Path Components of the Space of (Weighted) Composition Operators on Bergman Spaces

Abstract

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $A_{\alpha }^p$ with $p \in (0, \infty )$ and $\alpha \in (-1, \infty )$. In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

1 Introduction

Let $H({\mathbb {D}})$ be the space of all holomorphic functions on the unit disc ${\mathbb {D}}$. Given $p \in (0, \infty )$ and $\alpha \in (-1, \infty )$, the Bergman space $A_{\alpha }^p$ is defined as follows

$$\begin{aligned} A_{\alpha }^p := \left\{ f \in H({\mathbb {D}}): \Vert f\Vert _{p,\alpha } := \left( \int _{{\mathbb {D}}}|f(z)|^p dA_{\alpha }(z) \right) ^{\frac{1}{p}} < \infty \right\} \end{aligned}$$

with

$$\begin{aligned} dA_{\alpha }(z) := \dfrac{1+\alpha }{\pi } \left( 1-|z|^2\right) ^{\alpha }dA(z), \end{aligned}$$

where dA(z) is the Lebesgue measure on ${\mathbb {D}}$. It is well known that $A_{\alpha }^p$ with $1 \le p < \infty$ is a Banach space, while for $0< p < 1$, $A_{\alpha }^p$ is a complete metric space with the distance $d(f,g) := \Vert f-g\Vert _{p, \alpha }^p$.

Let ${{\mathcal {S}}}({\mathbb {D}})$ be the set of all holomorphic self-maps of ${\mathbb {D}}$. For two functions $\varphi \in {{\mathcal {S}}}({\mathbb {D}})$ and $\psi \in H({\mathbb {D}})$, a weighted composition operator $W_{\psi , \varphi }$ is defined by $W_{\psi , \varphi }f := \psi \cdot (f\circ \varphi ), \ f \in H({\mathbb {D}})$. In particular, when $\psi$ is identically 1, $W_{\psi , \varphi }$ reduces to a composition operator $C_{\varphi }$. According to Littlewood’s Subordinate Theorem, each composition operator $C_{\varphi }$ is bounded on Bergman spaces $A_{\alpha }^p$, while compactness of $C_{\varphi }$ on $A_{\alpha }^p$ was firstly characterized by MacCluer and Shapiro [16] in terms of angular derivative of $\varphi$. In details, the operator $C_{\varphi }$ is compact on $A_{\alpha }^p$ if and only if $\varphi$ has no finite angular derivative at any point $\zeta$ of $\partial {\mathbb {D}}$, which is equivalent to

$$\begin{aligned} \lim _{|z| \rightarrow 1^-} \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} = 0. \end{aligned}$$

(1.1)

Boundedness and compactness of weighted composition operators $W_{\psi , \varphi }$ on Bergman spaces $A_{\alpha }^p$ were investigated by C̆uc̆ković and Zhao [7] in terms of Berezin type transforms, which are rather difficult to use.

When the basic questions on topological properties of (weighted) composition operators were completely solved, many researchers have paid more attention to the study of the topological structure of the space of such operators endowed with the operator norm topology. This problem was initiated by Berkson [2] with his isolation result on composition operators on the Hardy space $H^2$, and then developed by MacCluer [15], Shapiro and Sundberg [19]. Thereafter, the topological structure problem has been intensively investigated on various function spaces during the past few decades (see, for instance, [9, 12] on Hardy spaces, [8, 17] on Bergman spaces, [13, 14] on the space $H^{\infty }$ of all bounded holomorphic functions on ${\mathbb {D}}$, [11] on Bloch spaces, [1] on weighted Banach spaces with sup-norm, and [20] on Fock spaces).

In this paper we are interested in the topological structure problem on Bergman spaces $A_{\alpha }^p$. Recall that the authors in [8, 15, 17] studied this problem only for composition operators on Hilbert Bergman spaces $A_{\alpha }^2$ and obtained some partial results. Firstly, MacCluer [15] gave a sufficient condition for isolated composition operators and a necessary condition for a composition operator to be in the path component of another one. Later, Moorhouse [17] established a sufficient condition under which two composition operators belong to the same path component. Recently, Dai [8] stated a criterion for two composition operators to be linearly connected. It is worth mentioning that in the Bergman space setting, till now there is no complete description of (path) components in the space of composition operators; moreover, the space of weighted composition operators has not yet been studied.

The aim of this paper is, firstly, to continue studying the topological structure problem for composition operators on Bergman spaces $A_{\alpha }^p$ with $p \in (0, \infty )$; secondly, to initiate this problem for weighted composition operators on these spaces $A_{\alpha }^p$. Note that the technique of adjoint operators on Hilbert Bergman spaces $A_{\alpha }^2$, played an essential role in [8, 15, 17], does not work for general spaces $A_{\alpha }^p$. So we develop a new approach based on Carleson measure.

The paper is organized as follows. In Sect.2 we recall some preliminary results on pseudo-hyperbolic distance and Carleson measure for Bergman spaces $A_{\alpha }^p$.

Section 3 is devoted to the topological structure of the space ${{\mathcal {C}}}(A_{\alpha }^p)$ of composition operators on $A_{\alpha }^p$ endowed with the operator norm topology. In Theorem 3.3, we prove that two operators $C_{\varphi }$ and $C_{\phi }$ are linearly connected in the space ${{\mathcal {C}}}(A_{\alpha }^p)$, i.e. the path $C_{\varphi _t}$ with $\varphi _t(z) := (1-t)\varphi (z) + t\phi (z), \ t \in [0,1]$, is continuous in ${{\mathcal {C}}}(A_{\alpha }^p)$, if and only if

$$\begin{aligned} \limsup _{\rho (\varphi (z), \phi (z)) \rightarrow 1} \left( \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} + \dfrac{1 - |z|^2}{1 - |\phi (z)|^2} \right) = 0, \end{aligned}$$

where $\rho (\varphi (z), \phi (z))$ is the pseudo-hyperbolic distance between $\varphi (z)$ and $\phi (z)$. This result implies that the set $[C_{\varphi }]$ of all composition operators that differ from the given one $C_{\varphi }$ by a compact operator is path connected in ${{\mathcal {C}}}(A_{\alpha }^p)$ (Theorem 3.6). Moreover, the set ${{\mathcal {C}}}_0(A_{\alpha }^p)$ of all compact composition operators on $A_{\alpha }^p$ forms a path component in ${{\mathcal {C}}}(A_{\alpha }^p)$ (Corollary 3.8). However, as in the setting of other function spaces, such as the Hardy space $H^2$, the space $H^{\infty }$, and weighted Banach space with sup-norm, the set of such a type, in general, is not always a component of ${{\mathcal {C}}}(A_{\alpha }^p)$ (Example 3.10). Finally, we also give sufficient conditions for isolated and non-isolated points in ${{\mathcal {C}}}(A_{\alpha }^p)$ (Proposition 3.11).

The space ${{\mathcal {C}}}_w(A_{\alpha }^p)$ of all nonzero weighted composition operators on $A_{\alpha }^p$ under the operator norm topology is studied in Sect. 4. We show that the set ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$ of all nonzero compact weighted composition operators on $A_{\alpha }^p$ is path connected in ${{\mathcal {C}}}_w(A_{\alpha }^p)$; nevertheless, it is not a component in this space (Theorem 4.2). Moreover, we provide two path connected sets of the same type in the space ${{\mathcal {C}}}_w(A_{\alpha }^p)$, one of which is a path component, while another one is not (Examples 4.5 and 4.6).

Notation. Throughout this paper we always assume that $p \in (0, \infty )$ and $\alpha \in (-1, \infty )$ unless otherwise is stated. We denote constants by $c, c_0, c_1,...$ to distinguish from composition operators $C_{z_0}$ induced by $\varphi (z) \equiv z_0$. We also use the notation $A \lesssim B$ (and $A > rsim B$) for nonnegative quantities A and B to mean that there is a constant $c> 0$, depending only on indexes $p, \alpha , \beta , \gamma$, such that $A \le c B$ (and $A \ge c B$, respectively). Finally, the notation $A \simeq B$ means that both $A \lesssim B$ and $B \lesssim A$ hold.

2 Preliminaries

In this section we recall some basic notation, definitions and facts which will be used in the sequel.

2.1 Test Functions

For each $\sigma > 0$ and $w \in {\mathbb {D}}$ fixed, we define

$$\begin{aligned} k_{w}(z) := \dfrac{\left( 1 - |w|^2\right) ^{\frac{\sigma }{p}}}{\left( 1 - {\overline{w}}z\right) ^{\frac{\sigma + \alpha + 2}{p}}}, \quad z \in {\mathbb {D}}. \end{aligned}$$

These functions $k_w$ play an important role in the study of Bergman spaces $A_{\alpha }^p$ and operators defined on them. From [21, Theorem 1.12] it follows that there is a constant $c_0$, depending only on $p, \alpha , \sigma$, such that

$$\begin{aligned} c_0^{-1} \le \Vert k_w\Vert _{p,\alpha } \le c_0 \quad \text { for all } w \in {\mathbb {D}}. \end{aligned}$$

(2.1)

2.2 Pseudo-Hyperbolic Distance

The pseudo-hyperbolic distance between z and $\zeta$ in ${\mathbb {D}}$ is given by

$$\begin{aligned} \rho (z,\zeta ) := \left| \frac{z-\zeta }{1 - {\overline{z}}\zeta } \right| . \end{aligned}$$

The pseudo-hyperbolic disc with center $z \in {\mathbb {D}}$ and radius $r \in (0,1)$ is defined by $\Delta (z,r) := \{\zeta \in {\mathbb {D}}: \rho (z,\zeta ) < r\}.$ For simplicity, we write $\Delta (z)$ instead of $\Delta (z, \frac{1}{2})$.

By [21, Lemma 2.24], there exists a constant $c_1 = c_1(p, \alpha )$ such that

$$\begin{aligned} |f(z)|^p \le \dfrac{c_1^p}{\left( 1 - |z|^2\right) ^{\alpha + 2}} \int _{\Delta (z)}|f(\zeta )|^p dA_{\alpha }(\zeta ), \end{aligned}$$

(2.2)

for every $f \in H({\mathbb {D}})$ and $z \in {\mathbb {D}}$. This implies that

$$\begin{aligned} |f(z)| \le \dfrac{c_1}{\left( 1 - |z|^2\right) ^{\frac{\alpha + 2}{p}}} \Vert f\Vert _{p,\alpha } \end{aligned}$$

(2.3)

and, changing the constant $c_1$ if necessary,

$$\begin{aligned} |f'(z)| \le \dfrac{c_1}{\left( 1 - |z|^2\right) ^{\frac{\alpha + 2}{p} + 1}} \Vert f\Vert _{p,\alpha }, \end{aligned}$$

(2.4)

for every $f \in A_{\alpha }^p$ and $z \in {\mathbb {D}}$.

Next, by [21, Lemma 2.20], for every $r \in (0,1)$, there is a constant $c_2 = c_2(r) > 0$ such that

$$\begin{aligned} c_2^{-1} \le \dfrac{1-|z|^2}{1 - |\zeta |^2} \le c_2 \ \text { for all } \zeta \in \Delta (z,r) \text { and } z \in {\mathbb {D}}. \end{aligned}$$

(2.5)

The next auxiliary result follows from [5, Lemma 3.2].

Lemma 2.1

For every $r \in (0, 1)$, there is a constant $c_3 = c_3(p, \alpha , r) > 0$ such that

$$\begin{aligned} |f(z) - f(\zeta )|^p \le c_3 \dfrac{\rho (z,\zeta )^p}{\left( 1-|z|^2\right) ^{\alpha + 2}} \int _{\Delta \left( z,\frac{r+1}{2}\right) }|f'(\omega )|^p \left( 1-|\omega |^2\right) ^p dA_{\alpha }(\omega ), \end{aligned}$$

for all $f \in H({\mathbb {D}})$ and $z, \zeta \in {\mathbb {D}}$ with $\zeta \in \Delta (z,r)$.

The following lemma is quite standard. It is originally noticed in [10] for the unit ball and given in [8, Lemma 3.2].

