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Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces

Abstract

Some criteria for the boundedness, as well as for the compactness, of the generalized weighted composition operator \(D^{n}_{\varphi, u}\) from α-Bloch spaces into weighted-type spaces are given. Estimates for the norm and the essential norm of the operator are also given. Our results extend and complement some results in the literature.

1 Introduction

Let \(\mathbb{D}\) be the unit disk of the complex plane \(\Bbb{C}\), \(H(\mathbb{D})\) the class of functions analytic on \(\mathbb{D}\), and \(H^{\infty}=H^{\infty}(\mathbb{D})\) the space of bounded analytic functions on \(\mathbb{D}\). For \(0<\alpha <\infty\), an \(f\in H(\mathbb{D})\) is said to belong to the α-Bloch space \(\mathcal{B}^{\alpha}=\mathcal{B}^{\alpha}(\mathbb{D})\) if

$$b_{\alpha}(f)=\sup_{z \in\mathbb{D}}\bigl(1-|z|^{2} \bigr)^{\alpha}\bigl\vert f'(z)\bigr\vert < \infty. $$

It is easy to check that \(\mathcal{B}^{\alpha}\) becomes a Banach space with the norm \(\|f\|_{\mathcal{B}^{\alpha}}=|f(0)|+b_{\alpha}(f)\). The little α-Bloch space \(\mathcal{B}^{\alpha}_{0}=\mathcal{B}^{\alpha}_{0}(\mathbb{D})\), is a subspace of \(\mathcal{B}^{\alpha}\) consisting of all \(f\in H(\mathbb{D})\) such that

$$\lim_{|z|\to1^{-}}\bigl(1-|z|^{2}\bigr)^{\alpha}\bigl\vert f'(z)\bigr\vert =0. $$

When \(\alpha=1\), \(\mathcal{B}^{1}=\mathcal{B}\) is the well-known Bloch space, while \(\mathcal{B}^{1}_{0}=\mathcal{B}_{0}\) is the well-known little Bloch space. For some results on the α-Bloch spaces and the little α-Bloch spaces, see, for example, [1].

A positive continuous function on \(\mathbb{D}\) is called a weight. Let \(\mu(z)\) be a weight. The weighted-type space on \(\mathbb{D}\) [2, 3], denoted by \(H^{\infty}_{\mu}=H^{\infty}_{\mu}(\mathbb{D})\), consists of all \(f\in H(\mathbb{D})\) such that

$$\|f\|_{ H^{\infty}_{\mu}}=\sup_{z\in\mathbb{D}} \mu(z)\bigl\vert f(z)\bigr\vert < \infty. $$

It is obvious that \(H^{\infty}_{0}=H^{\infty}\), while for \(\mu(z)=(1-|z|^{2})^{\beta}\), \(\beta>0\), is obtained the growth space \(H^{\infty}_{\beta}\) [4].

Let \(u \in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). The weighted composition operator \(uC_{\varphi}\), induced by φ and u, is defined by

$$(uC_{\varphi}f) (z) =u(z)\cdot f \bigl(\varphi(z)\bigr) ,\quad f \in H( \mathbb{D}), z\in\mathbb{D}. $$

When \(u(z)\equiv1\), then the weighted composition operator is reduced to the composition operator, usually denoted by \(C_{\varphi}\), while for \(\varphi(z)\equiv z\), it is reduced to the multiplication operator, usually denoted by \(M_{u}\).

A natural generalization of the weighted composition operator is the generalized weighted composition operator [5] or the weighted differentiation composition operator [6] \(D^{n}_{\varphi, u}\), which is defined as

$$\bigl(D^{n}_{\varphi, u} f\bigr) (z) =u(z)\cdot f^{(n)} \bigl(\varphi(z)\bigr) ,\quad f \in H(\mathbb{D}), z\in\mathbb{D}, $$

where \(n\in\mathbb{N}_{0}\), \(u \in H(\mathbb{D})\), and φ is an analytic self-map of \(\mathbb{D}\). Clearly, when \(n=0\) and \(u(z)=1\), \(D^{n}_{\varphi,u}\) is the composition operator \(C_{\varphi}\), if \(n=0\), then \(D^{n}_{\varphi,u}\) is the weighted composition operator \(uC_{\varphi}\). If \(n=1\) and \(u(z)=\varphi'(z)\), then \(D^{n}_{\varphi, u}= DC_{\varphi}\), which was studied, for example, in [3, 715], while for \(u(z)=1\), \(D^{n}_{\varphi, u}= C_{\varphi}D^{n}\), which was studied in [3, 13, 15, 16]. For some other results on the generalized weighted composition operator on various spaces of holomorphic functions, see, for example, [1722]. A fundamental problem concerning concrete operators is to relate function theoretic properties of their symbols to their operator theoretic properties (see, for example, [3, 529]).

It is well known that the composition operator is bounded on the Bloch space \(\mathcal{B}\). See, for example, [26, 28, 29] for the compactness and essential norm of the composition operator on \(\mathcal{B}\). In [28], it was shown that \(C_{\varphi}\) is compact on \(\mathcal{B}\) if and only if

$$\|C_{\varphi}p_{j}\|_{\mathcal{B}}=\bigl\Vert \varphi^{j}\bigr\Vert _{\mathcal{B}}\to0 \quad \mbox{as } j\to \infty, $$

where \(p_{j}(z)=z^{j}\), \(j\in\mathbb{N}_{0}\).

