1 INTRODUCTION

Let \(X\) and \(Y\) be Banach spaces. The essential norm of a bounded linear operator \(T:X\to Y\) is its distance to the set of compact operators \(K\) mapping \(X\) into \(Y\), that is,

$$||T||_{e,X\to Y}=\inf\{||T-K||_{X\to Y}:\ \text{K is compact}\}.$$

Let \(\mathbb{D}\) be the open unit disc in the complex plane \(\mathbb{C}\), \(H(\mathbb{D})\) the space of analytic functions on \(\mathbb{D}\) and \(H^{\infty}\) be the space of bounded analytic functions on \(\mathbb{D}\) with norm \(||f||_{\infty}=\sup_{z\in\mathbb{D}}|f(z)|\).

Let \(u\in H(\mathbb{D})\) and \(\varphi\in S(\mathbb{D})\), the set of self-maps of \(\mathbb{D}\). The weighted composition operator with symbols \(u\) and \(\varphi\), denoted by \(uC_{\varphi}\), is defined as follows

$$uC_{\varphi}f=M_{u}C_{\varphi}f=u(f\circ\varphi),\quad f\in H(\mathbb{D}),$$

where \(M_{u}\) is the multiplication operator with symbol \(u\) and \(C_{\varphi}\) is the composition operator. We refer the interested reader to [4] and [12] for the theory of the composition operators and to [2, 3, 6, 10, 14, 16, 20, 21, 22] for (weighted) composition on various spaces of analytic functions.

Let \(\mathcal{Z}\) denote the set of all functions \(f\in H(\mathbb{D})\cap C(\mathbb{\overline{D}})\) such that

$$||f||=\sup\frac{|f(e^{i(\theta+h)})+f(e^{i(\theta-h)})-2f(e^{i\theta})|}{h}<\infty,$$

where the supremum is taken over all \(\theta\in\mathbb{R}\) and \(h>0\). By Theorem 5.3 of [12] and the Closed Graph Theorem, we see that an analytic function \(f\) on \(\mathbb{D}\) belongs to \(\mathcal{Z}\) if and only if \(\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime\prime}(z)|<\infty\). Furthermore,

$$||f||\approx\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime\prime}(z)|.$$

The preceding quantity is seminorm for the space \(\mathcal{Z}\). The norm defined by

$$||f||_{\mathcal{Z}}=|f(0)|+|f^{\prime}(0)|+\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime\prime}(z)|$$

yields a Banach space structure on \(\mathcal{Z}\), which is called the Zygmund space.

Let \(0<\alpha<\infty\). A function \(f\in H(\mathbb{D})\) is said to belong to the Bloch type space \(\mathcal{B}^{\alpha}\) if

$$\beta_{f}=\sup_{z\in\mathbb{D}}(1-|z|^{2})^{\alpha}|f^{\prime}(z)|<\infty.$$

Under the seminorm \(f\to\beta_{f}\) , \(\mathcal{B}^{\alpha}\) is conformally invariant, and the norm defined by \(||f||_{\mathcal{B}^{\alpha}}=|f(0)|+\beta_{f}\) yields a Banach space structure on \(\mathcal{B}^{\alpha}\). It is well known that for \(0<\alpha<1\) \(\mathcal{B}^{\alpha}\) is a subspace of \(H^{\infty}\). When \(\alpha=1\), we get the classical Bloch space \(\mathcal{B}\).

The Lipschitz space \(\textrm{Lip}_{\alpha}\) \((\text{with}\quad 0<\alpha<1)\) is the space of functions \(f\in H(\mathbb{D})\) satisfying the Lipschitz condition of order \(\alpha\), i.e, there exists a constant \(C>0\) such that

$$|f(z)-f(w)|\leq C|z-w|^{\alpha},\quad z,w\in\mathbb{D}.$$

Such functions \(f\) extend continuously to the closure of the disc. The quantity

$$||f||_{\textrm{Lip}_{\alpha}}=|f(0)|+\sup\left\{\frac{|f(z)-f(w)|}{|z-w|^{\alpha}},\ z,w\in\mathbb{D},z\neq w\right\}$$

defines a norm on \(\textrm{Lip}_{\alpha}\). Let \(f\in\textrm{Lip}_{\alpha}\) and set

$$C=\sup\left\{\frac{|f(z)-f(w)|}{|z-w|^{\alpha}},\ z,w\in\mathbb{D},z\neq w\right\}.$$

Then, for \(z\in\mathbb{D}\), we have \(|f(z)|\leq|f(0)|+C|z|^{\alpha}\leq C|z-w|^{\alpha}\leq||f||_{\textrm{Lip}_{\alpha}}\).