Lemma 2.2

The following inequality holds

$$\begin{aligned} \rho \left( z_t,z_s\right) \le \dfrac{|t-s|}{1 - \left( 1-|t-s|\right) \rho (z,\zeta )}\rho (z,\zeta ), \end{aligned}$$

for every $z, \zeta \in {\mathbb {D}}$ and $t, s \in [0,1]$, where $z_t := (1-t)z + t\zeta$. In particular, $\rho (z_t, z_s) \le \rho (z,\zeta )$ for the same $z, \zeta , z_t$, and $z_s$.

2.3 Carleson Measure

A positive Borel measure $\mu$ on ${\mathbb {D}}$ is called an $\alpha$-Carleson measure, if the embedding operator $I_{\mu }: A_{\alpha }^p \rightarrow L^p({\mathbb {D}}, d\mu )$ is bounded, i.e. there exists a constant $c>0$ such that

$$\begin{aligned} \left( \int _{{\mathbb {D}}}|f(z)|^pd\mu (z)\right) ^{\frac{1}{p}} \le c \Vert f\Vert _{p,\alpha } \ \text { for all } f \in A_{\alpha }^p. \end{aligned}$$

Moreover, if the operator $I_{\mu }$ is compact, then $\mu$ on ${\mathbb {D}}$ is called a compact $\alpha$-Carleson measure. In this case we put $\Vert \mu \Vert _{\alpha }: = \Vert I_{\mu }\Vert ^p_{A_{\alpha }^p \rightarrow L^p({\mathbb {D}}, d\mu )}$. Here and below we omit the dependence on p of norms of measures and operators, since $p>0$ is always an arbitrary fixed number.

By [21, Theorems 2.25 and 2.26], a positive Borel measure $\mu$ is an $\alpha$-Carleson (respectively, a compact $\alpha$-Carleson) one if and only if

$$\begin{aligned} \sup _{z \in {\mathbb {D}}}\dfrac{\mu (\Delta (z,r))}{\left( 1 - |z|^2\right) ^{\alpha + 2}} < \infty \left( \text {respectively, } \lim _{|z| \rightarrow 1^-}\dfrac{\mu (\Delta (z,r))}{\left( 1 - |z|^2\right) ^{\alpha + 2}} = 0 \right) , \end{aligned}$$

for some number $r \in (0,1)$. Note that these conditions are independent of p and r. Moreover, for each $r \in (0,1)$, there is a constant $c_4 = c_4(\alpha ,r)$ such that

$$\begin{aligned} c_4^{-1} \sup _{z \in {\mathbb {D}}}\dfrac{\mu (\Delta (z,r))}{\left( 1 - |z|^2\right) ^{\alpha + 2}} \le \Vert \mu \Vert _{\alpha } \le c_4 \sup _{z \in {\mathbb {D}}}\dfrac{\mu (\Delta (z,r))}{\left( 1 - |z|^2\right) ^{\alpha + 2}}. \end{aligned}$$

(2.6)

On the other hand, in [16, Section 4], the $\alpha$-Carleson measure was also characterized in terms of semidiscs $S(\zeta , \delta )$, where $S(\zeta , \delta ) := \{z \in {\mathbb {D}}: |z - \zeta | < \delta \}$ with $\delta \in (0, 2]$ and $\zeta \in \partial {\mathbb {D}}$. By [16, Theorem 4.3], a positive Borel measure $\mu$ is an $\alpha$-Carleson (respectively, a compact $\alpha$-Carleson) one if and only if

$$\begin{aligned} \sup _{\delta \in (0,2], \zeta \in \partial {\mathbb {D}}} \dfrac{\mu (S(\zeta , \delta ))}{\delta ^{\alpha + 2}} < \infty \left( \text {respectively, } \lim _{\delta \rightarrow 0} \sup _{\zeta \in \partial {\mathbb {D}}} \dfrac{\mu (S(\zeta , \delta ))}{\delta ^{\alpha + 2}} = 0\right) . \end{aligned}$$

In addition,

$$\begin{aligned} \Vert \mu \Vert _{\alpha } \simeq \sup _{\delta \in (0,2], \zeta \in \partial {\mathbb {D}}} \dfrac{\mu (S(\zeta , \delta ))}{\delta ^{\alpha + 2}}. \end{aligned}$$

For a function $\varphi \in {{\mathcal {S}}}({\mathbb {D}})$ and a Borel function $v: {\mathbb {D}}\rightarrow [0, \infty )$, we define the pull-back measure $(v A_{\alpha }) \circ \varphi ^{-1}$ by

$$\begin{aligned} \left( v A_{\alpha }\right) \circ \varphi ^{-1}(E) := \int _{\varphi ^{-1}(E)} v(z)dA_{\alpha }(z), \end{aligned}$$

for each Borel set $E \subset {\mathbb {D}}$. Then, for each $\varphi \in {{\mathcal {S}}}({\mathbb {D}})$ and $f \in A_{\alpha }^p$,

$$\begin{aligned} \Vert C_{\varphi }f\Vert _{p,\alpha } = \left( \int _{{\mathbb {D}}} |f(\varphi (z))|^pdA_{\alpha }(z)\right) ^{\frac{1}{p}} = \left( \int _{{\mathbb {D}}} |f(z)|^pd(A_{\alpha }\circ \varphi ^{-1})(z)\right) ^{\frac{1}{p}}. \end{aligned}$$

From this and the boundedness of $C_{\varphi }$ on $A_{\alpha }^p$, it follows that $A_{\alpha }\circ \varphi ^{-1}$ is always an $\alpha$-Carleson measure and

$$\begin{aligned} \Vert A_{\alpha }\circ \varphi ^{-1}\Vert _{\alpha } = \Vert C_{\varphi }\Vert _{\alpha }^p \le \left( \dfrac{1 + |\varphi (0)|}{1 - |\varphi (0)|}\right) ^{\alpha + 2}. \end{aligned}$$

(2.7)

Moreover, $C_{\varphi }$ is compact on $A_{\alpha }^p$ if and only if $A_{\alpha }\circ \varphi ^{-1}$ is a compact $\alpha$-Carleson measure.

For each $\varphi \in {{\mathcal {S}}}({\mathbb {D}})$, $\psi \in H({\mathbb {D}})$, and $f \in A_{\alpha }^p$,

$$\begin{aligned} \Vert W_{\psi , \varphi }f\Vert _{p,\alpha }&= \left( \int _{{\mathbb {D}}} |\psi (z)|^p |f(\varphi (z))|^pdA_{\alpha }(z)\right) ^{\frac{1}{p}} \\&= \left( \int _{{\mathbb {D}}} |f(z)|^pd((|\psi |^pA_{\alpha })\circ \varphi ^{-1})(z)\right) ^{\frac{1}{p}}. \end{aligned}$$

This implies that $W_{\psi , \varphi }$ is bounded (respectively, compact) on $A_{\alpha }^p$ if and only if $(|\psi |^pA_{\alpha })\circ \varphi ^{-1}$ is an $\alpha$-Carleson measure (respectively, a compact $\alpha$-Carleson measure). Moreover, we have

$$\begin{aligned} \Vert \left( |\psi |^pA_{\alpha }\right) \circ \varphi ^{-1}\Vert _{\alpha } = \Vert W_{\psi , \varphi }\Vert ^p_{\alpha }. \end{aligned}$$

(2.8)

3 Path Components of the Space of Composition Operators

In this section we study the space ${{\mathcal {C}}}(A_{\alpha }^p)$ of all composition operators on $A_{\alpha }^p$ under the operator norm topology.

First, we establish a necessary and sufficient condition under which two composition operators $C_{\varphi }$ and $C_{\phi }$ are linearly connected in ${{\mathcal {C}}}(A_{\alpha }^p)$, i.e., according to [8], the path $C_{\varphi _t}$ with $\varphi _t(z) := (1 - t)\varphi (z) + t \phi (z), t \in [0,1]$, is continuous in ${{\mathcal {C}}}(A_{\alpha }^p)$. Since $\varphi _t(z)$ lies on a straight-line path between $\varphi (z)$ and $\phi (z)$,

$$\begin{aligned} \dfrac{1}{1 - |\varphi _t(z)|^s} \le \dfrac{1}{1 - |\varphi (z)|^s} + \dfrac{1}{1 - |\phi (z)|^s}, \end{aligned}$$

(3.1)

for every $z \in {\mathbb {D}}$, $t \in [0,1]$, and $s > 0$.

We need the following auxiliary lemmas.

Lemma 3.1

Let $p \in (0,\infty )$ and $\alpha , \gamma \in (-1, \infty )$ with $\beta = \alpha - \gamma \in (0, 1]$. For every two functions $\varphi , \phi$ from ${{\mathcal {S}}}({\mathbb {D}})$ and every Borel function $v: {\mathbb {D}}\rightarrow [0,1]$, the following inequality holds

$$\begin{aligned} \Vert \left( v A_{\alpha }\right) \circ \varphi _t^{-1}\Vert _{\alpha } \lesssim M_{v, \varphi ,\phi }^{\beta } \left( \dfrac{1}{1 - |\varphi (0)|} + \dfrac{1}{1 - |\phi (0)|}\right) ^{\gamma + 2}, \end{aligned}$$

for all $t \in [0,1]$, where, as above, $\varphi _t(z) := (1 - t)\varphi (z) + t \phi (z)$, and

$$\begin{aligned} M_{v, \varphi ,\phi } := \sup _{z \in {\mathbb {D}}} v(z) \left( \dfrac{1-|z|^2}{1-|\varphi (z)|^2} + \dfrac{1-|z|^2}{1-|\phi (z)|^2} \right) . \end{aligned}$$

Proof

Using (2.5) for $r = \frac{1}{2}$ and (3.1), we obtain

$$\begin{aligned}&(v A_{\alpha }) \circ \varphi _t^{-1}(\Delta (z)) = \int _{\varphi _t^{-1}(\Delta (z))} v(\omega )dA_{\alpha }(\omega ) \\&\quad = \; \frac{\alpha + 1}{\gamma + 1}\int _{\varphi _t^{-1}(\Delta (z))} v(\omega )^{1 - \beta } \left( v(\omega )(1-|\omega |^2) \right) ^{\beta }dA_{\gamma }(\omega ) \\&\quad \lesssim \; \sup _{\omega \in \varphi _t^{-1}(\Delta (z))} \left( v(\omega ) \dfrac{1 - |\omega |^2}{1 - |\varphi _t(\omega )|^2} \right) ^{\beta }\int _{\varphi _t^{-1}(\Delta (z))} (1-|\varphi _t(\omega )|^2)^{\beta }dA_{\gamma }(\omega ) \\&\quad \lesssim \; \sup _{\omega \in {\mathbb {D}}} \left( v(\omega ) \left( \dfrac{1 - |\omega |^2}{1 - |\varphi (\omega )|^2} + \dfrac{1 - |\omega |^2}{1 - |\phi (\omega )|^2} \right) \right) ^{\beta } \left( 1 - |z|^2\right) ^{\beta } \int _{\varphi _t^{-1}(\Delta (z))} dA_{\gamma }(\omega ) \\&\quad = \; M_{v,\varphi ,\phi }^{\beta } \left( 1 - |z|^2\right) ^{\beta } A_{\gamma }\circ \varphi _t^{-1} (\Delta (z)), \end{aligned}$$

for every $t \in [0,1]$ and $z \in {\mathbb {D}}$. Then, using (2.6) for $r = \frac{1}{2}$, (2.7) and (3.1), we get

$$\begin{aligned} \Vert (v A_{\alpha }) \circ \varphi _t^{-1}\Vert _{\alpha } \simeq \;&\sup _{z \in {\mathbb {D}}}\dfrac{(v A_{\alpha })\circ \varphi _t^{-1}(\Delta (z))}{\left( 1 - |z|^2\right) ^{\alpha + 2}} \\ \lesssim \;&M_{v,\varphi ,\phi }^{\beta } \sup _{z \in {\mathbb {D}}} \dfrac{\left( 1 - |z|^2\right) ^{\beta } A_{\gamma }\circ \varphi _t^{-1} (\Delta (z))}{\left( 1 - |z|^2\right) ^{\alpha + 2}} \\ = \;&M_{v,\varphi ,\phi }^{\beta } \sup _{z \in {\mathbb {D}}} \dfrac{A_{\gamma }\circ \varphi _t^{-1}(\Delta (z))}{\left( 1 - |z|^2\right) ^{\gamma + 2}} \simeq \; M_{v,\varphi ,\phi }^{\beta } \Vert A_{\gamma }\circ \varphi _t^{-1}\Vert _{\gamma } \\ \le \;&M_{v,\varphi ,\phi }^{\beta } \left( \dfrac{1 + |\varphi _t(0)|}{1 - |\varphi _t(0)|} \right) ^{\gamma + 2} \lesssim \; M_{v,\varphi ,\phi }^{\beta } \left( \dfrac{1}{1 - |\varphi _t(0)|} \right) ^{\gamma + 2} \\ \lesssim \;&M_{v,\varphi ,\phi }^{\beta } \left( \dfrac{1}{1 - |\varphi (0)|} + \dfrac{1}{1 - |\phi (0)|}\right) ^{\gamma + 2}, \end{aligned}$$

for every $t \in [0,1]$. $\square$

Lemma 3.2

For every functions $\varphi$ and $\phi$ from ${{\mathcal {S}}}({\mathbb {D}})$,

$$\begin{aligned} \Vert C_{\varphi } - C_{\phi }\Vert _{\alpha } \ge \dfrac{1}{2 c_0 c_1} \limsup _{\rho (\varphi (z),\phi (z)) \rightarrow 1} \left[ \left( \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} \right) ^{\frac{\alpha + 2}{p}} + \left( \dfrac{1 - |z|^2}{1 - |\phi (z)|^2} \right) ^{\frac{\alpha + 2}{p}} \right] , \end{aligned}$$

(3.2)

where $c_0, c_1$ are the constants defined in (2.1) and (2.2), respectively, and, by definition, the limit on the right-hand side of (3.2) is zero if $\rho (\varphi (z), \phi (z)) \le r_0$ for some $r_0 \in (0,1)$ and all $z \in {\mathbb {D}}$.

