounded. Then by letting $v = u$ in Theorem 2.1 it follows that the conditions $(2.5)$ and $(2.6)$ hold.

Conversely, assume that the conditions $(2.5)$ and $(2.6)$ hold. To prove that $T_{\varphi, \; \psi}$ is bounded, it suffices in view of Theorem 2.1 to prove that

We have

Using Theorem 2.1, we obtain the boundedness of $uC_\varphi- uC_\psi$: ${\mathcal N}_K(\mathbb{D}) \longrightarrow {\mathcal H}_\alpha^\infty(\mathbb{D}).$ The proof of the corollary is complete.

Remark 2.2. There exist non-bounded weighted composition operators such that their difference is bounded.

In the following example we give operators such that neither $W_{\varphi, \; u}$, $W_{\psi, \; v}$ and $T_{\varphi, \; \psi} = W_{\varphi, \; u}-W_{\psi, \; v}$ are bounded from $\mathcal{N}_K(\mathbb{D})$ to ${\cal H}_\alpha^{\infty}(\mathbb{D}).$

Example 2.1. By choosing the maps $u, v, \varphi$ and $\psi$ as follows:

A direct calculation shows

In view of Theorem 2.1, it follows that neither $W_{\varphi, \; u}$: $\mathcal{N}_{K}(\mathbb{D}) \longrightarrow {\cal H}_{\alpha}^{\infty}(\mathbb{D})$ nor $W_{\psi, \; v}$: $\mathcal{N}_{K}(\mathbb{D}) \longrightarrow H_{\alpha}^{\infty}(\mathbb{D})$ is bounded. However from condition $(2.1)$ or $(2.2)$ it is clear that the difference operator $W_{\varphi, u}-W_{\psi, \; v}: \mathcal{N}_{K}(\mathbb{D}) \longrightarrow {\cal H}_{\alpha}^{\infty}(\mathbb{D})$ is not bounded.

The following theorem characterize when the difference weighted composition operators $T_{\varphi, \; \psi}$ acting between weighted analytic type spaces $\mathcal{N}_{K}(\mathbb{D})$ and ${\cal H}_{\alpha}^{\infty}(\mathbb{D})$ are compact.

Theorem 2.2. Let $\varphi, \psi$: $\mathbb{D}\longrightarrow \mathbb{D}$ be two holomorphic functions, $u, v$: $\mathbb{D}\longrightarrow \mathbb{C}$ two holomorphic functions. Let further $W_{\varphi, u}$ and $W_{\psi, v}$ be two weighted composition operators acting from $\mathcal{N}_{K}(\mathbb{D})$ into $H_{\alpha, }^\infty(\mathbb{D})$. Then the operators $T_{\varphi, \psi} = W_{\varphi, u}-W_{\psi, v}$ is compact if and only if the following conditions hold.

where

Proof. We omit the proof, since the techniques are similar to those of [14,Theorem 2.4].

Remark 2.3. The statement $(2)$ of Theorem 1.1 follows easily for the simple case $v\equiv 0$ of Theorem 2.2.

Corollary 2.2. Let $K$: $[0, \; \infty)\longrightarrow [0, \; \infty)$ be a nondecreasing function, $\varphi$ and $\psi$ are holomorphic self-maps from $\mathbb{D}$ to $\mathbb{D}$. For $u \in \mathcal{O}(\mathbb{D})$ and $\alpha > 0,$ then, $uC_\varphi-uC_\psi$: ${\mathcal N}_K(\mathbb{D})\longrightarrow {\mathcal H}_\alpha^\infty(\mathbb{D})$ is compact if and only if the following two conditions hold:

and

Proof. Assume that $T_{\varphi, \; \psi}$ is compact. Then by letting $v = u$ in Theorem 2.2 it follows that the conditions $(2.10)$ and $(2.11)$ hold.

Conversely, assume that the conditions $(2.10)$ and $(2.11)$ hold. To prove that $T_{\varphi, \; \psi}$ is compact, it suffices in view of Theorem 2.2 to prove that the condition $(2.9)$ is holds. Since $u\equiv v$, then

Using Theorem 2.2, we obtain the compactness of $uC_\varphi- uC_\psi$: ${\mathcal N}_K(\mathbb{D}) \longrightarrow {\mathcal H}_\alpha^\infty(\mathbb{D}).$ The proof of the corollary is complete.

2.2.   Differences of weighted composition operators from ${\cal H}_{\alpha}^\infty(\mathbb{D})$ into $\mathcal{N}_{K}(\mathbb{D})$

In this section, we investigate the boundedness of differences weighted composition operators

Theorem 2.3. Let $K$: $[0, \; \infty)\longrightarrow [0, \; \infty)$ be a nondecreasing function, $\varphi$ and $\psi$ are holomorphic self-maps from $\mathbb{D}$ to $\mathbb{D}$. For $u, v \in \mathcal{O}(\mathbb{D})$ and $\alpha > 0.$ Then the operator $T_{\varphi, \; \psi}$: $H_{\alpha}^{\infty}(\mathbb{D}) \longrightarrow \mathcal{N}_{K}(\mathbb{D})$ is bounded if the following condition is satisfies$\max\big(I, \; J\big) < \infty$, where

and

Proof. Assume that the condition in the statement $(2)$ is holds and let $f \in {H}_{\alpha}^{\infty}(\mathbb{D}).$ We have

which shows that $T_{\varphi, \; \psi}$ is bounded form $H_{\alpha}^\infty(\mathbb{D})$ to $\mathcal{N}_{K}(\mathbb{D}).$ Finally, it seems to be natural to enquire a necessary and sufficient conditions for the boundedness and compactness of difference weighted composition operator

So it is left as an open question.

3.   Conclusions

Firstly, the boundedness and compactness of two differences weighted composition operators

are obtained. Secondly, we have investigated the boundedness of differences weighted composition operators

Acknowledgments

The author extends his appreciation to the Deanship of Scientific Research at Jouf University for funding this work through research grant No. (DSR-2020-05-2575).

We would like to thank reviewers for taking the necessary time and effort to review the manuscript. We sincerely appreciate all valuable comments and suggestions, which helped us to improve the quality of the article.

Conflict of interest

The author declares that there is no conflict of interest.