Thus, taking the supremum over \(\mathbb{D}\), we obtain \(||f||_{\infty}\leq||f||_{\textrm{Lip}_{\alpha}}\). By a theorem of Hardy and Littlewood [5], the elements of \(\textrm{Lip}_{\alpha}\) are characterized by the following Bloch-type condition: A function \(f\in H(\mathbb{D})\) belongs to \(\textrm{Lip}_{\alpha}\) if and only if

$$\alpha(f)=\sup_{z\in\mathbb{D}}(1-|z|^{2})^{1-\alpha}|f^{\prime}(z)|<\infty.$$

Moreover,

$$||f||_{\textrm{Lip}_{\alpha}}\approx|f(0)|+\alpha(f).$$
(1.1)

Composition operators, weighted composition operators, and related operators between the Zygmund space and some various spaces of analytic functions have been studied in [7, 8, 9, 13, 19]. In [8], Li and Stević defined the generalization composition operator \(C_{\varphi}^{g}\) as follows

$$(C_{\varphi}^{g}f)(z)=\int\limits_{0}^{z}f^{\prime}(\varphi(\xi))g(\xi)d\xi.$$
(1.2)

Li and Stević studied the boundedness and compactness of the generalized composition operator on the Zygmund space and the Bloch type space and the little Bloch type space in [8]. In this paper, we study boundedness and compactness of the generalization composition operator \(C_{\varphi}^{g}\) from \(\textrm{Lip}_{\alpha}\) to \(\mathcal{Z}\). Also we give some estimates for the essential norm of this operator. Weighted composition operators \(uC_{\varphi}\) between \(\textrm{Lip}_{\alpha}\) and \(\mathcal{Z}\) spaces were studied by Colonna and Li in [2]. Some characterizations of the boundedness and compactness of the composition operator, as well as Volterra type operator, on the Bloch type space and the Zygmund space can be found in [1, 5, 17].

The notation \(a\preceq b\) means that there is a positive constant \(C\) such that \(a\leq Cb\). We say that \(a\approx b\) if both \(a\preceq b\) and \(b\preceq a\) hold.

2 BOUNDEDNESS OF THE OPERATOR \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)

In this section, we give necessary and sufficient conditions for the boundedness of the operator \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\).

Theorem 2.1.Let \(0<\alpha<1\), \(g\in H(\mathbb{D})\)and \(\varphi\in S(\mathbb{D})\).Then the operator \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)is bounded if only if the following quantities are finite:

$$M_{1}=\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|g^{\prime}(z)|}{(1-|\varphi(z)|^{2})^{1-\alpha}}$$

and

$$M_{2}=\sup_{z\in\mathbb{D}}\frac{(1-|z|^{2})|g(z)\varphi^{\prime}(z)|}{(1-|\varphi(z)|^{2})^{2-\alpha}}.$$

Proof. For any \(f\in\textrm{Lip}_{\alpha}\),

$$(1-|z|^{2})|(C_{\varphi}^{g}f)^{\prime\prime}(z)|=(1-|z|^{2})|(f^{\prime}(\varphi(z))g(z))^{\prime}|\leq(1-|z|^{2})|f^{\prime}(\varphi(z)||g^{\prime}(z))|$$
$${}+(1-|z|^{2})|(\varphi^{\prime}(z))||g(z)||(f^{\prime\prime}(\varphi(z))|\leq C||f||_{\textrm{Lip}_{\alpha}}\left(\frac{(1-|z|^{2})|g(z)|}{(1-|\varphi(z)|^{2})^{1-\alpha}}+\frac{(1-|z|^{2})|g^{\prime}(z)||\varphi(z)|}{(1-|\varphi(z)|^{2})^{2-\alpha}}\right),$$