Proof

Obviously, it is enough to consider the case when

$$\begin{aligned} \limsup _{\rho \left( \varphi (z),\phi (z)\right) \rightarrow 1} \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} \ge \limsup _{\rho (\varphi (z),\phi (z)) \rightarrow 1} \dfrac{1 - |z|^2}{1 - |\phi (z)|^2} \end{aligned}$$

and

$$\begin{aligned} \limsup _{\rho (\varphi (z),\phi (z)) \rightarrow 1} \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} > 0. \end{aligned}$$

Then taking a sequence $(z_n)_n \subset {\mathbb {D}}$ so that $\rho (\varphi (z_n), \phi (z_n)) \rightarrow 1$ as $n \rightarrow \infty$ and

$$\begin{aligned} \lim _{n \rightarrow \infty } \dfrac{1 - |z_n|^2}{1 - |\varphi \left( z_n\right) |^2} = \limsup _{\rho \left( \varphi (z),\phi (z)\right) \rightarrow 1} \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} , \end{aligned}$$

we easily verify that

$$\begin{aligned} \limsup _{n \rightarrow \infty } \dfrac{1 - |\varphi \left( z_n\right) |^2}{1 - |\phi (z_n)|^2} \le 1. \end{aligned}$$

Applying this and the well-known identity

$$\begin{aligned} 1 - \rho \left( \varphi \left( z_n\right) , \phi \left( z_n\right) \right) ^2 = \dfrac{\left( 1 - |\varphi \left( z_n\right) |^2\right) \left( 1 - |\phi \left( z_n\right) |^2\right) }{\left| 1 - \overline{\varphi \left( z_n\right) }\phi \left( z_n\right) \right| ^2}, \end{aligned}$$

we get

$$\begin{aligned} \limsup _{n \rightarrow \infty } \dfrac{\left( 1 - |\varphi \left( z_n\right) |^2\right) ^2}{\left| 1 - \overline{\varphi \left( z_n\right) }\phi \left( z_n\right) \right| ^2} \le \lim _{n \rightarrow \infty } \left( 1 - \rho \left( \varphi \left( z_n\right) , \phi \left( z_n\right) \right) ^2\right) = 0, \end{aligned}$$

and hence,

$$\begin{aligned} \limsup _{n \rightarrow \infty } \dfrac{1 - |\varphi \left( z_n\right) |^2}{\left| 1 - \overline{\varphi \left( z_n\right) }\phi \left( z_n\right) \right| } = 0. \end{aligned}$$

(3.3)

Next, for every $n \in {\mathbb {N}}$, using (2.1) and (2.2), we have

$$\begin{aligned}&\Vert C_{\varphi } - C_{\phi }\Vert _{\alpha } \ge \; \dfrac{1}{c_0} \Vert C_{\varphi }k_{\varphi \left( z_n\right) } - C_{\phi }k_{\varphi \left( z_n\right) }\Vert _{p,\alpha } \\&\quad = \dfrac{1}{c_0} \left( \int _{{\mathbb {D}}} |k_{\varphi \left( z_n\right) }(\varphi (z)) - k_{\varphi \left( z_n\right) }(\phi (z))|^p dA_{\alpha }(z) \right) ^{\frac{1}{p}} \\&\quad \ge \dfrac{1}{c_0} \left( \int _{\Delta \left( z_n\right) } |k_{\varphi \left( z_n\right) }(\varphi (z)) - k_{\varphi \left( z_n\right) }(\phi (z))|^p dA_{\alpha }(z) \right) ^{\frac{1}{p}} \\&\quad \ge \dfrac{1}{c_0 c_1} (1 - |z_n|^2)^{\frac{\alpha + 2}{p}} |k_{\varphi \left( z_n\right) }\left( \varphi \left( z_n\right) \right) - k_{\varphi \left( z_n\right) }(\phi \left( z_n\right) )| \\&\quad = \dfrac{1}{c_0 c_1} (1 - |z_n|^2)^{\frac{\alpha + 2}{p}} \left| \dfrac{1}{\left( 1 - |\varphi \left( z_n\right) |^2\right) ^{\frac{\alpha + 2}{p}}} - \dfrac{\left( 1 - |\varphi \left( z_n\right) |^2\right) ^{\frac{\sigma }{p}}}{\left( 1 - \overline{\varphi \left( z_n\right) }\phi \left( z_n\right) \right) ^{\frac{\sigma + \alpha + 2}{p}}} \right| \\&\quad \ge \dfrac{1}{c_0 c_1} \left( \dfrac{1 - |z_n|^2}{1 - |\varphi \left( z_n\right) |^2}\right) ^{\frac{\alpha + 2}{p}} \left| 1 - \left( \dfrac{1 - |\varphi \left( z_n\right) |^2}{\left| 1 - \overline{\varphi \left( z_n\right) }\phi \left( z_n\right) \right| } \right) ^{\frac{\sigma + \alpha + 2}{p}} \right| . \end{aligned}$$

Letting $n \rightarrow \infty$ in the last inequality and using (3.3), we get

$$\begin{aligned} \Vert C_{\varphi } - C_{\phi }\Vert _{\alpha } \ge \dfrac{1}{c_0 c_1}\lim _{n \rightarrow \infty }\left( \dfrac{1 - |z_n|^2}{1 - |\varphi \left( z_n\right) |^2}\right) ^{\frac{\alpha + 2}{p}}, \end{aligned}$$

which implies (3.2). $\square$

Theorem 3.3

For every functions $\varphi$ and $\phi$ from ${{\mathcal {S}}}({\mathbb {D}})$, the operators $C_{\varphi }$ and $C_{\phi }$ are linearly connected in ${{\mathcal {C}}}(A_{\alpha }^p)$ if and only if

$$\begin{aligned} \limsup _{\rho (\varphi (z), \phi (z)) \rightarrow 1} \left( \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} + \dfrac{1 - |z|^2}{1 - |\phi (z)|^2} \right) = 0. \end{aligned}$$

(3.4)

Proof

As above, let $\varphi _t(z) := (1 - t) \varphi (z) + t \phi (z), t \in [0,1]$.

(a) Necessity. Following the proof of [8, Theorem 3.2], assume (3.4) does not hold. Then, similarly to the proof of the previous Lemma 3.2, we can find a sequence $(z_n)_n$ in ${\mathbb {D}}$ so that (3.3) holds and

$$\begin{aligned} \limsup _{n \rightarrow \infty } \dfrac{1 - |z_n|^2}{1 - |\phi \left( z_n\right) |^2} \le \lim _{n \rightarrow \infty } \dfrac{1 - |z_n|^2}{1 - |\varphi \left( z_n\right) |^2} = \varepsilon _0 \in (0, \infty ). \end{aligned}$$

Then for every $t \in (0,1]$, by (3.3), we get

$$\begin{aligned} \dfrac{|1 - \overline{\varphi _t\left( z_n\right) }\varphi \left( z_n\right) |}{1 - |\varphi \left( z_n\right) |^2}&= \dfrac{|(1-t)(1 - |\varphi \left( z_n\right) |^2) + t( 1 - \overline{\phi \left( z_n\right) }\varphi \left( z_n\right) )|}{1 - |\varphi \left( z_n\right) |^2} \\&\ge t \dfrac{|1 - \overline{\varphi \left( z_n\right) }\phi \left( z_n\right) |}{1 - |\varphi \left( z_n\right) |^2} - (1 - t) \rightarrow \infty \text { as } n \rightarrow \infty . \end{aligned}$$

Therefore, for every $t \in (0,1]$,

$$\begin{aligned} 1 - \rho (\varphi \left( z_n\right) , \varphi _t\left( z_n\right) )^2&= \dfrac{(1 - |\varphi \left( z_n\right) |^2)(1 - |\varphi _t\left( z_n\right) |^2)}{\left| 1 - \overline{\varphi \left( z_n\right) }\varphi _t\left( z_n\right) \right| ^2} \\&\le 2 \dfrac{1 - |\varphi \left( z_n\right) |^2}{\left| 1 - \overline{\varphi \left( z_n\right) }\varphi _t\left( z_n\right) \right| } \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

Consequently, by Lemma 3.2,

$$\begin{aligned} \Vert C_{\varphi } - C_{\varphi _t}\Vert _{\alpha }&\ge \dfrac{1}{2 c_0 c_1} \limsup _{n \rightarrow \infty } \left[ \left( \dfrac{1 - |z_n|^2}{1 - |\varphi \left( z_n\right) |^2} \right) ^{\frac{\alpha + 2}{p}} + \left( \dfrac{1 - |z_n|^2}{1 - |\varphi _t\left( z_n\right) |^2} \right) ^{\frac{\alpha + 2}{p}} \right] \\&\ge \dfrac{\varepsilon _0^{\frac{\alpha + 2}{p}}}{2 c_0 c_1} \text { for all } t \in (0,1]. \end{aligned}$$

From this it follows that the path $C_{\varphi _t}, t \in [0,1],$ is not continuous at $t = 0$, which is a contradiction.

(b) Sufficiency. Suppose that (3.4) holds. We prove that the map $t \mapsto C_{\varphi _t}$ is continuous in ${{\mathcal {C}}}(A_{\alpha }^p)$, i.e., $\displaystyle \lim _{s \rightarrow t} \Vert C_{\varphi _s} - C_{\varphi _{t}}\Vert _{\alpha } = 0$ for each $t \in [0,1]$ fixed.

We take an arbitrary number $r \in (0,1)$ and put

$$\begin{aligned} E_r := \left\{ z \in {\mathbb {D}}: \rho (\varphi (z),\phi (z)) \le r\right\} \text { and } E_r^c := {\mathbb {D}}\setminus E_r. \end{aligned}$$

For every $s \in [0,1]$ and $f \in A_{\alpha }^p$, we write

$$\begin{aligned} \Vert C_{\varphi _s}f - C_{\varphi _{t}}f\Vert _{p,\alpha }^p&= \int _{{\mathbb {D}}} |f\left( \varphi _s(z)\right) - f\left( \varphi _{t}(z)\right) |^pdA_{\alpha }(z) \\&= \left( \int _{E_r} + \int _{E_r^c} \right) |f\left( \varphi _s(z)\right) - f\left( \varphi _{t}(z)\right) |^pdA_{\alpha }(z), \end{aligned}$$

and estimate the integrals in the right-hand side separately.