Motivated by this result, in [22], the author proved that \(D^{n}_{\varphi,u}: \mathcal{B} \to H^{\infty}_{\beta}\) is compact if and only if it is bounded and

$$\lim_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(p_{j}) \bigr\Vert _{H^{\infty}_{\beta}}=0. $$

Following the line of the above mentioned investigations, in this work, we consider the operators \(D^{n}_{\varphi, u} :\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\), and show that \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\) is bounded (respectively, compact) if and only if the sequence \((j^{\alpha -1}\|D^{n}_{\varphi,u} (p_{j})\|_{H_{\mu}^{\infty}})_{j=n}^{\infty}\) is bounded (respectively, convergent to 0 as \(j\to\infty\)). Moreover, we give some estimates for the norm, as well as for the essential norm of the operator \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\ (\mbox{or } \mathcal {B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\). Recall that the essential norm of the operator \(T:X\rightarrow Y\) is its distance to the set of compact operators K mapping X to Y, that is,

$$\|T\|_{e, X\rightarrow Y}=\inf\bigl\{ \Vert T-K\Vert _{X\rightarrow Y}: K \mbox{ is compact}\bigr\} , $$

where X and Y are Banach spaces and \(\|\cdot\|_{X\rightarrow Y}\) is the operator norm. Consequently, \(\|T\|_{e,X\rightarrow Y}=0\) if and only if T is compact.

Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. We write \(P\preceq Q\) if there exists a positive constant C independent of the quantities P and Q such that \(P\leq CQ\). The symbol \(P\approx Q\) means that \(P\preceq Q\preceq P\).

2 Boundedness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\to H^{\infty}_{\mu}\)

For \(w\in\mathbb{D}\), set

$$f_{w}(z)=\frac{1-|w|^{2}}{ (1-\overline{w} z)^{\alpha}},\quad z\in\mathbb{D} . $$

Note that

$$ f^{(n)}_{w}(z)=\frac{(1-|w|^{2})\overline{w}^{n}}{ (1-\overline{w} z)^{\alpha +n}}\prod _{j=0}^{n-1}(\alpha +j),\quad z\in \mathbb{D}, n\in \mathbb{N}. $$
(1)

In this section, we will use this family of functions, as well as the sequence of functions \((j^{\alpha -1} p_{j})_{j\in\mathbb{N}}\) to characterize the boundedness and compactness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\to H^{\infty}_{\mu}\).

Theorem 2.1

Let n be a positive integer, \(\alpha >0\), μ a weight, \(u\in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Then the following statements are equivalent.

  1. (a)

    The operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded.

  2. (b)

    The operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}\) is bounded.

  3. (c)

    \(M_{1}:= \sup_{j \geq n} j^{\alpha -1} \| D^{n}_{\varphi, u}(p_{j})\|_{H^{\infty}_{\mu}}<\infty\).

  4. (d)

    \(M_{2}:=\sup_{w\in\mathbb{D}}\|D^{n}_{\varphi,u} f_{\varphi (w)}\|_{H^{\infty}_{\mu}}<\infty\) and \(u\in H^{\infty}_{\mu}\).

  5. (e)

    \(M_{3}:= \sup_{z\in\mathbb{D} } \frac{\mu(z)|u(z) |}{(1-|\varphi(z)|^{2})^{ n+\alpha -1}} <\infty \) and \(u\in H^{\infty}_{\mu}\).

Moreover, if the operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded, then the following asymptotic relations hold:

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{\mathcal{B}^{\alpha}\to H^{\infty}_{\mu}}\approx \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}\approx M_{1}\approx\max\bigl\{ M_{2},\Vert u\Vert _{H^{\infty}_{\mu}} \bigr\} \approx M_{3}. $$
(2)

Proof

(a) (b) Since \(\mathcal{B}^{\alpha}_{0}\subset \mathcal{B}^{\alpha}\), this implication, as well as the inequality

$$ \bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}\leq \bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}\to H^{\infty}_{\mu}}, $$
(3)

is obvious.

(b) (c) It is easy to see that the sequence \((j^{\alpha -1} p_{j})_{j\in\mathbb{N}}\) is bounded in \(\mathcal {B}^{\alpha}_{0}\) and

$$\| p_{j}\|_{\mathcal{B}^{\alpha}}=j \biggl(\frac{2\alpha}{j-1+2\alpha } \biggr)^{\alpha}\biggl(\frac{j-1}{j-1+2\alpha} \biggr)^{\frac {j-1}{2}}, \quad \mbox{for } j\in\mathbb{N}, $$

which implies that \(\|j^{\alpha -1}p_{j}\|_{\mathcal{B}^{\alpha}}\approx1\). Notice that \((D^{n}_{\varphi, u} p_{n})(z)=u(z)n! \), \(z\in\mathbb{D}\), while for \(j< n\), \(D^{n}_{\varphi, u} (p_{j}) =0\). Therefore, by the boundedness of \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}\), we get

$$j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} = \bigl\Vert D^{n}_{\varphi, u} \bigl(j^{\alpha -1}p_{j}\bigr)\bigr\Vert _{H^{\infty}_{\mu}} \leq C\bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}< \infty, $$

for every \(j\in \mathbb{N}\), proving (c), as well as the asymptotic relation

$$ M_{1}\preceq\bigl\Vert D^{n}_{\varphi, u} \bigr\Vert _{\mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}}. $$
(4)

(c) (a) If \(\|\varphi\|_{\infty}=\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1\), then by Proposition 8 in [1], we have

$$\bigl\| D^{n}_{\varphi, u} f\bigr\| _{H_{\mu}^{\infty}}\preceq \frac{\|u\|_{H_{\mu}^{\infty}}\|f\|_{{\mathcal{B}}^{\alpha}}}{(1-\|\varphi\|_{\infty}^{2})^{n+\alpha-1}}, $$

from which the boundedness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) follows in this case.