where in the last inequality we have used (1.1) and the following well known characterization of Bloch type functions (see [18]):

$$\sup_{z\in\mathbb{D}}(1-|z|^{2})^{1-\alpha}|f^{\prime}(z)|\approx|f^{\prime}(0)|+\sup_{z\in\mathbb{D}}(1-|z|^{2})^{2-\alpha}|f^{\prime\prime}(z)|.$$

Conversely, assume that \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded. For a fixed \(a\in\mathbb{D}\) and for \(z\in\mathbb{D}\) and \(1\leq j\leq 3\), set

$$f_{a,j}(z)=\frac{(1-|a|^{2})^{j}}{(1-\overline{a}z)^{j-\alpha}}.$$

A direct calculation shows that

$$f_{a,j}(a)=(1-|a|^{2})^{\alpha},\quad f^{\prime}_{a,j}(a)=\frac{(j-\alpha)\overline{a}}{(1-|a|^{2})^{1-\alpha}},\quad f_{a,j}^{\prime\prime}(a)=\frac{(j-\alpha)(j+1-\alpha)\overline{a}^{2}}{(1-|a|^{2})^{2-\alpha}}.$$

Then, for \(w\in\mathbb{D}\), we get,

$$(C_{\varphi}^{g}f_{\varphi(w),1})^{\prime\prime}(w)=\frac{(1-\alpha)g(w)\overline{\varphi(w)}}{(1-|\varphi(w)|^{2})^{1-\alpha}}+\frac{(1-\alpha)(2-\alpha)g(w)\varphi^{\prime}(w)\overline{\varphi(w)}^{2}}{(1-|\varphi(w)|^{2})^{2-\alpha}},$$
(2.1)
$$(C_{\varphi}^{g}f_{\varphi(w),2})^{\prime\prime}(w)=\frac{(2-\alpha)g(w)\overline{\varphi(w)}}{(1-|\varphi(w)|^{2})^{1-\alpha}}+\frac{(2-\alpha)(3-\alpha)g(w)\varphi^{\prime}(w)\overline{\varphi(w)}^{2}}{(1-|\varphi(w)|^{2})^{2-\alpha}}$$
(2.2)

and

$$(C_{\varphi}^{g}f_{\varphi(w),3})^{\prime\prime}(w)=\frac{(3-\alpha)g(w)\overline{\varphi(w)}}{(1-|\varphi(w)|^{2})^{1-\alpha}}+\frac{(3-\alpha)(4-\alpha)g(w)\varphi^{\prime}(w)\overline{\varphi(w)}^{2}}{(1-|\varphi(w)|^{2})^{2-\alpha}}.$$
(2.3)

Subtracting (2.1) from (2.2), we get

$$(C_{\varphi}^{g}f_{\varphi(w),2})^{\prime\prime}(w)-(C_{\varphi}^{g}f_{\varphi(w),1})^{\prime\prime}(w)=\frac{g(w)\overline{\varphi(w)}}{(1-|\varphi(w)|^{2})^{1-\alpha}}+\frac{(4-2\alpha)g(w)\varphi(w)^{2}\overline{\varphi(w)}^{2}}{(1-|\varphi(w)|^{2})^{2-\alpha}}.$$
(2.4)

On the other hand, subtracting (2.1) from (2.3), we obtain

$$(C_{\varphi}^{g}f_{\varphi(w),3})^{\prime\prime}(w)-(C_{\varphi}^{g}f_{\varphi(w),1})^{\prime\prime}(w)=\frac{2g(w)\overline{\varphi(w)}}{(1-|\varphi(w)|^{2})^{1-\alpha}}+\frac{(10-4\alpha)g(w)\varphi(w)^{2}\overline{\varphi(w)}^{2}}{(1-|\varphi(w)|^{2})^{2-\alpha}}.$$