First, we estimate the integral

$$\begin{aligned} {{\mathcal {I}}}(f,r,s) := \int _{E_r}|f\left( \varphi _s(z)\right) - f\left( \varphi _{t}(z)\right) |^pdA_{\alpha }(z). \end{aligned}$$

For each $z \in E_r$, by Lemma 2.2,

$$\begin{aligned} \rho \left( \varphi _s(z),\varphi _{t}(z)\right) \le \rho \left( \varphi (z),\phi (z)\right) \le r \text { for every } s \in [0,1]. \end{aligned}$$

By this and Lemma 2.1, for some constant $c_3 = c_3(p, \alpha , r)$ and each $z \in E_r$, we have

$$\begin{aligned} |f\left( \varphi _s(z)\right) - f\left( \varphi _{t}(z)\right) |^p \le c_3 \dfrac{\rho \left( \varphi _s(z),\varphi _{t}(z)\right) ^p}{(1 - |\varphi _{t}(z)|^2)^{\alpha + 2}} \int _{\Delta }|f'(\omega )|^p\left( 1 - |\omega |^2\right) ^p dA_{\alpha }(\omega ), \end{aligned}$$

where, for simplicity, we write $\Delta$ instead of $\Delta (\varphi _{t}(z),\frac{r+1}{2})$. By Lemma 2.2, for each $z \in E_r$, we obtain

$$\begin{aligned} |f(\varphi _s(z)) - f(\varphi _{t}(z))|^p&\le c_3 |s - t|^p \dfrac{\rho (\varphi (z),\phi (z))^p}{\left( 1 - (1 - |s - t|)\rho (\varphi (z),\phi (z))\right) ^p} \\&\quad \times \dfrac{1}{\left( 1 - |\varphi _{t}(z)|^2\right) ^{\alpha + 2}} \int _{\Delta }|f'(\omega )|^p(1 - |\omega |^2)^p dA_{\alpha }(\omega ) \\&\le c_3 |s - t|^p \dfrac{r^p}{\left( 1 - (1 - |s - t|)r\right) ^p} \\&\quad \times \dfrac{1}{(1 - |\varphi _{t}(z)|^2)^{\alpha + 2}} \int _{\Delta }|f'(\omega )|^p\left( 1 - |\omega |^2\right) ^p dA_{\alpha }(\omega ). \end{aligned}$$

Using this, Fubini’s theorem, (2.5) and (2.6), for every $s \in [0,1]$ and $f \in A_{\alpha }^p$, we get

$$\begin{aligned} {{\mathcal {I}}}(f,r,s) =&\int _{E_r} |f(\varphi _s(z)) - f(\varphi _{t}(z))|^pdA_{\alpha }(z) \\ \le \;&c_3 |s - t|^p \dfrac{r^p}{\left( 1 - (1 - |s - t|)r\right) ^p} \\&\times \int _{{\mathbb {D}}} \dfrac{1}{(1 - |\varphi _{t}(z)|^2)^{\alpha + 2}} \left( \int _{\Delta }|f'(\omega )|^p(1 - |\omega |^2)^p dA_{\alpha }(\omega )\right) dA_{\alpha }(z) \\ = \;&c_3 |s - t|^p \dfrac{r^p}{\left( 1 - (1 - |s - t|)r\right) ^p} \\&\times \int _{{\mathbb {D}}} |f'(\omega )|^p(1 - |\omega |^2)^p \left( \int _{\varphi _{t}^{-1}(\Delta (\omega ,\frac{r+1}{2}))} \dfrac{1}{(1 - |\varphi _{t}(z)|^2)^{\alpha + 2}} dA_{\alpha }(z)\right) dA_{\alpha }(\omega ) \\ \le&\; c_2^{\alpha + 2} c_3 |s - t|^p \dfrac{r^p}{\left( 1 - (1 - |s - t|)r\right) ^p} \\&\times \int _{{\mathbb {D}}} |f'(\omega )|^p(1 - |\omega |^2)^p \dfrac{A_{\alpha } \circ \varphi _{t}^{-1}(\Delta (\omega ,\frac{r+1}{2}))}{(1 - |\omega |^2)^{\alpha + 2}} dA_{\alpha }(\omega )\\ \le&\; c_2^{\alpha + 2} c_3 c_4 |s - t|^p \dfrac{r^p}{\left( 1 - \left( 1 - |s - t|\right) r\right) ^p} \Vert A_{\alpha } \circ \varphi _{t}^{-1}\Vert _{\alpha } \\&\times \int _{{\mathbb {D}}} |f'(\omega )|^p(1 - |\omega |^2)^p dA_{\alpha }(\omega ). \end{aligned}$$

On the other hand, by (2.7) and (3.1),

$$\begin{aligned} \Vert A_{\alpha } \circ \varphi _{t}^{-1}\Vert _{\alpha }&\le \left( \dfrac{1 + |\varphi _{t}(0)|}{1 - |\varphi _{t}(0)|} \right) ^{\alpha + 2} \le \left( \dfrac{2}{1 - |\varphi (0)|} + \dfrac{2}{1 - |\phi (0)|} \right) ^{\alpha + 2}. \end{aligned}$$

Moreover, from [21, Theorem 2.16] and (2.3), it follows that

$$\begin{aligned} \int _{{\mathbb {D}}} |f'(\omega )|^p\left( 1 - |\omega |^2\right) ^p dA_{\alpha }(\omega )&\simeq \int _{{\mathbb {D}}} |f(\omega ) - f(0)|^p dA_{\alpha }(\omega ) \\&\lesssim \left( \Vert f\Vert ^p_{p,\alpha } + |f(0)|^p\right) \lesssim \Vert f\Vert ^p_{p,\alpha }. \end{aligned}$$

Thus,

$$\begin{aligned} {{\mathcal {I}}}(f,r,s) \lesssim&\; c_2^{\alpha + 2} c_3 c_4 |s - t|^p \dfrac{r^p}{\left( 1 - (1 - |s - t|)r\right) ^p} \\&\quad \times \left( \dfrac{2}{1 - |\varphi (0)|} + \dfrac{2}{1 - |\phi (0)|} \right) ^{\alpha + 2} \Vert f\Vert ^p_{p,\alpha }, \end{aligned}$$

for every $s \in [0,1]$ and $f \in A_{\alpha }^p$.

Next, we estimate the integral

$$\begin{aligned} {{\mathcal {J}}}(f,r,s) := \int _{E_r^c}|f\left( \varphi _s(z)\right) - f\left( \varphi _{t}(z)\right) |^pdA_{\alpha }(z). \end{aligned}$$

We have

$$\begin{aligned} {{\mathcal {J}}}(f, r, s) =&\int _{E_r^c}|f\left( \varphi _s(z)\right) -f\left( \varphi _{t}(z)\right) |^pdA_{\alpha }(z)\\ \lesssim&\int _{E_r^c}|f\left( \varphi _s(z)\right) |^pdA_{\alpha }(z) + \int _{E_r^c}|f\left( \varphi _{t}(z)\right) |^pdA_{\alpha }(z)\\ =&\int _{{\mathbb {D}}}|f(\varphi _s(z))|^p \chi _{E_r^c}(z)dA_{\alpha }(z) + \int _{{\mathbb {D}}}|f(\varphi _{t}(z))|^p\chi _{E_r^c}(z)dA_{\alpha }(z) \\ =&\int _{{\mathbb {D}}}|f(z)|^p d(\chi _{E_r^c}A_{\alpha })\circ \varphi _s^{-1}(z) + \int _{{\mathbb {D}}}|f(z)|^pd\left( \chi _{E_r^c}A_{\alpha }\right) \circ \varphi _t^{-1}(z) \\ \le \;&\Vert (\chi _{E_r^c}A_{\alpha }) \circ \varphi _s^{-1}\Vert _{\alpha } \Vert f\Vert _{p,\alpha }^p + \Vert \left( \chi _{E_r^c}A_{\alpha }\right) \circ \varphi _{t}^{-1}\Vert _{\alpha } \Vert f\Vert _{p,\alpha }^p, \end{aligned}$$

for every $s \in [0,1]$ and $f \in A_{\alpha }^p$, where $\chi _{E}$ denotes the characteristic function of a Borel subset $E \subset {\mathbb {D}}$. Applying Lemma 3.1 to two functions $\varphi , \phi$ from ${{\mathcal {S}}}({\mathbb {D}})$, and the characteristic function $\chi _{E_r^c}$, we obtain

$$\begin{aligned} \Vert \left( \chi _{E_r^c}A_{\alpha }\right) \circ \varphi _s^{-1}\Vert _{\alpha } \lesssim M_{\chi _{E_r^c}, \varphi , \phi }^{\beta } \left( \dfrac{1}{1 - |\varphi (0)|} + \dfrac{1}{1 - |\psi (0)|}\right) ^{\gamma +2} \end{aligned}$$

for all $s \in [0,1]$. Here, $\beta$ and $\gamma$ are the same as in Lemma 3.1.

Combining the above estimates for ${{\mathcal {I}}}(f,r,s)$ and ${{\mathcal {J}}}(f,r,s)$, yields

$$\begin{aligned} \Vert C_{\varphi _s} - C_{\varphi _{t}}\Vert ^p_{\alpha } \lesssim \;&c_2^{\alpha + 2} c_3 c_4 |s - t|^p \dfrac{r^p}{\left( 1 - \left( 1 - |s - t|\right) r\right) ^p}\\&\times \left( \dfrac{2}{1 - |\varphi (0)|} + \dfrac{2}{1 - |\phi (0)|} \right) ^{\alpha + 2} \\&+ \; 2 M_{\chi _{E_r^c}, \varphi , \phi }^{\beta } \left( \dfrac{1}{1 - |\varphi (0)|} + \dfrac{1}{1 - |\phi (0)|}\right) ^{\gamma +2}, \end{aligned}$$

for every $s \in [0,1]$. Then, letting $s \rightarrow t$, we get

$$\begin{aligned} \lim _{s \rightarrow t} \Vert C_{\varphi _s} - C_{\varphi _{t}}\Vert ^p_{\alpha } \lesssim M_{\chi _{E_r^c}, \varphi , \phi }^{\beta } \left( \dfrac{1 }{1 - |\varphi (0)|} + \dfrac{1}{1 - |\phi (0)|}\right) ^{\gamma +2}, \end{aligned}$$

for every $r \in (0,1)$. Moreover, by (3.4),

$$\begin{aligned} M_{\chi _{E_r^c}, \varphi , \phi } = \sup _{\rho \left( \varphi (z), \phi (z)\right) \ge r} \left( \dfrac{1-|z|^2}{1 - |\varphi (z)|^2} + \dfrac{1-|z|^2}{1 - |\phi (z)|^2} \right) \rightarrow 0 \text { as } r \rightarrow 1^-, \end{aligned}$$

which implies that $\displaystyle \lim _{s \rightarrow t} \Vert C_{\varphi _s} - C_{\varphi _{t}}\Vert _{\alpha } = 0$ and completes the proof. $\square$

From Theorem 3.3 we immediately get the following result.

Corollary 3.4

Let two functions $\varphi$ and $\phi$ from ${{\mathcal {S}}}({\mathbb {D}})$ satisfy $\rho (\varphi (z), \phi (z)) \le r_0$ for some number $r_0 \in (0,1)$ and every $z \in {\mathbb {D}}$. Then $C_{\varphi }$ and $C_{\phi }$ are linearly connected in ${{\mathcal {C}}}(A_{\alpha }^p)$.

Remark 3.5

Theorem 3.3 and Corollary 3.4 extend the results of [8, Theorem 3.3] and, respectively, [17, Theorem 8] on Hilbert Bergman spaces $A^2_{\alpha }$ to all Bergman spaces $A_{\alpha }^p$ with $p \in (0, \infty )$.

To describe path components of the space ${{\mathcal {C}}}(A_{\alpha }^p)$, we introduce the following notation. We say that two composition operators $C_{\varphi }$ and $C_{\phi }$ are equivalent in ${{\mathcal {C}}}(A_{\alpha }^p)$, if their difference $C_{\varphi } - C_{\phi }$ is a compact operator on $A_{\alpha }^p$. Obviously, this is an equivalence relation in ${{\mathcal {C}}}(A_{\alpha }^p)$. Let denote by $[C_{\varphi }]$ the equivalence class of all composition operators that are equivalent to the given one $C_{\varphi }$. Then the set ${{\mathcal {C}}}_0(A_{\alpha }^p)$ of all compact composition operators on $A_{\alpha }^p$ is the equivalence class $[C_0]$ of all operators from ${{\mathcal {C}}}(A_{\alpha }^p)$ that are equivalent to the operator $C_0: f \mapsto f(0)$.