Now assume that \(\|\varphi\|_{\infty}=1\). Let \(\mathbb{D}_{j}=\{z\in \mathbb{D}: r_{j} \leq |\varphi(z)|< r_{j+1}\}\) where \(r_{j}=(j-n)/(j+\alpha-1)\) for \(j\ge n\). Then from Lemma 1 in [16], which also holds for \(m=0\), i.e., \(n=1\) in our case, we have that there is a \(\delta>0\) such that

$$\min_{z\in \mathbb{D}_{j}}j^{\alpha-1}j(j-1)\cdots(j-n+1)\bigl|\varphi(z)\bigr|^{j-n} \bigl(1-\bigl|\varphi(z)\bigr|\bigr)^{\alpha+n-1}\geq \delta, $$

for every \(j\ge k+1\), where k is the smallest natural number such that \(\mathbb{D}_{k}\ne\emptyset\).

Fix \(N\ge k+1\). Then, clearly \(N\ge n+1\) and we have

$$ \bigl\| D^{n}_{\varphi, u}f\bigr\| _{H^{\infty}_{\mu}}\leq \sup_{|\varphi(z)|< \frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr|+ \sup_{|\varphi(z)|\ge \frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr|. $$
(5)

The finiteness of \(M_{1}\) implies \(u\in H_{\mu}^{\infty}\). Hence, as in the first case, we have

$$ \sup_{|\varphi(z)|< \frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr| \preceq \|u\|_{H^{\infty}_{\mu}}\|f\|_{{\mathcal{B}}^{\alpha}}. $$
(6)

On the other hand, since \(\mathbb{D}\setminus \{|\varphi(z)|<\frac{N-n}{N+\alpha-1}\}= \bigcup_{j\ge N}\mathbb{D}_{j}\), we get

$$\begin{aligned}& \sup_{|\varphi(z)|\ge\frac{N-n}{N+\alpha-1} }\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr| \\& \quad =\sup_{j\ge N}\sup_{z\in \mathbb{D}_{j}}\mu(z)\bigl|u(z)\bigr|\bigl|f^{(n)}\bigl(\varphi(z)\bigr)\bigr| \\& \quad =\sup_{j\ge N}\sup_{z\in \mathbb{D}_{j}}\mu(z)\bigl|u(z)\bigr| \frac{j^{\alpha-1}j(j-1)\cdots(j-n+1) |\varphi(z)|^{j-n} |f^{(n)}(\varphi(z))|(1-|\varphi(z)|)^{\alpha+n-1}}{j^{\alpha-1}j(j-1)\cdots(j-n+1)(1-|\varphi(z)|)^{\alpha+n-1}|\varphi(z)|^{j-n}} \\& \quad \preceq\frac{\|f\|_{{\mathcal{B}}^{\alpha}}}{\delta} \sup_{j\ge N} j^{\alpha-1} \bigl\| D^{n}_{\varphi, u}(p_{j})\bigr\| _{H^{\infty}_{\mu}} \leq \frac{M_{1}}{\delta}\|f\|_{{\mathcal{B}}^{\alpha}}< \infty. \end{aligned}$$
(7)

From (5), (6) and (7), the boundedness of \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) follows.

(c) (d) First note that (c) implies that \(u\in H^{\infty}_{\mu}\). Further, since

$$\sup_{z\in{\mathbb{D}}}\bigl(1-|z|^{2}\bigr)^{\alpha}\bigl|f'_{w}(z)\bigr|=\sup_{z\in {\mathbb{D}}} \bigl(1-|z|^{2}\bigr)^{\alpha}\frac{|\alpha \overline{w}|(1-|w|^{2})}{|1-\overline{w} z|^{\alpha +1}}\leq |\alpha |2^{\alpha +1}, \quad w\in{\mathbb{D}}, $$

the family of functions \((f_{w})_{w\in\mathbb{D}}\) is uniformly bounded in \(\mathcal{B}^{\alpha}\). Furthermore

$$f_{w}(z)=\bigl(1-|w|^{2}\bigr) \sum _{j=0}^{\infty}\frac{\Gamma(j+\alpha)}{j!\Gamma(\alpha )}\overline{w}^{j}z^{j},\quad z \in \mathbb{D}. $$

By Stirling’s formula, we have \(\frac{\Gamma(j+\alpha)}{j!\Gamma(\alpha)}\approx j^{\alpha-1} \) as \(j\rightarrow\infty\). Using this fact, the linearity and continuity of the operator, we get

$$\bigl\Vert D^{n}_{\varphi, u} f_{w}\bigr\Vert _{H^{\infty}_{\mu}} \leq C\bigl(1-|w|^{2}\bigr)\sum _{j=n}^{\infty}|w|^{j}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u} (p_{j})\bigr\Vert _{H^{\infty}_{\mu}}\preceq M_{1}< \infty,\quad w\in\mathbb{D}. $$

Consequently, \(\sup_{w\in\mathbb{D}} \|D^{n}_{\varphi, u} f_{\varphi(w)}\|_{H^{\infty}_{\mu}}\preceq M_{1}\), and along with the inequality \(n^{\alpha -1}n!\|u\|_{H^{\infty}_{\mu}}\leq M_{1}\), obtained by considering \(\|D^{n}_{\varphi, u}(n^{\alpha -1}p_{n})\|_{H^{\infty}_{\mu}}\), we also have

$$ \max\bigl\{ M_{2},\Vert u\Vert _{H^{\infty}_{\mu}}\bigr\} \preceq M_{1}. $$
(8)

(d) (e) For \(\lambda\in\mathbb{D}\), it follows from (d) and (1) that

$$\begin{aligned} M_{2} \geq \bigl\Vert D^{n}_{\varphi,u} f_{\varphi(\lambda)}\bigr\Vert _{H^{\infty}_{\mu}} \geq\frac{\mu(\lambda)|u(\lambda)| |\varphi(\lambda)|^{n}\prod_{j=0}^{n-1}(\alpha +j)}{(1-|\varphi(\lambda)|^{2})^{n+\alpha -1}}. \end{aligned}$$
(9)