Subtracting (2.3) from (2.4), we get

$$\frac{2g(w)\varphi^{\prime}(w)\overline{\varphi(w)}^{2}}{(1-|\varphi(w)|^{2})^{2-\alpha}}=(C_{\varphi}^{g}f_{\varphi(w),1})^{\prime\prime}(w)-2(C_{\varphi}^{g}f_{\varphi(w),2})^{\prime\prime}(w)+(C_{\varphi}^{g}f_{\varphi(w),3})^{\prime\prime}(w),$$

which implies that

$$\frac{(1-|w|^{2})|g(w)\varphi^{\prime}(w)||\varphi(w)|^{2}}{(1-|\varphi(w)|^{2})^{2-\alpha}}$$
$${}=\frac{1}{2}|(C_{\varphi}^{g}f_{\varphi(w),1})^{\prime\prime}(w)|+|(C_{\varphi}^{g}f_{\varphi(w),2})^{\prime\prime}(w)|+\frac{1}{2}|(C_{\varphi}^{g}f_{\varphi(w),3})^{\prime\prime}(w)|$$
$${}\leq\frac{1}{2}||(C_{\varphi}^{g}f_{\varphi(w),1})(w)||_{\mathcal{Z}}+||(C_{\varphi}^{g}f_{\varphi(w),2})(w)||_{\mathcal{Z}}+\frac{1}{2}||(C_{\varphi}^{g}f_{\varphi(w),3})(w)||_{\mathcal{Z}}\leq C.$$
(2.5)

Fix \(r\in(0,1)\). If \(|\varphi(w)|>r\), then by (2.5), we have

$$\frac{(1-|w|^{2})|g^{\prime}(w)|\varphi(w)|}{(1-|\varphi(w)|^{2})^{2-\alpha}}\leq\frac{C}{r}.$$

Taking the functions \(f(z)=z\) and \(f(z)=z^{2}\) respectively, we obtain

$$N_{1}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)|<\infty$$
(2.6)

and \(N_{2}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)\varphi(z)+g(z)\varphi^{\prime}(z)|<\infty.\)

If \(|\varphi(w)|<r\), then by (2.6) we get

$$M_{1}=\frac{(1-|w|^{2})|g^{\prime}(w)|}{(1-|\varphi(w)|^{2})^{2-\alpha}}\leq\frac{N_{1}}{(1-r^{2})^{1-\alpha}},$$
(2.7)

which, combined with (2.7), implies that \(M_{1}<\infty\). Arguing similarly, we get

$$\frac{(1-|w|^{2})|g^{\prime}(w)|}{(1-|\varphi(w)|^{2})^{2-\alpha}}\leq C$$

and

$$M_{2}=\frac{(1-|w|^{2})|g(w)\varphi^{\prime}(w)|}{(1-|\varphi(w)|^{2})^{2-\alpha}}\leq\frac{N_{2}}{(1-r^{2})^{2-\alpha}}<\infty.$$

This means \(M_{2}<\infty\). \(\Box\)

3 COMPACTNESS OF THE OPERATOR \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)

In this section, we study the compactness of the operator \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\rightarrow\mathcal{Z}\). We begin with the following lemma.

Lemma 3.1 ([11], Lemma 3.7). Let \(0<\alpha<1\)and \(T\)be a bounded linear operator from \(\textrm{Lip}_{\alpha}\)into a normed linear space \(Y\).Then \(T\)is compact if and only if \(||Tf_{n}||_{Y}\rightarrow 0\)whenever \(\{f_{n}\}\)is a norm-bounded sequence in \(\textrm{Lip}_{\alpha}\)that converges to 0 uniformly on \(\bar{\mathbb{D}}\).