Recall, by [5, Theorem 1.1], [4, Proposition 4.1], and [17, Theorem 4], that for every functions $\varphi$ and $\phi$ from ${{\mathcal {S}}}({\mathbb {D}})$, the difference $C_{\varphi } - C_{\phi }$ is compact on $A_{\alpha }^p$ if and only if

$$\begin{aligned} \lim _{|z| \rightarrow 1^-} \rho \left( \varphi (z),\phi (z)\right) \left( \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} + \dfrac{1 - |z|^2}{1 - |\phi (z)|^2}\right) = 0. \end{aligned}$$

(3.5)

Theorem 3.6

Each equivalence class $[C_{\varphi }]$ is path connected in the space ${{\mathcal {C}}}(A_{\alpha }^p)$.

Proof

Let $C_{\phi }$ be an arbitrary operator in $[C_{\varphi }]$, i.e. $C_{\varphi } - C_{\phi }$ is compact on $A_{\alpha }^p$ and, hence, (3.5) holds. We can verify that all operators $C_{\varphi _t}$, where, as above, $\varphi _t(z) := (1-t)\varphi (z) + t \phi (z)$ for $t \in [0,1]$, belong to the class $[C_{\varphi }]$. Indeed, using (3.1) and Lemma 2.2, we get $\rho (\varphi (z),\varphi _t(z)) \le \rho (\varphi (z),\phi (z))$ and

$$\begin{aligned} \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} + \dfrac{1 - |z|^2}{1 - |\varphi _t(z)|^2} \le 2 \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} + \dfrac{1 - |z|^2}{1 - |\phi (z)|^2}, \end{aligned}$$

for every $t \in [0,1]$ and $z \in {\mathbb {D}}$. Thus, by (3.5),

$$\begin{aligned} \lim _{|z| \rightarrow 1^-}\rho \left( \varphi (z),\varphi _t(z)\right) \left( \dfrac{1 - |z|^2}{1 - |\varphi (z)|^2} + \dfrac{1 - |z|^2}{1 - |\varphi _t(z)|^2}\right) = 0, \end{aligned}$$

for every $t \in [0,1]$. This and (3.5) imply that $C_{\varphi } - C_{\varphi _t}$ is compact on $A_{\alpha }^p$ for every $t \in [0,1]$.

On the other hand, it is easy to see that (3.5) implies (3.4). Then, by Theorem 3.3, $C_{\varphi }$ and $C_{\phi }$ are in the same path component of the space ${{\mathcal {C}}}(A_{\alpha }^p)$ via the path $C_{\varphi _t}, t \in [0,1]$, in $[C_{\varphi }]$.

From this the assertion follows. $\square$

Now we show that the set ${{\mathcal {C}}}_0(A_{\alpha }^p)$ forms a path component in the space ${{\mathcal {C}}}(A_{\alpha }^p)$. To do this we need some additional facts concerning angular derivatives of a function $\varphi \in {{\mathcal {S}}}({\mathbb {D}})$ (for more information we refer the reader to [15]). Let $\zeta \in \partial {\mathbb {D}}$. The condition

$$\begin{aligned} \liminf _{z \rightarrow \zeta } \dfrac{1 - |\varphi (z)|}{1 - |z|} < \infty \end{aligned}$$

(3.6)

is necessary and sufficient for the existence of the finite angular derivative $\varphi '(\zeta )$ at $\zeta$. Moreover, the limit in (3.6) is equal to $|\varphi '(\zeta )|$ and

$$\begin{aligned} |\varphi '(\zeta )| \ge \frac{1 - |\varphi (0)|}{1 + |\varphi (0)|} > 0. \end{aligned}$$

If the limit if infinite, we put $\varphi '(\zeta ) := \infty$. Following [15] we say that $\varphi$ and $\phi$ from ${{\mathcal {S}}}({\mathbb {D}})$ have the same data at $\zeta$ if they have radial limits of modulus 1 at $\zeta$ with $\varphi (\zeta ) = \phi (\zeta )$ and $|\varphi '(\zeta )| = |\phi '(\zeta )|$. Obviously, in that case $\varphi '(\zeta ) = \phi '(\zeta )$.

From [4, Theorem 3.5] it follows that there is a constant $c > 0$ depending only on $\alpha$ such that

$$\begin{aligned} \Vert C_{\varphi } - C_{\phi }\Vert _{\alpha } \ge c |\varphi '(\zeta )|^{- \frac{\alpha + 2}{p}}, \end{aligned}$$

(3.7)

for every point $\zeta \in \partial {\mathbb {D}}$, at which $\varphi$ and $\phi$ do not have the same data.

The following result is an analog of [15, Theorem 2.4] and proved in the similar way by using inequality (3.7).

Proposition 3.7

If the operator $C_{\phi }$ belongs to the path component of ${{\mathcal {C}}}(A_{\alpha }^p)$ containing $C_{\varphi }$, then $\varphi$ and $\phi$ have the same data at every point $\zeta \in \partial {\mathbb {D}}$ for which $\varphi '(\zeta )$ is finite.

Corollary 3.8

The set ${{\mathcal {C}}}_0(A_{\alpha }^p)$ of all compact composition operators on $A_{\alpha }^p$ is a path component in ${{\mathcal {C}}}(A_{\alpha }^p)$.

Proof

By Theorem 3.6, the set ${{\mathcal {C}}}_0(A_{\alpha }^p)$ is path connected in ${{\mathcal {C}}}(A_{\alpha }^p)$.

Recall, by [16, Theorem 3.5], that an operator $C_{\varphi }$ is compact on $A_{\alpha }^p$ if and only if $\varphi$ has no finite angular derivative at any point of $\partial {\mathbb {D}}$. Thus for any non-compact operator $C_{\phi }$ on $A_{\alpha }^p$, there is at least one point $\zeta \in \partial {\mathbb {D}}$ with $\phi '(\zeta )$ finite. So a compact operator $C_{\varphi }$ and a non-compact operator $C_{\phi }$ do not belong to the same path component of ${{\mathcal {C}}}(A_{\alpha }^p)$ by Proposition 3.7, since $\varphi$ and $\phi$ do not have the same data at $\zeta$. $\square$

Remark 3.9

Corollary 3.8 extends the result on Hilbert Bergman spaces $A^2_{\alpha }$ in [6, Corollary 9.19] to Bergman spaces $A_{\alpha }^p$ for all $p \in (0, \infty )$. It should be noted that in the setting of Hardy spaces, the set of all compact composition operators on $H^2$ is path connected in the space ${{\mathcal {C}}}(H^2)$ (see, [15, Proposition 2.1]); however, it does not form a path component in this space (see, [9, Main Theorem]).

The next result shows that similarly to the Hardy space $H^2$ [3, 18], the space $H^{\infty }$ [14], and weighted Banach spaces with sup-norm [1], the well-known Shapiro-Sundberg conjecture (see [19, Page 149]) is also false for all Bergman spaces $A_\alpha ^p$. Since this example is constructed by the same reasons as in [1, Example 3.4] and [14, Examples 1 and 2], we only sketch its proof.

Example 3.10

Let $\varphi _0(z) := 1 + a(z-1)$ with $0< a < 1$. Then the class $[C_{\varphi _0}]$ is not a path component in ${{\mathcal {C}}}(A_{\alpha }^p)$.

Proof

Let $\delta := \frac{a(1-a)}{10}$. For each $t \in [-\delta , \delta ]$, we put

$$\begin{aligned} \varphi _t(z) := \varphi _0(z) + t (z-1)^2. \end{aligned}$$

Similarly to [1, Example 3.4] and [14, Example 1], we get $\varphi _t \in {{\mathcal {S}}}({\mathbb {D}})$ and $C_{\varphi _t} \in {{\mathcal {C}}}(A_{\alpha }^p)$ for all $t \in [-\delta , \delta ]$ and consider a sequence $(z_n) \subset {\mathbb {D}}$ such that $z_n \rightarrow 1$ along the arc $|1-z|^2 = 1 - |z|^2$. Then, as in [1, Example 3.4], for all $n \ge 1$ and $t \in [-\delta , \delta ]$,

$$\begin{aligned} \rho \left( \varphi _0\left( z_n\right) ,\varphi _t\left( z_n\right) \right) \ge \dfrac{|t|}{a(2-a) + |t|} \end{aligned}$$

and hence,

$$\begin{aligned}&\rho \left( \varphi _0\left( z_n\right) ,\varphi _t\left( z_n\right) \right) \\&\quad \left( \dfrac{1 - |z_n|^2}{1 - |\varphi _0(z_n)|^2} + \dfrac{1 - |z_n|^2}{1 - |\varphi _t(z_n)|^2}\right) \\&\quad \ge \dfrac{|t|}{a(2-a)\left( a(2-a) + |t|\right) }. \end{aligned}$$

This and (3.5) imply that $C_{\varphi _t} - C_{\varphi _0}$ is not compact on $A_{\alpha }^p$ and so $C_{\varphi _t}\notin [C_{\varphi _0}]$ for all $0<|t|\le \delta$.

On the other hand, as in [14, Example 2], for every $z \in {\mathbb {D}}$,

$$\begin{aligned} \rho \left( \varphi _{-\delta }(z),\varphi _{\delta }(z)\right) \le \dfrac{2\delta }{a(1-a) - 2 \delta - 4\delta ^2} < \dfrac{1}{2}. \end{aligned}$$

From this and Corollary 3.4, it follows that the path $C_{\phi _s}$ with

$$\begin{aligned} \phi _{s}(z)&:= (1 - s)\varphi _{-\delta }(z) + s \varphi _{\delta }(z) = \varphi _0(z) + \delta \left( 2 s - 1\right) \left( z - 1\right) ^2 \\&= \varphi _{\delta (2 s - 1)}(z), s \in [0,1], \end{aligned}$$

is a continuous path connecting $C_{\varphi _{-\delta }}$ and $C_{\varphi _{\delta }}$ in ${{\mathcal {C}}}(A_{\alpha }^p)$. Thus, the assertion follows. $\square$

We end this section with the following characterizations for isolated and non-isolated points in the space ${{\mathcal {C}}}(A_{\alpha }^p)$.

Proposition 3.11

Let $\varphi$ be a function from ${{\mathcal {S}}}({\mathbb {D}})$.

• (a) If $\varphi$ has a finite angular derivative on a set of positive measure, then $C_{\varphi }$ is isolated in the space ${{\mathcal {C}}}(A_{\alpha }^p)$.

• (b) If

  $$\begin{aligned} \int _{0}^{2\pi }\log (1 - |\varphi (e^{i\theta })|)d\theta > -\infty , \end{aligned}$$

  (3.8)

  then $C_{\varphi }$ is not isolated in ${{\mathcal {C}}}(A_{\alpha }^p)$.

Proof

The part (a) follows from (3.7) by the arguments in the proof of [15, Corollary 2.3].

(b). We put

$$\begin{aligned} \phi (z) := \exp \left( \dfrac{1}{2\pi } \int _0^{2\pi } \dfrac{e^{i\theta }+z}{e^{i\theta }-z}\log (1 - |\varphi (e^{i\theta })|)d\theta \right) , z \in {\mathbb {D}}. \end{aligned}$$

Then $\phi$ is a bounded outer function in ${\mathbb {D}}$ with $|\phi | \le 1 - |\varphi |$ in ${\mathbb {D}}$ and $|\phi | = 1 - |\varphi |$ almost everywhere on $\partial {\mathbb {D}}$.

For each $t \in (-1,1)$, we define $\phi _t(z) := \varphi (z) + t \phi (z)$. Obviously, $\phi _t \in {{\mathcal {S}}}({\mathbb {D}})$ for every $t \in (-1,1)$. We claim that the path $C_{\phi _t}, t \in [-\delta , \delta ]$ with $\delta = \frac{1}{6}$, is continuous in ${{\mathcal {C}}}(A_{\alpha }^p)$. Hence, $C_{\varphi }$ is not isolated in ${{\mathcal {C}}}(A_{\alpha }^p)$.