For any fixed \(r\in(0,1)\), from (9), we have

$$ \sup_{|\varphi(\lambda)|>r} \frac{\mu(\lambda)|u(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{ n+\alpha -1}} \leq \sup_{|\varphi(\lambda)|>r} \frac{ |\varphi(\lambda)|^{n }}{r^{n }} \frac{\mu(\lambda) |u(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{ n+\alpha -1}} \preceq\frac{M_{2}}{r^{n }}. $$
(10)

On the other hand, from \(u \in H^{\infty}_{\mu}\), we have

$$\begin{aligned} \sup_{|\varphi(\lambda)|\leq r}\frac{\mu(\lambda)|u(\lambda)| }{(1-|\varphi(\lambda)|^{2})^{ n+\alpha -1}} & \leq\frac{\sup_{|\varphi(\lambda)|\leq r} \mu (\lambda) |u(\lambda)|}{(1-r^{2})^{ n+\alpha -1}} \\ &\leq \frac{\|u\|_{H^{\infty}_{\mu}}}{(1-r^{2})^{ n+\alpha -1}}< \infty. \end{aligned}$$
(11)

Therefore, (10) and (11) yield the inequality of (e), as well as the asymptotic relation

$$ M_{3}\preceq \max\bigl\{ M_{2},\Vert u\Vert _{H^{\infty}_{\mu}}\bigr\} . $$
(12)

(e) (a) By Proposition 8 in [1], if \(f \in \mathcal{B}^{\alpha}\) and \(k\in\mathbb{N}\), we see that

$$\sup_{z\in\mathbb{D}}\bigl(1-|z|^{2}\bigr)^{k+\alpha -1}\bigl\vert f^{(k)}(z)\bigr\vert \leq C\|f\| _{\mathcal {B}^{\alpha}}, $$

for some constant C independent of f. Therefore, for \(z\in\mathbb{D}\), we have

$$\begin{aligned} \mu(z)\bigl\vert \bigl(D^{n}_{\varphi,u} f\bigr) (z) \bigr\vert =&\mu(z)\bigl\vert u(z)\bigr\vert \bigl\vert f^{(n)}\bigl( \varphi(z)\bigr)\bigr\vert \\ \leq& C\frac{\mu(z)|u(z)| }{(1-|\varphi(z)|^{2})^{n+\alpha -1}}\|f\|_{\mathcal{B}^{\alpha}}, \end{aligned}$$
(13)

where C is independent of f. Taking the supremum in (13) over \(\mathbb{D}\) and then using the first condition in (e) we see that \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is bounded, and

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{\mathcal{B}^{\alpha}\to H^{\infty}_{\mu}}\preceq M_{3}. $$
(14)

If the operator \(D^{n}_{\varphi, u} : \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded, then from (3), (4), (8), (12), and (14), we obtain (2), completing the proof. □

3 Compactness and essential norm of \(D^{n}_{\varphi,u}: \mathcal {B}^{\alpha}\ (\mbox{or } \mathcal{B}^{\alpha}_{0})\rightarrow H^{\infty}_{\mu}\)

In this section we will give an estimate for the essential norm of the operator \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\), as well as of \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}\). For this purpose, we state several lemmas, which will be used in the proof of the main result.

Lemma 3.1

[16]

Let \(\alpha >0\), \(m\geq n+1\), where \(n\in\mathbb{N}\). Define the function \(H_{m,\alpha }:[0,1]\rightarrow[0,\infty)\) by

$$ H_{m,\alpha}(x)=\frac{m!}{(m-n-1)!}x^{m-n-1}(1-x)^{n+\alpha }. $$

Then the following statements hold:

  1. (i)
    $$\max_{0\leq x\leq1}H_{m, \alpha}(x)=H_{m,\alpha}(r_{m})= \left \{ \textstyle\begin{array}{l@{\quad}l} (n+1)!, & m=n+1, \\ \frac{m!}{(m-n-1)!} (\frac{m-n-1}{m+\alpha-1} )^{m-n-1} (\frac{n+\alpha}{m+\alpha-1} )^{\alpha+n} , & m > n+1 , \end{array}\displaystyle \right . $$

    where

    $$r_{m}= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, & m=n+1, \\ \frac{m-n-1}{m+\alpha-1} , & m > n+1 . \end{array}\displaystyle \right . $$
  2. (ii)

    For \(m>n+1\), \(H_{m,\alpha }\) is decreasing on \([r_{m}, r_{m+1}]\), and so

    $$ \min_{r_{m}\leq x\leq r_{m+1}}H_{m, \alpha}(x)=H_{m,\alpha}(r_{m+1})= \frac{m!}{(m-n-1)!} \biggl(\frac {m-n}{m+\alpha} \biggr)^{m-n-1} \biggl( \frac{n+\alpha}{m+\alpha} \biggr)^{\alpha+n} . $$

    Consequently,

    $$ \lim_{m\rightarrow\infty} m^{\alpha-1} \min_{r_{m}\leq x\leq r_{m+1}}H_{m, \alpha}(x) = \frac{(n+\alpha)^{n+\alpha}}{e^{n+\alpha}}. $$

Denote by \(K_{r}f(z)=f(rz)\) for \(r\in(0,1)\) and \(z\in\mathbb{D}\). Then \(K_{r}\) is a compact operator on \({\mathcal{B}}^{\alpha}\) for every \(\alpha >0\), and \(\|K_{r}\|\leq1\) (see, e.g., Proposition 1.3 in [24] and [27]). Let I denote the identity operator. The following three lemmas can be found in [25] (see also [16]).

Lemma 3.2

Let \(0<\alpha<1\). Then there is a sequence \((r_{k})_{k\in\mathbb{N}}\), with \(0< r_{k}<1\) tending to 1, such that the sequence of compact operators \(L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}\), \(j\in\mathbb{N}\), on \(\mathcal {B}_{0}^{\alpha}\) satisfies the following.

  1. (i)

    For any \(t\in(0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0\).