Theorem 3.2.Let \(0<\alpha<1\), \(\varphi\in S(\mathbb{D})\)and \(g\in H(\mathbb{D})\).Then \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)is compact if and only if bounded,

$$\lim_{|\varphi(z_{k})|\to 1}\frac{(1-|z_{k}|^{2})|g^{\prime}(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{1-\alpha}}=0$$
(3.1)

and

$$\lim_{|\varphi(z_{k})|\to 1}\frac{(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}=0.$$

Proof. Let \((z_{k})_{k\in\mathbb{N}}\) be a sequence in \(\mathbb{D}\) such that \(|\varphi(z_{k})|\to 1\) as \(k\to\infty\). Let \(f_{k,j}=\frac{(1-|\varphi(z_{k})|^{2})^{j}}{(1-\overline{\varphi(z_{k}})z)^{j-\alpha}},k\in\mathbb{N}\). Then \(f_{k,j}\in\textrm{Lip}_{\alpha}\), \(\sup_{k\in\mathbb{N}}||f_{k,j}||_{\textrm{Lip}_{\alpha}}<\infty\) and \(f_{k,j}\to 0\) uniformly on \(\mathbb{\overline{D}}\) as \(k\to\infty\). Let \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) be compact. By Lemma 3.1 it gives \(\lim_{k\to\infty}||C_{\varphi}^{g}f_{k,j}||_{\mathcal{Z}}=0\). Note that

$$f_{k,j}^{\prime}(\varphi(z_{k}))=\frac{(j-\alpha)\overline{\varphi(z_{k}})}{(1-|\varphi(z_{k}|^{2})^{1-\alpha}},\quad f_{a,j}^{\prime\prime}(\varphi(z_{k}))=\frac{(j-\alpha)(j+1-\alpha)\overline{\varphi^{2}(z_{k})}}{(1-|\varphi(z_{k}|^{2})^{2-\alpha}}.$$

We have

$$||C_{\varphi}^{g}f_{k,j}||_{\mathcal{Z}}\geq\left.\frac{(j-\alpha)(1-|z_{k}|^{2})|g^{\prime}(z_{k})||\varphi(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{1-\alpha}}\right|$$
$${}-\left.\frac{(j-\alpha)(j+1-\alpha)(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})||\varphi(z_{k})|^{2}}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}\right|.$$

Consequently,

$$\lim_{|\varphi(z_{k})|\to 1}\frac{(j-\alpha)(1-|z_{k}|^{2})|g^{\prime}(z_{k})||\varphi(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{1-\alpha}}$$
$$=\lim_{|\varphi(z_{k})|\to 1}\frac{(j-\alpha)(j+1-\alpha)(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})||\varphi(z_{k})|^{2}}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}$$
(3.2)

if one of these two limits exists. Next, set

$$h_{k,j}=\frac{(1-|\varphi(z_{k})|^{2})^{j}}{(1-\overline{\varphi(z_{k}})z)^{j-\alpha}}-\frac{j-\alpha}{j+1-\alpha}\frac{(1-|\varphi(z_{k})|^{2})^{j+1}}{(1-\overline{\varphi(z_{k}})z)^{j+1-\alpha}}.$$

Then \(h_{k,j}^{\prime}(\varphi(z_{k}))=0\), \(\sup_{k\in\mathbb{N}}||h_{k,j}||_{\mathcal{Z}}<\infty\) and \(h_{k,j}\) converges to \(0\) uniformly on \(\mathbb{\overline{D}}\) as \(k\to\infty\). Since \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is compact, we have \(\lim_{k\to\infty}||C_{\varphi}^{g}h_{k,j}||_{\mathcal{Z}}=0\). On the other hand,

$$||C_{\varphi}^{g}h_{k,j}||_{\mathcal{Z}}\geq\frac{(j-\alpha)(j+1-\alpha)(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})||\varphi(z_{k})|^{2}}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}.$$

Hence,

$$\lim_{k\to\infty}\frac{(j-\alpha)(j+1-\alpha)(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})||\varphi(z_{k})|^{2}}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}=0.$$

Therefore,

$$\lim_{|\varphi(z_{k})|\to 1}\frac{(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}=\lim_{k\to\infty}\frac{(j-\alpha)(j+1-\alpha)(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})||\varphi(z_{k})|^{2}}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}=0.$$

This together with (3.2) imply that

$$\lim_{|\varphi(z_{k})|\to 1}\frac{(j-\alpha)(1-|z_{k}|^{2})|g^{\prime}(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{1-\alpha}}=0.$$