It remains to prove the claim. For each $z \in {\mathbb {D}}$, we have

$$\begin{aligned} \rho \left( \phi _{-\delta }(z),\phi _{\delta }(z)\right)&= \left| \dfrac{\phi _{-\delta }(z) - \phi _{\delta }(z)}{1 - \overline{\phi _{-\delta }(z)} \phi _{\delta }(z)} \right| \\&\le \dfrac{2 \delta |\phi (z)|}{1 - |\varphi (z)|^2 - 2 \delta |\varphi (z)||\phi (z)| - \delta ^2 |\phi (z)|^2} \\&= \dfrac{2 \delta }{\frac{1 - |\varphi (z)|^2}{|\phi (z)|} - 2 \delta |\varphi (z)| - \delta ^2 |\phi (z)|} \\&\le \dfrac{2 \delta }{\frac{1 - |\varphi (z)|}{|\phi (z)|} - 2 \delta - \delta ^2} \le \dfrac{2 \delta }{1 - 2 \delta - \delta ^2} \le \dfrac{2}{3}. \end{aligned}$$

From this and Corollary 3.4, it follows that the path $C_{\varphi _{s}}$ with

$$\begin{aligned} \varphi _{s}(z)&:= (1 - s)\phi _{-\delta }(z) + s \phi _{\delta }(z) \\&= \varphi (z) + \delta (2s - 1) \phi (z) = \phi _{\delta (2s-1)}(z), s \in [0,1], \end{aligned}$$

is a continuous path connecting $C_{\phi _{-\delta }}$ and $C_{\phi _{\delta }}$ in ${{\mathcal {C}}}(A_{\alpha }^p)$. Thus, the claim follows. $\square$

Remark 3.12

Part (a) of Proposition 3.11 is an extension of [15, Corollary 2.3], which was stated only for Hilbert Bergman spaces $A_{\alpha }^2$.

Furthermore, (3.8) is a sufficient condition for the operator $C_{\varphi }$ to be non-isolated in the space of composition operators on the Hardy space $H^2$ [19, Theorem 3.1], on the space $H^{\infty }$ [14, Corollary 9], and on weighted Banach spaces with sup-norm [1, Proposition 3.6]. Part (b) of Proposition 3.11 extends this result to all Bergman spaces $A_{\alpha }^p$.

4 Path Components of the Space of Weighted Composition Operators

In this section we study the topological structure of the space ${{\mathcal {C}}}_w(A_{\alpha }^p)$ of all nonzero bounded weighted composition operators on $A_{\alpha }^p$ under the operator norm topology. For simplicity, we write $W_{\psi ,\varphi } \sim W_{\chi ,\phi }$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$ if the operators $W_{\psi ,\varphi }$ and $W_{\chi ,\phi }$ are in the same path component of ${{\mathcal {C}}}_w(A_{\alpha }^p)$.

In our further considerations we use the following simple fact, which is proved similarly to [20, Lemma 4.8].

Lemma 4.1

Every operator $W_{\psi ,\varphi }\in {{\mathcal {C}}}_w(A_{\alpha }^p)$ is path connected with the operator $C_{\varphi }$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$.

The main result of this section is as follows.

Theorem 4.2

The set ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$ of all nonzero compact weighted composition operators on $A_{\alpha }^p$ is path connected in the space ${{\mathcal {C}}}_{w}(A_{\alpha }^p)$; but it is not a path component in this space.

Proof

(a) To prove that the set ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$ is path connected in the space ${{\mathcal {C}}}_{w}(A_{\alpha }^p)$, it suffices to show that every operator $W_{\psi ,\varphi }$ in ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$ and the operator $C_0: f \mapsto f(0)$ belong to the same path component of ${{\mathcal {C}}}_{w}(A_{\alpha }^p)$ via a path in ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$.

If $\psi (z) \equiv const$, then the assertion follows from Lemma 4.1 and Corollary 3.8.

Now suppose that $\psi$ is non-constant. Obviously, $\psi = W_{\psi ,\varphi }(\mathbf{1} ) \in A_{\alpha }^p$. For each $t\in [0,1]$, we put

$$\begin{aligned} \psi _t(z) := 1 - t + t\psi (z) \text { and } \varphi _t(z) := t\varphi (z), z \in {\mathbb {D}}. \end{aligned}$$

Then, for every $t \in [0,1)$, $\psi _t$ is a nonzero function in $A_{\alpha }^p$ and $\overline{\varphi _t({\mathbb {D}})} \subset t \overline{\varphi ({\mathbb {D}})} \subset {\mathbb {D}}$. From this it follows that all operators $W_{\psi _t,\varphi _t}, t \in [0,1),$ are compact on $A_{\alpha }^p$. Indeed, for every bounded sequence $(f_n)_n$ in $A^p_{\alpha }$ converging to 0 uniformly on compact sets of ${\mathbb {D}}$, we get

$$\begin{aligned} \Vert W_{\psi _t,\varphi _t}f_n\Vert _{p,\alpha }&= \left( \int _{{\mathbb {D}}} |\psi _t(z)|^p |f_n(t\varphi (z))|^p dA_{\alpha }(z)\right) ^{\frac{1}{p}} \\&\le \Vert \psi _t\Vert _{p, \alpha } \sup _{|z| \le t} |f_n(z)| \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

From this and a slight modification of [6, Proposition 3.11], the assertion follows. Thus $W_{\psi _t,\varphi _t} \in {{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$ for all $t \in [0,1]$; moreover, $W_{\psi _0,\varphi _0} = C_0$ and $W_{\psi _1,\varphi _1} = W_{\psi ,\varphi }$. We claim that the map

$$\begin{aligned}{}[0,1] \rightarrow {{\mathcal {C}}}_w(A_{\alpha }^p), t \mapsto W_{\psi _t,\varphi _t}, \end{aligned}$$

is continuous on [0, 1]. Then $W_{\psi ,\varphi } \sim C_0$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$ via a path $W_{\psi _t,\varphi _t}, t \in [0,1]$, in ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$.

It remains to prove the claim. Obviously, $W_{\psi _t,\varphi _t} = (1 - t)C_{t\varphi } + W_{t\psi ,t\varphi }$, and hence,

$$\begin{aligned} \Vert W_{\psi _s,\varphi _s} - W_{\psi _t,\varphi _t}\Vert _{\alpha } \le \Vert (1 - s)C_{s\varphi } - (1 - t)C_{t\varphi }\Vert _{\alpha } + \Vert W_{s\psi ,s\varphi } - W_{t\psi ,t\varphi }\Vert _{\alpha }, \end{aligned}$$

for every $t, s \in [0,1]$. Consequently, to prove the claim, it is enough to show that, for every $t \in [0,1]$ fixed,

$$\begin{aligned} \mathbf (i) \ \lim _{s \rightarrow t} \Vert (1 - s)C_{s\varphi } - (1 - t)C_{t\varphi }\Vert _{\alpha } = 0 \text { and}\, \mathbf (ii) \ \lim _{s \rightarrow t} \Vert W_{s\psi ,s\varphi } - W_{t\psi ,t\varphi }\Vert _{\alpha } = 0. \end{aligned}$$

In our further demonstration we use the obvious inequality for functions $f\in H({\mathbb {D}})$:

$$\begin{aligned} |f(sz) - f(tz)| \le |t-s| |z| \max _{\tau \in [s, t]} |f'(\tau z)|, \ z\in {\mathbb {D}}, \ t,s\in [0,1], \end{aligned}$$

(4.1)

where we briefly write [s, t] for the interval between s and t.

First, we prove (i). If $t = 1$, then, by (2.7),

$$\begin{aligned} \Vert (1 - s)C_{s\varphi }\Vert _{\alpha } = (1 - s) \Vert C_{s\varphi }\Vert _{\alpha } \le (1 - s) \left( \dfrac{1 + s|\varphi (0)|}{1 - s|\varphi (0)|} \right) ^{\frac{\alpha + 2}{p}} \rightarrow 0, s \rightarrow 1. \end{aligned}$$

Let now $t \in [0,1)$ and $t_0 \in (t,1)$. For every $s \in [0,t_0)$ and $f \in A_{\alpha }^p$, using (2.4), (2.7) and (4.1), we get

$$\begin{aligned}&\Vert (1 - s)C_{s\varphi }f - (1 - t)C_{t\varphi }f\Vert _{p, \alpha }^p \\&\quad = \; \int _{{\mathbb {D}}} |(1 - s)f(s\varphi (z)) - (1 - t)f(t\varphi (z))|^p dA_{\alpha }(z)\\&\quad \lesssim \; (1 - s)^p\int _{{\mathbb {D}}} |f(s\varphi (z)) - f(t\varphi (z))|^pdA_{\alpha }(z) + |s - t|^p \int _{{\mathbb {D}}} |f(t\varphi (z))|^p dA_{\alpha }(z)\\&\quad \le \; |s - t|^p\int _{{\mathbb {D}}} |\varphi (z)|^p \max _{\tau \in [s,t]}|f'(\tau \varphi (z))|^p dA_{\alpha }(z) + |s - t|^p \Vert C_{t\varphi }f\Vert _{p, \alpha }^p\\&\quad \le \; c_1^p |s - t|^p \Vert f\Vert _{p,\alpha }^p \int _{{\mathbb {D}}} \max _{\tau \in [s,t]}\dfrac{1}{(1 - |\tau \varphi (z)|^2)^{\alpha + 2 + p}}dA_{\alpha }(z) + |s - t|^p \Vert f\Vert _{p,\alpha }^p \Vert C_{t\varphi }\Vert _{\alpha }^p\\&\quad \le \; \dfrac{c_1^p}{(1 - t_0^2)^{\alpha + p + 2}} |s - t|^p \Vert f\Vert _{p,\alpha }^p + |s - t|^p \left( \dfrac{1 + t|\varphi (0)|}{1 - t|\varphi (0)|} \right) ^{\alpha + 2} \Vert f\Vert _{p, \alpha }^p. \end{aligned}$$

Therefore, for every $s \in [0, t_0)$,

$$\begin{aligned} \Vert (1 - s)C_{s\varphi } - (1 - t)C_{t\varphi }\Vert _{\alpha }^p \lesssim \dfrac{c_1^p}{(1 - t_0^2)^{\alpha + p + 2}} |s - t|^p + |s - t|^p \left( \dfrac{1 + |\varphi (0)|}{1 - |\varphi (0)|} \right) ^{\alpha + 2}. \end{aligned}$$

Thus, $\Vert (1 - s)C_{s\varphi } - (1 - t)C_{t\varphi }\Vert _{\alpha } \rightarrow 0$ as $s \rightarrow t$, which completes the proof for (i).

Next, we prove (ii). For every $s, t \in [0, 1]$ and $f \in A_{\alpha }^p$, we have

$$\begin{aligned} \Vert W_{s\psi ,s\varphi }f - W_{t\psi ,t\varphi }f\Vert _{p, \alpha }^p&= \int _{{\mathbb {D}}} |s\psi (z)f(s\varphi (z)) - t\psi (z)f(t\varphi (z))|^p dA_{\alpha }(z) \\&\lesssim \; |s|^p \int _{{\mathbb {D}}} |\psi (z)(f(s\varphi (z)) - f(t\varphi (z)))|^p dA_{\alpha }(z) \\&\qquad + |s-t|^p\int _{{\mathbb {D}}} |\psi (z)f(t\varphi (z))|^p dA_{\alpha }(z). \end{aligned}$$

To continue, we need several auxiliary estimates in the following cases.

Case 1. $t \in [0, 1)$. We fix a number $t_0 \in (t, 1)$.