  2. (ii)

    \(\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}^{\alpha}}\leq1}\sup_{z\in\mathbb{D}}|(I-L_{j})f(z)|=0\).

  3. (iii)

    \(\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1\).

Furthermore, these statements hold as well for the sequence of biadjoints \(L^{**}_{j}\) on \(\mathcal{B}^{\alpha}\).

Lemma 3.3

Let \(\alpha=1\). Then there is a sequence \((r_{k})_{k\in\mathbb{N}}\), with \(0< r_{k}<1\) tending to 1, such that the sequence of compact operators \(L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}\), \(j\in\mathbb{N}\), on \(\mathcal{B}_{0}\) satisfies the following.

  1. (i)

    For any \(t\in[0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0\).

  2. (iia)

    \(\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}}\leq1}\sup_{|z|>s}|(I-L_{j})f(z)| (\log\frac {1}{1-|z|^{2}} )^{-1}\leq1\), for s sufficiently close to 1.

  3. (iib)

    \(\lim_{j\rightarrow\infty}\sup_{\|f\| _{\mathcal{B}}\leq1}\sup_{|z|\leq s}|(I-L_{j})f(z)|=0\) for the above s.

  4. (iii)

    \(\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1\).

Furthermore, these statements hold as well for the sequence of biadjoints \(L^{**}_{j}\) on \(\mathcal{B}\).

Lemma 3.4

Let \(\alpha>1\). Then there is a sequence \((r_{k})_{k\in\mathbb{N}}\), with \(0< r_{k}<1\) tending to 1, such that the sequence of compact operators \(L_{j}=\frac{1}{j}\sum^{j}_{k=1}K_{r_{k}}\), \(j\in\mathbb{N}\), on \(\mathcal {B}_{0}^{\alpha}\) satisfies the following.

  1. (i)

    For any \(t\in[0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|((I-L_{j})f)'(z)|=0\).

  2. (ii)

    For any \(t\in[0,1)\), \(\lim_{j\rightarrow \infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{|z|\leq t}|(I-L_{j})f(z)|=0\).

  3. (iii)

    \(\limsup_{j\rightarrow\infty}\|I-L_{j}\| \leq1\).

Furthermore, these statements hold as well for the sequence of biadjoints \(L^{**}_{j}\) on \(\mathcal{B}^{\alpha}\).

To study the compactness, we also need the following lemma, which can be proved in a standard way (see, for example, Proposition 3.11 in [23]).

Lemma 3.5

Let n be a nonnegative integer, \(\alpha >0\), μ a weight, \(u \in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). Then \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\ (\textit{or } \mathcal{B}^{\alpha}_{0})\rightarrow H^{\infty}_{\mu}\) is compact if and only if \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\ (\textit{or } \mathcal {B}^{\alpha}_{0}) \rightarrow H^{\infty}_{\mu}\) is bounded and for any bounded sequence \((f_{k})_{k\in{\mathbb{N}}}\) in \(\mathcal{B}^{\alpha}\), which converges to zero uniformly on compact subsets of \(\mathbb{D}\),

$$\lim_{k\to\infty}\bigl\Vert D^{n}_{\varphi,u} f_{k}\bigr\Vert _{H^{\infty}_{\mu}}=0. $$

Now we are ready to state and prove the main results in this section.

Theorem 3.6

Let n be a positive integer, \(\alpha >0\), μ a weight, \(u\in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Suppose that \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is bounded. Then

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}\approx \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}\approx \limsup_{j\rightarrow\infty}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} . $$
(15)

Proof

First note that the inequality

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}\leq \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} $$
(16)

obviously holds.

Now we give a lower estimate for the essential norm \(\|D^{n}_{\varphi, u}\|_{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}\). Without loss of generality, we assume that \(j\geq n\). Choose the sequence of functions \(q_{j}=j^{\alpha-1} p_{j}\in\mathcal{B}^{\alpha}_{0} \), \(j\in\mathbb{N}\). Then \(\|q_{j}\|_{\mathcal{B}^{\alpha}}\approx1\), and \((q_{j})_{j\in\mathbb{N}}\) converges to zero weakly on \(\mathcal{B}^{\alpha}_{0}\) as \(j\rightarrow\infty\) (see, for example, Theorem 7.5 in [30]). Since by a well-known theorem, for any compact operator \(\widehat{K}:X\to Y\), where X and Y are Banach spaces, the weak convergence \(x_{n}\stackrel{w}{\to}x_{0}\) implies the norm convergence \(\widehat{K}x_{n}\to\widehat{K}x_{0}\) [31], we have

$$ \lim_{j\rightarrow\infty}\|Kq_{j}\|_{H^{\infty}_{\mu}}=0, $$
(17)

for any given compact operator K from \(\mathcal{B}^{\alpha}_{0}\) to \(H^{\infty}_{\mu}\).

Hence

$$\bigl\Vert D^{n}_{\varphi, u}-K\bigr\Vert _{\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}} \succeq\bigl\Vert \bigl(D^{n}_{\varphi, u}-K\bigr)q_{j} \bigr\Vert _{H^{\infty}_{\mu}}\geq\bigl\Vert D^{n}_{\varphi, u} q_{j}\bigr\Vert _{H^{\infty}_{\mu}}-\Vert Kq_{j}\Vert _{H^{\infty}_{\mu}}. $$

Letting \(j\to\infty\) in the last relation and using (17), we obtain

$$\bigl\Vert D^{n}_{\varphi, u}-K\bigr\Vert _{\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}} \succeq\limsup_{j\rightarrow\infty}\bigl\Vert D^{n}_{\varphi, u} q_{j}\bigr\Vert _{H^{\infty}_{\mu}}=\limsup_{j\rightarrow\infty}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{j})\bigr\Vert _{H^{\infty}_{\mu}}, $$

and consequently

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}_{0}\rightarrow H^{\infty}_{\mu}}=\inf _{K} \bigl\Vert D^{n}_{\varphi, u}-K\bigr\Vert _{\mathcal{B}^{\alpha}_{0} \rightarrow H^{\infty}_{\mu}}\succeq\limsup_{j\rightarrow\infty}j^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{j})\bigr\Vert _{H^{\infty}_{\mu}} . $$
(18)