Conversely, assume that \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded and (3.1) holds. Since \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded, we have \(||C_{\varphi}^{g}f||_{\mathcal{Z}}\leq C||f||_{\textrm{Lip}_{\alpha}}\) for all \(f\in\textrm{Lip}_{\alpha}\). Taking the functions \(f(z)=z\) and \(f(z)=z^{2}\) respectively, we obtain

$$\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)|<\infty$$
(3.3)

and

$$\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)\varphi(z)+g(z)\varphi^{\prime}(z)|<\infty.$$
(3.4)

Using these facts and the boundedness of the function \(\varphi(z)\), we get

$$\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)|<\infty.$$

Then,

$$C_{1}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g(z)||\varphi^{\prime}(z)|<\infty$$
(3.5)

and

$$C_{2}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g(z)||\varphi^{\prime}(z)|<\infty.$$

On the other hand, from (3.1), for every \(\epsilon>0\), there is a \(\delta\in(0,1)\) such that

$$\frac{(1-|z_{k}|^{2})|g^{\prime}(z_{k})||\varphi(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{1-\alpha}}<\epsilon\quad\textrm{and}\quad\frac{(1-|z_{k}|^{2})|\varphi^{\prime}(z_{k})|g(z_{k})|}{(1-|\varphi(z_{k})|^{2})^{2-\alpha}}<\epsilon,$$
(3.6)

whenever \(\delta<|\varphi(z)|<1\). Assume that \((f_{k})_{k\in\mathbb{N}}\) is a sequence in \(\textrm{Lip}_{\alpha}\) such that \(\sup_{k\in\mathbb{N}}||f_{k}||_{\textrm{Lip}_{\alpha}}<\infty\) and \((f_{k})\) converges to \(0\) uniformly on \(\mathbb{\overline{D}}\) as \(k\to\infty\). Let \(U=\{z\in\mathbb{D}:|\varphi(z)|\leq\delta\}\). Then by (3.5) and (3.6), it follows that

$$\sup_{z\in\mathbb{D}}(1-|z|^{2})|(C_{\varphi}^{g}f_{k})^{\prime\prime}(z)|\leq\sup_{z\in\mathbb{D}}(1-|z|^{2})|f_{k}^{\prime}(\varphi(z))||g^{\prime}(z)|+\sup_{z\in\mathbb{D}}(1-|z|^{2})|(\varphi^{\prime}(z))||g(z)||f_{k}^{\prime\prime}(\varphi(z))|$$
$${}+\sup_{z\in\mathbb{D}\backslash U}(1-|z|^{2})|f_{k}^{\prime}(\varphi(z))||g^{\prime}(z)|+\sup_{z\in\mathbb{D}\backslash U}(1-|z|^{2})|(\varphi^{\prime}(z))||g(z)||f_{k}^{\prime\prime}(\varphi(z))|$$
$${}\leq C_{1}\sup_{z\in\mathbb{D}}|f_{k}^{\prime}(\varphi(z))|+\sup_{z\in\mathbb{D}\backslash U}\frac{(1-|z|^{2})|g(z)|}{(1-|\varphi(z)|^{2})^{1-\alpha}}||f||_{\textrm{Lip}_{\alpha}}+C_{2}\sup_{z\in\mathbb{D}}|f_{k}^{\prime\prime}(\varphi(z))|$$
$${}+\sup_{z\in\mathbb{D}\backslash U}\frac{(1-|z|^{2})|g^{\prime}(z)||\varphi(z)|}{(1-|\varphi(z)|^{2})^{2-\alpha}}||f||_{\textrm{Lip}_{\alpha}}\leq C_{1}\sup_{|\lambda|\leq\delta}|f_{k}^{\prime}(\lambda)|+C_{2}\sup_{|\lambda|\leq\delta}|f_{k}^{\prime\prime}(\lambda)|+2C_{\epsilon}||f||_{\textrm{Lip}_{\alpha}}.$$