Estimate 1.1. By (2.3), we have

$$\begin{aligned} \int _{{\mathbb {D}}} |\psi (z)f(t\varphi (z))|^p dA_{\alpha }(z)&\le c_1^p \Vert f\Vert _{p, \alpha }^p \int _{{\mathbb {D}}} \dfrac{1}{(1 - |t\varphi (z)|^2)^{\alpha + 2}} |\psi (z)|^p dA_{\alpha }(z)\\&\le \dfrac{c_1^p}{(1 - t_0^2)^{\alpha + 2}} \Vert f\Vert _{p, \alpha }^p \Vert \psi \Vert _{p, \alpha }^p. \end{aligned}$$

Estimate 1.2. By (2.4) and (4.1), for every $s \in [0, t_0)$,

$$\begin{aligned}&\int _{{\mathbb {D}}} |\psi (z)(f(s\varphi (z)) - f(t\varphi (z)))|^p dA_{\alpha }(z) \\&\quad \le |s - t|^p \int _{{\mathbb {D}}} |\psi (z)\varphi (z)|^p \max _{\tau \in [s,t]}|f'(\tau \varphi (z))|^p dA_{\alpha }(z) \\&\quad \le c_1^p |s - t|^p \Vert f\Vert _{p, \alpha }^p \int _{{\mathbb {D}}} |\psi (z)|^p \max _{\tau \in [s, t]} \dfrac{1}{(1 - |\tau \varphi (z)|^2)^{\alpha + 2 + p}} dA_{\alpha }(z)\\&\quad \le \dfrac{c_1^p}{(1 - t_0^2)^{\alpha + 2 + p}} |s - t|^p \Vert f\Vert _{p, \alpha }^p \Vert \psi \Vert _{p, \alpha }^p. \end{aligned}$$

Using the above estimates, for every $s \in [0, t_0)$, we obtain

$$\begin{aligned}&\Vert W_{s\psi ,s\varphi } - W_{t\psi ,t\varphi }\Vert _{\alpha }^p \\&\quad \lesssim \dfrac{c_1^p}{\left( 1 - t_0^2\right) ^{\alpha + 2 + p}} |s - t|^p \Vert \psi \Vert _{p, \alpha }^p + \dfrac{c_1^p}{\left( 1 - t_0^2\right) ^{\alpha + 2}} |s - t|^p \Vert \psi \Vert _{p, \alpha }^p. \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{s \rightarrow t} \Vert W_{s\psi ,s\varphi } - W_{t\psi ,t\varphi }\Vert _{\alpha } = 0. \end{aligned}$$

Case 2. $t = 1$.

Estimate 2.1. We have

$$\begin{aligned} \int _{{\mathbb {D}}} |\psi (z)f(\varphi (z))|^p dA_{\alpha }(z) = \Vert W_{\psi , \varphi }f\Vert _{p, \alpha }^ p \le \Vert W_{\psi , \varphi }\Vert _{\alpha }^p \Vert f\Vert _{p, \alpha }^p. \end{aligned}$$

Estimate 2.2. For each $r \in (0,1)$, we put

$$\begin{aligned} E_r := \{z \in {\mathbb {D}}: |\varphi (z)| \le r \} \text { and } E_r^c := {\mathbb {D}}\setminus E_r. \end{aligned}$$

By (2.4) and (4.1), for every $s \in [0, 1)$, $r \in (0,1)$, and $f \in A_{\alpha }^p$, we have

$$\begin{aligned}&\int _{{\mathbb {D}}} |\psi (z)(f(s\varphi (z)) - f(\varphi (z)))|^p dA_{\alpha }(z) \\&\quad = \left( \int _{E_r} + \int _{E_r^c} \right) |\psi (z)(f(s\varphi (z)) - f(\varphi (z)))|^p dA_{\alpha }(z) \\&\quad \lesssim (1 - s)^p \int _{E_r} |\psi (z)\varphi (z)|^p \max _{\tau \in [s,1]}|f'(\tau \varphi (z))|^p dA_{\alpha }(z) \\&\qquad + \int _{E_r^c} \left( |\psi (z)f(s\varphi (z))|^p + |\psi (z)f(\varphi (z))|^p \right) dA_{\alpha }(z) \\&\quad \le c_1^p (1 - s)^p \Vert f\Vert _{p, \alpha }^p \int _{E_r} |\psi (z)|^p \max _{\tau \in [s, 1]} \dfrac{1}{(1 - |\tau \varphi (z)|^2)^{\alpha + 2 + p}} dA_{\alpha }(z)\\&\qquad + \Vert (\chi _{E_r^c} |\psi |^p A_{\alpha }) \circ (s\varphi )^{-1}\Vert _{\alpha } \Vert f\Vert _{p, \alpha }^p + \Vert (\chi _{E_r^c} |\psi |^p A_{\alpha }) \circ \varphi ^{-1}\Vert _{\alpha } \Vert f\Vert _{p, \alpha }^p\\&\quad \le \dfrac{c_1^p}{(1 - r^2)^{\alpha + 2 + p}} (1 - s)^p \Vert f\Vert _{p, \alpha }^p \Vert \psi \Vert _{p, \alpha }^p \\&\qquad + \Vert \left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ (s\varphi )^{-1}\Vert _{\alpha } \Vert f\Vert _{p, \alpha }^p + \Vert \left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ \varphi ^{-1}\Vert _{\alpha } \Vert f\Vert _{p, \alpha }^p. \end{aligned}$$

Thus, for every $s \in [0, 1)$ and $r \in (0,1)$,

$$\begin{aligned} \Vert W_{s\psi ,s\varphi } - W_{\psi ,\varphi }\Vert _{\alpha }^p&\lesssim (1 - s)^p \Vert W_{\psi , \varphi }\Vert _{\alpha }^p + \dfrac{c_1^p}{(1 - r^2)^{\alpha + 2 + p}} (1 - s)^p \Vert \psi \Vert _{p, \alpha }^p \\&\qquad + \Vert \left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ (s\varphi )^{-1}\Vert _{\alpha } + \Vert \left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ \varphi ^{-1}\Vert _{\alpha }. \end{aligned}$$

To complete the proof, we take an arbitrary number $\varepsilon > 0$. Since $W_{\psi , \varphi }$ is compact on $A_{\alpha }^p$, $(|\psi |^p A_{\alpha }) \circ \varphi ^{-1}$ is a compact $\alpha$-Carleson measure. Then there is a number $\delta _0 = \delta _0(\varepsilon ) \in (0,1)$ such that

$$\begin{aligned} \left( |\psi |^p A_{\alpha }\right) \circ \varphi ^{-1} \left( S(\zeta , \delta )\right) < \varepsilon ^p \left( \dfrac{\delta }{2}\right) ^{\alpha + 2} \end{aligned}$$

for every $\delta < \delta _0$ and $\zeta \in \partial {\mathbb {D}}$.

For every $s \in [\frac{1}{2}, 1]$, $\delta \in (0,2]$, and $\zeta \in \partial {\mathbb {D}}$, by some geometric arguments, we can see that

$$\begin{aligned} \frac{1}{s}S(\zeta , \delta ) \cap {\mathbb {D}}\subset S(\zeta , \frac{\delta }{s}), \end{aligned}$$

Then, for every $r \in (0,1)$, $s \in [\frac{1}{2}, 1]$, $\delta < \frac{\delta _0}{2}$, and $\zeta \in \partial {\mathbb {D}}$, we get

$$\begin{aligned}&\left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ (s\varphi )^{-1}(S(\zeta , \delta )) = \int _{(s\varphi )^{-1}\left( S(\zeta , \delta )\right) }\chi _{E_r^c}(\omega ) |\psi (\omega )|^p dA_{\alpha }(\omega ) \\&\quad \le \int _{\varphi ^{-1}(\frac{1}{s}S(\zeta , \delta ) \cap {\mathbb {D}})} |\psi (\omega )|^p dA_{\alpha }(\omega ) \le \int _{\varphi ^{-1}\left( S(\zeta , \frac{\delta }{s})\right) } |\psi (\omega )|^p dA_{\alpha }(\omega ) \\&\quad = \; \left( |\psi |^p A_{\alpha }\right) \circ \varphi ^{-1}\left( S(\zeta , \frac{\delta }{s})\right) < \varepsilon ^p \left( \dfrac{\delta }{2s}\right) ^{\alpha + 2} \le \varepsilon ^p \delta ^{\alpha + 2}. \end{aligned}$$

Since $(|\psi |^p A_{\alpha }) \circ \varphi ^{-1}(B_r^c) \rightarrow 0$ as $r \rightarrow 1^-$ with $B_r^c := \{z \in {\mathbb {D}}: |z| > r\}$, there exists a number $r_0 \in (0, 1)$ such that

$$\begin{aligned} \left( |\psi |^p A_{\alpha }\right) \circ \varphi ^{-1}\left( B_r^c\right) < \varepsilon ^p \left( \dfrac{\delta _0}{2}\right) ^{\alpha + 2} \text { for every } r > r_0. \end{aligned}$$

Then, for every $r > r_0$, $s \in [\frac{1}{2}, 1]$, $\delta \in [\frac{\delta _0}{2}, 2]$, and $\zeta \in \partial {\mathbb {D}}$, we have

$$\begin{aligned}&\left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ (s\varphi )^{-1}(S(\zeta , \delta )) = \int _{(s\varphi )^{-1}(S(\zeta , \delta ))} \chi _{E_r^c}(\omega ) |\psi (\omega )|^p dA_{\alpha }(\omega ) \\&\quad = \int _{\varphi ^{-1}\left( \frac{1}{s}S(\zeta , \delta ) \cap {\mathbb {D}}\right) \cap E_{r}^c} |\psi (\omega )|^p dA_{\alpha }(\omega ) \le \int _{E_r^c} |\psi (\omega )|^p dA_{\alpha }(\omega ) \\&\quad = \; \left( |\psi |^p A_{\alpha }\right) \circ \varphi ^{-1}\left( B_r^c\right) < \varepsilon ^p \left( \dfrac{\delta _0}{2}\right) ^{\alpha + 2} \le \varepsilon ^p \delta ^{\alpha + 2}. \end{aligned}$$

Consequently, for every $r > r_0$ and $s \in [\frac{1}{2}, 1]$,

$$\begin{aligned} \Vert \left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ (s\varphi )^{-1}\Vert _{\alpha } = \sup _{\delta \in (0,2], \zeta \in \partial {\mathbb {D}}} \dfrac{\left( \chi _{E_r^c} |\psi |^p A_{\alpha }\right) \circ (s\varphi )^{-1}(S(\zeta , \delta ))}{\delta ^{\alpha + 2}} \le \varepsilon ^p. \end{aligned}$$

Therefore, for every $r > r_0$ and $s \in [\frac{1}{2}, 1)$, we get

$$\begin{aligned} \Vert W_{s\psi ,s\varphi } - W_{\psi ,\varphi }\Vert _{\alpha }^p&\lesssim \left( 1 - s\right) ^p \Vert W_{\psi , \varphi }\Vert _{\alpha }^p + \dfrac{c_1^p}{\left( 1 - r^2\right) ^{\alpha + 2 + p}} (1 - s)^p \Vert \psi \Vert _{p, \alpha }^p + 2 \varepsilon ^p. \end{aligned}$$

From this it follows that

$$\begin{aligned} \limsup _{s \rightarrow 1^-}\Vert W_{s\psi ,s\varphi } - W_{\psi ,\varphi }\Vert _{\alpha }^p \lesssim \varepsilon ^p, \text { hence, } \lim _{s \rightarrow 1^-}\Vert W_{s\psi ,s\varphi } - W_{\psi ,\varphi }\Vert _{\alpha } = 0. \end{aligned}$$

Thus, (ii) is proved.

(b) Now we consider the operators $W_{\psi _0,\varphi _0}$ and $C_{\varphi _0}$, where $\psi _0(z) := 1 - z$ and $\varphi _0(z) := 1 + a(z - 1)$ with $0< a < 1$. Obviously, $W_{\psi _0, \varphi _0}$ and $C_{\varphi _0}$ belong to ${{\mathcal {C}}}_w(A_{\alpha }^p)$. However, it is easy to check that $W_{\psi _0, \varphi _0}$ is compact, while $C_{\varphi _0}$ is not compact on $A_{\alpha }^p$.

Indeed, for all $r \in (0,1)$,

$$\begin{aligned} \dfrac{1 - r^2}{1 - |1 + a(r - 1)|^2} \ge 1. \end{aligned}$$

Hence, by (1.1), $C_{\varphi _0}$ is not compact on $A_{\alpha }^p$.

Next, for any sequence $(z_n)_n$ in ${\mathbb {D}}$ with $|z_n| \rightarrow 1^-$ as $n \rightarrow \infty$, without loss of generality, we suppose that $z_n \rightarrow \eta \in \partial {\mathbb {D}}$. If $\eta \ne 1$, then $\varphi _0(z_n) \rightarrow 1 + a(\eta - 1) \in {\mathbb {D}}$ as $n \rightarrow \infty$, hence,

$$\begin{aligned} \dfrac{|\psi _0\left( z_n\right) |\left( 1 - |z_n|^2\right) }{1 - |\varphi _0\left( z_n\right) |^2} \le 2 \dfrac{1 - |z_n|^2}{1 - |\varphi _0\left( z_n\right) |^2} \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

If $\eta = 1$, then $\psi _0(z_n) \rightarrow 0$ as $n \rightarrow \infty$, hence, using [6, Corollary 2.40], we get

$$\begin{aligned} \dfrac{|\psi _0\left( z_n\right) |\left( 1 - |z_n|^2\right) }{1 - |\varphi _0\left( z_n\right) |^2}&\le |\psi _0\left( z_n\right) | \sup _{z \in {\mathbb {D}}} \dfrac{1 - |z|^2}{1 - |\varphi _0(z)|^2} \\&\le \dfrac{2(2 - a)}{a} |\psi _0\left( z_n\right) | \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

Consequently,

$$\begin{aligned} \lim _{|z|\rightarrow 1^-}\dfrac{|\psi _0(z)|\left( 1 - |z|^2\right) }{1 - |\varphi _0(z)|^2} = 0, \end{aligned}$$

which implies, by [17, Corollary 1], that $W_{\psi _0, \varphi _0}$ is compact on $A_{\alpha }^p$.