Now, we give the upper estimates for the essential norm \(\|D^{n}_{\varphi, u}\|_{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}\). For the case of \(\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1\), there is a number \(\delta\in(0,1)\) such that \(\sup_{z\in{\mathbb{D}}}|\varphi (z)|<\delta\). In this case, the operator \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is compact. Indeed, choose a bounded sequence \((f_{j})_{j\in\mathbb{N}}\) in \(\mathcal {B}^{\alpha}\) which converges to zero uniformly on compact subsets of \({\mathbb {D}}\). From Cauchy’s integral formula, \((f^{(n)}_{j})_{j\in\mathbb{N}}\) also converges to zero on compact subsets of \({\mathbb{D}}\) as \(j\rightarrow\infty \). Hence

$$\begin{aligned} \lim_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u} f_{j}\bigr\Vert _{H^{\infty}_{\mu}} =& \lim_{j\rightarrow\infty} \sup _{z\in{\mathbb{D}}}\mu(z) \bigl\vert u(z)f_{j}^{(n)} \bigl(\varphi(z)\bigr)\bigr\vert \\ \leq&\Vert u\Vert _{H^{\infty}_{\mu}} \lim_{j\rightarrow \infty} \sup _{z\in{\mathbb{D}}}\bigl\vert f_{j}^{(n)}\bigl(\varphi(z) \bigr)\bigr\vert \\ =&\Vert u\Vert _{H^{\infty}_{\mu}}\lim_{j\rightarrow\infty} \sup _{|w|\leq \delta}\bigl\vert f_{j}^{(n)}(w)\bigr\vert =0. \end{aligned}$$

From this and by Lemma 3.5 we see that the operator \(D^{n}_{\varphi, u}:\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}\) is compact. This also shows that

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}=0. $$
(19)

From (16), (18), and (19), we get the desired result in the case \(\sup_{z\in{\mathbb{D}}}|\varphi(z)|<1\).

Next, we assume that \(\sup_{z\in{\mathbb{D}}}|\varphi(z)|=1\). Let \((L_{j})_{j\in\mathbb{N}}\) be the sequence of operators given in Lemmas 3.2-3.4. Since \(L_{j}^{**}\) is compact on \(\mathcal{B}^{\alpha}\), for every \(j\in\mathbb{N}\), and \(D^{n}_{\varphi, u}\) is bounded from \(\mathcal {B}^{\alpha}\) to \(H^{\infty}_{\mu}\), then \(D^{n}_{\varphi, u} L_{j}^{**}\) is also compact from \(\mathcal{B}^{\alpha}\) to \(H^{\infty}_{\mu}\). Hence

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \leq& \limsup_{j\rightarrow\infty}\bigl\Vert D^{n}_{\varphi, u} -D^{n}_{\varphi, u} L_{j}^{**}\bigr\Vert _{\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \\ =&\limsup_{j\rightarrow\infty}\bigl\Vert D^{n}_{\varphi, u} \bigl(I-L_{j}^{**}\bigr)\bigr\Vert _{\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \\ =&\limsup_{j\rightarrow\infty}\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq 1}\bigl\Vert D^{n}_{\varphi, u}\bigl(I-L_{j}^{**}\bigr)f\bigr\Vert _{H^{\infty}_{\mu}} \\ =&\limsup_{j\rightarrow\infty}\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq 1}\sup _{z\in{\mathbb{D}}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert . \end{aligned}$$

For each positive integer \(i\geq n\), we define \({\mathbb{D}}_{i}=\{z\in {\mathbb{D}}: r_{i}\leq|\varphi(z)|< r_{i+1}\}\), where \(r_{i}\) is given in Lemma 3.1. Let k be the smallest positive integer such that \({\mathbb{D}}_{k}\neq\emptyset\). Since \(\sup_{z\in{\mathbb {D}}}|\varphi(z)|=1\), \({\mathbb{D}}_{i}\) is not empty for every integer \(i\geq k\) and \(\mathbb{D}=\bigcup^{\infty}_{i=k}{\mathbb{D}}_{i}\), we have

$$\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq 1}\sup_{z\in{\mathbb{D}}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**}\bigr)f\bigr)^{(n)}\bigl( \varphi(z)\bigr)\bigr\vert =I_{1}+I_{2}, $$

where

$$I_{1}=\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{k\leq i\leq N-1}\sup _{z\in{\mathbb{D}}_{i}}\mu (z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert $$

and

$$I_{2}=\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq1}\sup_{N\leq i}\sup _{z\in {\mathbb{D}}_{i}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert . $$

Here N is a positive integer determined as follows.

By Lemma 3.1, \(\lim_{i\rightarrow\infty} \frac{i^{1-\alpha }}{H_{i,\alpha}(r_{i+1})}= \frac{e^{n+\alpha}}{(n+\alpha)^{n+\alpha}} \). Hence, for any given \(\varepsilon>0\), there exists an \(N\in\mathbb{N}\) such that

$$\frac{i^{ 1-\alpha}}{H_{ i,\alpha}(r_{i+1})}\leq \frac{e^{n+\alpha}}{(n+\alpha)^{n+\alpha}} +\varepsilon $$