So,

$$||C_{\varphi}^{g}f_{k}||_{\mathcal{Z}}=|f_{k}^{\prime}(\varphi(0))||g(0)|+\sup_{z\in\mathbb{D}}(1-|z|^{2})|(C_{\varphi}^{g}f_{k})^{\prime\prime}(z)|$$
$${}\leq C_{1}\sup_{|\lambda|\leq\delta}|f_{k}^{\prime}(\lambda)|+C_{2}\sup_{|\lambda|\leq\delta}|f_{k}^{\prime\prime}(\lambda)|+2C_{\epsilon}||f||_{\textrm{Lip}_{\alpha}}+|f_{k}^{\prime}(\varphi(0))||g(0)|.$$

The proof is complete. \(\Box\)

4 ESSENTIAL NORM OF \(C_{\varphi}^{g}f:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)

In this section, we give some estimates for the essential norm of operator \(C_{\varphi}^{g}f:\textrm{Lip}_{\alpha}\to\mathcal{Z}\).

Theorem 4.1. Let \(\varphi\in S(\mathbb{D})\) and \(g\in H(\mathbb{D})\) such that \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded. Then

$$||C_{\varphi}^{g}f||_{e,\textrm{Lip}_{\alpha}\to\mathcal{Z}}\approx\max\{A_{1},A_{2}\},$$

where

$$A_{j}:=\limsup_{|a|\to 1}||C_{\varphi}^{g}\left(\frac{(1-|a|^{2})^{j}}{(1-\overline{a}z)^{j-\alpha}}\right)||_{\mathcal{Z}},\quad j=1,2.$$

Proof. First we prove that \(\max\{{A_{1},A_{2}}\}\leq||C_{\varphi}^{g}||_{e,\textrm{Lip}_{\alpha}\to\mathcal{Z}}\). Let \(a\in\mathbb{D}\). Define

$$f_{a,j}(z)=\frac{(1-|a|^{2})^{j}}{(1-\overline{a}z)^{j-\alpha}}.$$

It is easy to check that \(f_{a,j}\in\textrm{Lip}_{\alpha}\) for all \(a\in\mathbb{D}\) and \(f_{a,j}\) converges uniformly to \(0\) on compact subset of \(\textrm{Lip}_{\alpha}\) as \(|a|\to 1\) Thus, for any compact operator \(T:\textrm{Lip}_{\alpha}\to\mathcal{Z}\), we have \(\lim_{|a|\to 1}||Tf_{a,j}||_{\mathcal{Z}}=0,\quad j=1,2\). Hence,

$$||C_{\varphi}^{g}-T||_{\textrm{Lip}_{\alpha}\to\mathcal{Z}}\gtrsim\limsup_{|a|\to 1}||C_{\varphi}^{g}-Tf_{a,j}||_{\mathcal{Z}}\gtrsim\limsup_{|a|\to 1}||C_{\varphi}^{g}f_{a,j}||_{\mathcal{Z}}-\limsup_{|a|\to 1}||Tf_{a,j}||_{\mathcal{Z}}=A_{j}.$$

Therefore, based on the definition of the essential norm, we obtain

$$||C_{\varphi}^{g}||_{e,\textrm{Lip}_{\alpha}\to\mathcal{Z}}=\inf_{k}||C_{\varphi}^{g}-T||_{\textrm{Lip}_{\alpha}\to\mathcal{Z}}\gtrsim A_{j},\quad j=1,2.$$

Now, we prove that \(||C_{\varphi}^{g}f||_{e,\textrm{Lip}_{\alpha}\to\mathcal{Z}}\lesssim\max\{{A_{1},A_{2}}\}.\) For \(r\in[0,1)\), set \(K_{r}:H(\mathbb{D})\to H(\mathbb{D})\) by \((K_{r}f)(z)=f_{r}(z)=f(rz)\). It is obvious that \(f_{r}-f\to 0\) uniformly on compact subsets of \(\mathbb{D}\) as \(r\to 1\). Moreover, the operator \(K_{r}\) is compact on \(\mathcal{B}\) and \(||K_{r}||_{\mathcal{B}\to\mathcal{B}}\leq 1\)(see [10]). By a similar argument it can be proved that the operator \(K_{r}\) is compact on \(\textrm{Lip}_{\alpha}\) and \(||K_{r}||_{\textrm{Lip}_{\alpha}\to\textrm{Lip}_{\alpha}}\leq 1\). Let \(\{r_{j}\}\subset(0,1)\) be a sequence such that \(r_{j}\to 1\) as \(j\to\infty\). Then for all positive integer \(j\), the operator \(C_{\varphi}^{g}K_{r_{j}}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is compact. By the definition of the essential norm, we get