It remains to note that, by Lemma 4.1, $W_{\psi _0,\varphi _0} \sim C_{\varphi _0}$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$. From this it follows that the set ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$ is not a path component of ${{\mathcal {C}}}_w(A_{\alpha }^p)$. $\square$

From Lemma  4.1 and the results in Sect. 3, we get the following result for weighted composition operators.

Proposition 4.3

Suppose that two functions $\varphi$ and $\phi$ from ${{\mathcal {S}}}({\mathbb {D}})$ satisfy either of the following conditions:

• (i) the difference $C_\varphi -C_\phi$ is compact on $A_{\alpha }^p$,

• (ii) there is a number $r_0 \in (0,1)$ such that $\rho (\varphi (z),\phi (z)) \le r_0$ for all $z \in {\mathbb {D}}$.

Then all the operators $W_{\psi ,\varphi }$ and $W_{\chi ,\phi }$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$ belong to the same path component of ${{\mathcal {C}}}_w(A_{\alpha }^p)$.

Proof

By Lemma 4.1, $W_{\psi ,\varphi } \sim C_{\varphi }$ and $W_{\chi ,\phi } \sim C_{\phi }$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$. On the other hand, by Theorem 3.6 and Corollary 3.4, $C_{\phi } \sim C_{\varphi }$ in ${{\mathcal {C}}}(A_{\alpha }^p)$, and hence, in ${{\mathcal {C}}}_w(A_{\alpha }^p)$. Consequently, $W_{\chi ,\phi } \sim W_{\psi ,\varphi }$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$. $\square$

In view of this proposition, for each function $\varphi \in {{\mathcal {S}}}({\mathbb {D}})$, we denote by ${{\mathcal {W}}}([C_{\varphi }])$ the set of all weighted composition operators $W_{\psi ,\phi }\in {{\mathcal {C}}}_w(A_{\alpha }^p)$ with $C_{\phi } \in [C_{\varphi }]$. The following result follows immediately from Proposition 4.3.

Corollary 4.4

Each set ${{\mathcal {W}}}([C_{\varphi }])$ with $\varphi \in {{\mathcal {S}}}({\mathbb {D}})$ is path connected in ${{\mathcal {C}}}_w(A_{\alpha }^p)$.

Now we show that the sets ${{\mathcal {W}}}([C_{\varphi }])$ may be path components of the space ${{\mathcal {C}}}_w(A_{\alpha }^p)$ and may be not. To do this, we give the following examples.

Example 4.5

For $\varphi _0(z) := 1 + a(z-1)$ with $0< a < 1$, the set ${{\mathcal {W}}}([C_{\varphi _0}])$ is not a path component of ${{\mathcal {C}}}_w(A_{\alpha }^p)$. More precisely, ${{\mathcal {W}}}([C_{\varphi _0}])$ is a proper subset of the path component of ${{\mathcal {C}}}_w(A_{\alpha }^p)$ containing ${{\mathcal {C}}}_{w,0}(A_{\alpha }^p)$.

Proof

By part (b) in the proof of Theorem 4.2, the operator $W_{\psi _0,\varphi _0}$ with $\psi _0(z) := 1 - z$ and $\varphi _0(z) := 1 + a(z-1)$ is compact, while $C_{\varphi _0}$ is not compact on $A_{\alpha }^p$. Then, by Theorem 4.2 again, $W_{\psi _0,\varphi _0} \sim C_0$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$. But $C_{\varphi _0} - C_0$ is not compact on $A_{\alpha }^p$, which implies that the operator $C_0$ does not belong to ${{\mathcal {W}}}([C_{\varphi _0}])$ and completes the proof. $\square$

Example 4.6

For $\varphi _1(z) := z$, the set ${{\mathcal {W}}}([C_{\varphi _1}])$ is a path component of ${{\mathcal {C}}}_w(A_{\alpha }^p)$.

Proof

By Proposition 3.11(a), $C_{\varphi _1}$ is isolated in ${{\mathcal {C}}}(A_{\alpha }^p)$, which, by Theorem 3.6, implies that $[C_{\varphi _1}] = \{C_{\varphi _1}\}$. Then

$$\begin{aligned} {{\mathcal {W}}}\left( \left[ C_{\varphi _1}\right] \right) = \left\{ W_{\psi ,\varphi _1}: 0< \Vert \psi \Vert _{\infty } < \infty \right\} , \end{aligned}$$

where, as usual, $\displaystyle \Vert \psi \Vert _{\infty } := \sup _{z \in {\mathbb {D}}}|\psi (z)|$. Indeed, by (2.2), (2.6) and (2.8), for each $z \in {\mathbb {D}}$, we get

$$\begin{aligned} |\psi (z)|^p&\le \dfrac{c_1^p}{\left( 1 - |z|^2\right) ^{\alpha + 2}} \int _{\Delta (z)} |\psi (\omega )|^p dA_{\alpha }(\omega ) \\&= c_1^p \dfrac{ \left( |\psi |^p A_{\alpha }\right) \circ \varphi _1^{-1}(\Delta (z))}{\left( 1 - |z|^2\right) ^{\alpha + 2}} \lesssim \Vert \left( |\psi |^p A_{\alpha }\right) \circ \varphi _1^{-1}\Vert _{\alpha } < \infty . \end{aligned}$$

Now we will prove that ${{\mathcal {W}}}([C_{\varphi _1}])$ is simultaneously open and closed in ${{\mathcal {C}}}_w(A_{\alpha }^p)$, from which the assertion follows.

Let $(W_{\psi _n,\varphi _1})_n$ be a sequence in ${{\mathcal {W}}}([C_{\varphi _1}])$ converging to some operator $W_{\chi ,\phi }$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$. Then $W_{\psi _n,\varphi _1}(f) \rightarrow W_{\chi ,\phi }(f)$ in $A_{\alpha }^p$ for all $f\in A_{\alpha }^p$. In particular, taking here $f(z)\equiv 1$ and $f(z)\equiv z$, we obtain that $\psi _n\rightarrow \chi$ and $\psi _n \varphi _1\rightarrow \chi \phi$ in $A_{\alpha }^p$ as $n\rightarrow \infty$. Therefore,

$$\begin{aligned} \chi \left( \varphi _1 - \phi \right) = \left( \chi - \psi _n\right) \varphi _1 + \left( \psi _n \varphi _1 - \chi \phi \right) \rightarrow 0 \text { in } A_{\alpha }^p. \end{aligned}$$

Since $\chi \not \equiv 0$, this implies that $\phi = \varphi _1$. Thus, the set ${{\mathcal {W}}}([C_{\varphi _1}])$ is closed in ${{\mathcal {C}}}_w(A_{\alpha }^p)$. The fact that it is open in ${{\mathcal {C}}}_w(A_{\alpha }^p)$ follows immediately from the following auxiliary lemma. $\square$

Lemma 4.7

Let $W_{\psi ,\varphi _1}$ be an operator in ${{\mathcal {W}}}([C_{\varphi _1}])$. Then the inequality $\Vert W_{\psi ,\varphi _1} - W_{\chi ,\phi }\Vert _{\alpha } > rsim \Vert \psi \Vert _e$ holds for every operator $W_{\chi , \phi }$ in ${{\mathcal {C}}}_w(A_{\alpha }^p)$ with $\phi \ne \varphi _1$, where

$$\begin{aligned} \Vert \psi \Vert _e := \inf \left\{ \varepsilon > 0: F(\psi ,\varepsilon ) \text { has Lebesgue zero}\right\} \end{aligned}$$

with

$$\begin{aligned} F\left( \psi , \varepsilon \right) := \left\{ \zeta \in \partial {\mathbb {D}}: |\psi (\zeta )| \ge \varepsilon \right\} . \end{aligned}$$

Proof

Using (2.1) and (2.2), for every $w, z \in {\mathbb {D}}$, we get

$$\begin{aligned} \Vert W_{\psi , \varphi _1} - W_{\chi ,\phi }\Vert _{\alpha }&\ge \dfrac{1}{c_0} \Vert W_{\psi , \varphi _1}k_w - W_{\chi ,\phi }k_w\Vert _{p,\alpha } \\&= \dfrac{1}{c_0} \left( \int _{{\mathbb {D}}} |\psi (\zeta )k_w(\zeta ) - \chi (\zeta )k_w(\phi (\zeta ))|^p dA_{\alpha }(\zeta ) \right) ^{\frac{1}{p}} \\&\ge \dfrac{1}{c_0} \left( \int _{\Delta (z)} |\psi (\zeta )k_w(\zeta ) - \chi (\zeta )k_w(\phi (\zeta ))|^p dA_{\zeta }(\omega ) \right) ^{\frac{1}{p}} \\&\ge \dfrac{1}{c_0 c_1} \left( 1 - |z|^2\right) ^{\frac{\alpha + 2}{p}} |\psi (z)k_w(z) - \chi (z)k_w\left( \phi (z)\right) |. \end{aligned}$$

In particular, with $w = z$, we have

$$\begin{aligned} \Vert W_{\psi , \varphi _1} - W_{\chi ,\phi }\Vert _{\alpha }&\ge \dfrac{1}{c_0 c_1} \left( 1 - |z|^2\right) ^{\frac{\alpha + 2}{p}} |\psi (z)k_z(z) - \chi (z)k_z(\phi (z))| \\&\ge \dfrac{1}{c_0 c_1} \left( |\psi (z)| - |\chi (z)| \left| \dfrac{1 - |z|^2}{1 - {\overline{z}}\phi (z)} \right| ^{\frac{\sigma + \alpha + 2}{p}} \right) , \end{aligned}$$

for every $z \in {\mathbb {D}}$.

On the other hand, obviously, $\Vert \psi \Vert _e > 0$. Fix an arbitrary number $r \in (0, \Vert \psi \Vert _e)$. Then $F(\psi ,r)$ has positive Lebesgue measure.

Since $\phi \ne \varphi _1$, the set $\{\zeta \in \partial {\mathbb {D}}: \phi (\zeta ) = \zeta \}$ has Lebesgue measure zero. So there exist a point $\zeta \in F(\psi ,r)$ and a sequence $(z_n)_n \subset {\mathbb {D}}$ such that $z_n \rightarrow \zeta$, $|\psi (z_n)| \rightarrow |\psi (\zeta )| \ge r$, and $\phi (z_n) \rightarrow \eta \ne \zeta$ as $n \rightarrow \infty$. Then for each $n \in {\mathbb {N}}$, using (2.3), we get

$$\begin{aligned} |\chi (z_n)| \left| \dfrac{1 - |z_n|^2}{1 - \overline{z_n}\phi (z_n)} \right| ^{\frac{\sigma + \alpha + 2}{p}}&\le c_1 \Vert \chi \Vert _{p, \alpha } \dfrac{(1 - |z_n|^2)^{\frac{\sigma }{p}}}{|1 - \overline{z_n}\phi (z_n)|^{\frac{\sigma + \alpha + 2}{p}}} \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

Thus,

$$\begin{aligned} \Vert W_{\psi ,\varphi _1} - W_{\chi ,\phi }\Vert _{\alpha } \ge \dfrac{1}{c_0 c_1} \limsup _{n\rightarrow \infty } |\psi (z_n)| \ge \dfrac{r}{c_0 c_1} \end{aligned}$$

and so $\Vert W_{\psi ,\varphi _1} - W_{\chi ,\phi }\Vert _{\alpha } \ge \dfrac{\Vert \psi \Vert _e}{c_0 c_1}$. From this the assertion follows. $\square$