when \(i\geq N\). For such N it follows that

$$\begin{aligned} I_{2} =&\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq1}\sup_{N\leq i} \sup_{z\in{\mathbb{D}}_{i}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \\ =& \sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq1}\sup_{N\leq i}\sup _{z\in {\mathbb{D}}_{i}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \frac{H_{ i,\alpha}(|\varphi(z)|)}{i^{1-\alpha}} \frac{i^{1-\alpha } }{H_{ i,\alpha}(|\varphi(z)|)} \\ \preceq& \biggl(\frac{e^{n+\alpha}}{(n+\alpha)^{n+\alpha }}+\varepsilon \biggr)\sup_{\Vert f\Vert _{\mathcal{B}^{\alpha}}\leq 1} \bigl\Vert \bigl(I-L_{j}^{**}\bigr)f\bigr\Vert _{\mathcal{B}^{\alpha}}\sup_{N\leq i}\sup_{z\in {\mathbb{D}}_{i}} \mu(z) \bigl\vert u(z)\bigr\vert \frac{i!}{(i-n)!}\frac{|\varphi(z)|^{i-n}}{i^{ 1-\alpha }} \\ \preceq&\bigl\Vert I-L_{j}^{**}\bigr\Vert \sup _{N\leq i}i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i}) \bigr\Vert _{H^{\infty}_{\mu}} . \end{aligned}$$

Thus

$$ \limsup_{j\rightarrow\infty}I_{2}\preceq\sup_{i\geq N}i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i})\bigr\Vert _{H^{\infty}_{\mu}} . $$
(20)

By Lemmas 3.2, 3.3, 3.4, and Cauchy’s integral formula, we have

$$\begin{aligned} \limsup_{j\rightarrow\infty}I_{1} =& \limsup _{j\rightarrow\infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq 1}\sup_{k\leq i\leq N-1} \sup_{z\in{\mathbb{D}}_{i}}\mu (z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \\ \leq&\|u\|_{H^{\infty}_{\mu}}\limsup_{j\rightarrow\infty}\sup _{\|f\| _{\mathcal{B}^{\alpha}}\leq 1}\sup_{|\varphi(z)|< r_{N}}\bigl\vert \bigl( \bigl(I-L_{j}^{**}\bigr)f\bigr)^{(n)}\bigl(\varphi(z) \bigr)\bigr\vert =0, \end{aligned}$$

which together with (20) implies that

$$\begin{aligned}& \limsup_{j\rightarrow\infty}\sup_{\|f\|_{\mathcal{B}^{\alpha}}\leq 1}\sup _{z\in{\mathbb{D}}}\mu(z)\bigl\vert u(z) \bigl(\bigl(I-L_{j}^{**} \bigr)f\bigr)^{(n)}\bigl(\varphi (z)\bigr)\bigr\vert \\& \quad \preceq \sup _{i\geq N}i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i}) \bigr\Vert _{H^{\infty}_{\mu}}. \end{aligned}$$

Therefore

$$\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}} \preceq \sup_{i\geq N} i^{\alpha-1}\bigl\Vert D^{n}_{\varphi,u}(p_{i}) \bigr\Vert _{H^{\infty}_{\mu}}. $$

From the last relation we get

$$ \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}^{\alpha}\rightarrow H^{\infty}_{\mu}}\preceq \limsup_{i\rightarrow\infty} i^{\alpha-1} \bigl\Vert D^{n}_{\varphi, u}(p_{i})\bigr\Vert _{H^{\infty}_{\mu}}. $$
(21)

From (16), (18), and (21), the asymptotic relations in (15) follow, completing the proof of the theorem. □

From Theorem 3.6, letting \(\alpha =1\) we deduce the following result.

Corollary 3.7

Let n be a positive integer, μ a weight, \(u \in H(\mathbb{D})\), and φ be an analytic self-map of \(\mathbb{D}\). Suppose that \(D^{n}_{\varphi, u}:\mathcal{B} \rightarrow H^{\infty}_{\mu}\) is bounded. Then

$$\bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B} \rightarrow H^{\infty}_{\mu}}\approx \bigl\Vert D^{n}_{\varphi, u}\bigr\Vert _{e,\mathcal{B}_{0} \rightarrow H^{\infty}_{\mu}}\approx \limsup_{j\rightarrow\infty} \bigl\Vert D^{n}_{\varphi, u}(p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} . $$

Theorem 3.8

Let n be a positive integer, \(\alpha >0\), μ a weight, \(u\in H(\mathbb{D})\) and φ be an analytic self-map of \(\mathbb{D}\). If \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is bounded, then the following statements are equivalent.

  1. (a)

    The operator \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}\to H^{\infty}_{\mu}\) is compact.

  2. (b)

    The operator \(D^{n}_{\varphi,u}: \mathcal{B}^{\alpha}_{0} \to H^{\infty}_{\mu}\) is compact.

  3. (c)

    \(\lim_{j\rightarrow\infty} j^{\alpha -1}\|D^{n}_{\varphi, u} (p_{j})\|_{H^{\infty}_{\mu}}=0\).

  4. (d)

    \(\lim_{|\varphi(w)|\rightarrow1} \|D^{n}_{\varphi,u} f_{\varphi (w)}\|_{H^{\infty}_{\mu}}=0\).

  5. (e)

    \(\lim_{|\varphi(z)|\rightarrow 1}\frac{\mu(z)|u(z)| }{(1-|\varphi(z)|^{2})^{n+\alpha -1}}=0\).

Proof

The equivalence of statements (a)-(c) follows from Theorem 3.6.

(c) (d) From (c), we see that, for every \(\varepsilon>0\), there is an \(N\in\mathbb{N}\) such that

$$j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}}< \varepsilon/2, $$

for all \(j\geq N\).

Let \((z_{k})_{k\in\mathbb{N}}\subset{\mathbb{D}}\) be an arbitrary sequence such that \(|\varphi (z_{k})|\to1\) as \(k\to\infty\) (if such a sequence does not exist then the equality in (d) vacuously holds). Similarly to the proof of Theorem 2.1, we have

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}f_{\varphi (z_{k})}\bigr\Vert _{H^{\infty}_{\mu}} \leq & C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{2}\bigr)\sum_{j=n}^{\infty}\bigl\vert \varphi (z_{k})\bigr\vert ^{j}j^{\alpha -1} \bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} \\ =&C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{2} \bigr)\sum_{j=n}^{N-1} \bigl\vert \varphi (z_{k})\bigr\vert ^{j}j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} \\ &{}+C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{2} \bigr)\sum_{j=N}^{\infty} \bigl\vert \varphi (z_{k})\bigr\vert ^{j}j^{\alpha -1}\bigl\Vert D^{n}_{\varphi, u} (p_{j}) \bigr\Vert _{H^{\infty}_{\mu}} \\ \leq & 2C\bigl(1-\bigl\vert \varphi (z_{k})\bigr\vert ^{N}\bigr)M_{0}+C\varepsilon, \end{aligned}$$
(22)

for \(k\in\mathbb{N}\), where \(M_{0}=\max_{n\leq j \leq N-1}j^{\alpha -1}\|D^{n}_{\varphi, u} (p_{j}) \|_{H^{\infty}_{\mu}}\).

Since \(|\varphi (z_{k})|\to1\) as \(k\to\infty\), from (22), we deduce that

$$\limsup_{k\to\infty}\bigl\Vert D^{n}_{\varphi,u}f_{\varphi (z_{k})} \bigr\Vert _{H^{\infty}_{\mu}} \leq C\varepsilon. $$

Since ε is an arbitrary positive number, the implication follows.

(d) (e) Let \((z_{k})_{k\in\mathbb{N}}\) be a sequence in \(\mathbb{D}\) such that \(\lim_{k\to\infty}|\varphi (z_{k})|=1\) (if such a sequence does not exist then the implication vacuously holds). Since the sequence \((f_{\varphi (z_{k})})_{k\in\mathbb{N}}\) is bounded in \(\mathcal{B}^{\alpha}\) and converges to 0 uniformly on compact subsets of \(\mathbb{D}\), by (9) and Lemma 3.5, we have

$$\frac{\mu(z_{k}) \vert u(z_{k})\vert |\varphi(z_{k})|^{n}\prod_{j=0}^{n-1}(\alpha +j)}{(1-|\varphi (z_{k})|^{2})^{n+\alpha -1}}\leq \bigl\Vert D^{n}_{\varphi,u} f_{\varphi (z_{k})} \bigr\Vert _{H^{\infty}_{\mu}} \rightarrow0\quad \mbox{as } k \rightarrow\infty. $$

Therefore

$$ \lim_{k\to\infty}\frac{\mu(z_{k}) |u(z_{k})| }{(1-|\varphi(z_{k})|^{2})^{n+\alpha -1}}= \lim_{k\to\infty} \frac{\mu(z_{k}) |u(z_{k})| |\varphi(z_{k})|^{n } }{(1-|\varphi(z_{k})|^{2})^{n+\alpha -1}}=0, $$
(23)

which implies (e).

(e) (a) Assume \((f_{k})_{k\in\mathbb{N}}\) is a bounded sequence in \(\mathcal{B}^{\alpha}\) converging to 0 uniformly on compact subsets of \(\mathbb{D}\). By the assumption, for any \(\varepsilon>0\), there exists a \(\delta\in(0,1)\) such that

$$ \frac{\mu(z)|u(z)|}{(1-|\varphi(z)|^{2})^{n+\alpha -1}}< \varepsilon $$
(24)

when \(\delta<|\varphi(z)|<1\).

Therefore, since \(u\in H^{\infty}_{\mu}\) we have

$$\begin{aligned} \bigl\Vert D^{n}_{\varphi,u}f_{k}\bigr\Vert _{H^{\infty}_{\mu}} =& \sup_{z\in\mathbb {D}} \mu(z)\bigl\vert \bigl(D^{n}_{\varphi,u}f_{k}\bigr) (z)\bigr\vert \\ \leq& \sup_{z\in\Omega_{\delta}}\mu(z)\bigl\vert u(z)\bigr\vert \bigl\vert f_{k}^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert \\ &{}+C\sup _{z\in\mathbb{D}\setminus\Omega_{\delta}} \frac{\mu(z)|u(z)| }{(1-|\varphi(z)|^{2})^{n+\alpha -1}} \Vert f_{k}\Vert _{\mathcal{B}^{\alpha}} \\ \leq& \Vert u\Vert _{H^{\infty}_{\mu}}\sup_{z\in\Omega_{\delta}} \bigl\vert f_{k}^{(n)}\bigl(\varphi(z)\bigr)\bigr\vert +C \varepsilon \Vert f_{k}\Vert _{\mathcal{B}^{\alpha}}, \end{aligned}$$
(25)

where \(\Omega_{\delta}=\{ z\in\mathbb{D}:|\varphi(z)| \leq\delta\}\).

Since \((f_{k})_{k\in\mathbb{N}}\) converges to 0 uniformly on compact subsets of \(\mathbb{D}\), by Cauchy’s estimate so do the sequences \((f^{(n)}_{k})_{k\in\mathbb{N}}\) for every \(n\in\mathbb{N}\). Letting \(k\to \infty\) in (25) and using the fact that ε is an arbitrary positive number, we obtain \(\lim_{k\to\infty}\|D^{n}_{\varphi,u} f_{k}\|_{H^{\infty}_{\mu}}=0\). By Lemma 3.5, we deduce that \(D^{n}_{\varphi,u}:\mathcal{B}^{\alpha}\to H_{\mu}^{\infty}\) is compact. □

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Acknowledgements

Songxiao Li was supported by the Macao Science and Technology Development Fund (No. 083/2014/A2). Stevo Stević was supported by the Serbian Ministry of Science projects III 41025 and III 44006.

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Li, S., Stević, S. Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces. J Inequal Appl 2015, 265 (2015). https://doi.org/10.1186/s13660-015-0770-9

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