$$||C_{\varphi}^{g}||_{e,\textrm{Lip}_{\alpha}\to\mathcal{Z}}\leq\limsup_{j\to\infty}||C_{\varphi}^{g}-C_{\varphi}^{g}K_{r_{j}}||_{\textrm{Lip}_{\alpha}\to\mathcal{Z}}.$$

For any \(f\in\textrm{Lip}_{\alpha}\) such that \(||f||_{\textrm{Lip}_{\alpha}}\leq 1\),

$$||(C_{\varphi}^{g}-C_{\varphi}^{g}K_{r_{j}})f||_{\mathcal{Z}}\leq|(C_{\varphi}^{g}f(0)|+|(f-f_{r_{j}})^{\prime}(\varphi(0))g(0)|+\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)|(f-f_{r_{j}})^{\prime}(\varphi(z))|$$
$${}+\sup_{z\in\mathbb{D}}(1-|z|^{2})|g(z)\varphi^{\prime}(z)|(f-f_{r_{j}})^{\prime\prime}(\varphi(z))|$$
$${}\leq\underbrace{\limsup_{j\to\infty}\sup_{|\varphi(z)|\leq r_{N}}(1-|z|^{2})|g^{\prime}(z)|(f-f_{r_{j}})^{\prime}(\varphi(z))|}_{M_{1}}$$
$${}+\underbrace{\limsup_{j\to\infty}\sup_{|\varphi(z)|>r_{N}}(1-|z|^{2})|g^{\prime}(z)|(f-f_{r_{j}})^{\prime}(\varphi(z))|}_{M_{2}}$$
$${}+\underbrace{\limsup_{j\to\infty}\sup_{|\varphi(z)|\leq r_{N}}(1-|z|^{2})|g(z)\varphi^{\prime}(z)|(f-f_{r_{j}})^{\prime\prime}(\varphi(z))|}_{M_{3}}$$
$${}+\underbrace{\limsup_{j\to\infty}\sup_{|\varphi(z)|>r_{N}}(1-|z|^{2})|g(z)\varphi^{\prime}(z)|(f-f_{r_{j}})^{\prime\prime}(\varphi(z))|}_{M_{4}},$$

where \(N\in\mathbb{N}\) is large enough such that \(r_{j}\geq\frac{1}{2}\) for all \(j\in\mathbb{N}\). Since \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded, by (3.3) and (3.4), we have

$$\widetilde{F_{1}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)|<\infty,\quad\widetilde{F_{2}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)\varphi(z)+g(z)\varphi^{\prime}(z)|<\infty.$$

Since \(r_{j}f_{r_{j}}\to f^{\prime}\) uniformly on compact subsets of \(\mathbb{D}\) as \(j\to\infty\), so

$$M_{1}\leq\widetilde{F_{1}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)|=0,\quad M_{3}\leq\widetilde{F_{2}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)\varphi(z)+g(z)\varphi^{\prime}(z)|=0.$$

Next we consider \(M_{2}\). We have \(M_{2}\leq\limsup_{j\to\infty}(Q_{1}+Q_{2})\), where

$$Q_{1}=\sup_{|\varphi(z)|>r_{N}}(1-|z|^{2})|(f^{\prime}(\varphi(z))||g(z)\varphi^{\prime}(z)|,\quad Q_{2}=\sup_{|\varphi(z)|>r_{N}}(1-|z|^{2})r_{j}|(f^{\prime}(\varphi(z))||g(z)\varphi^{\prime}(z)|.$$

Using the fact that \(||f||_{\textrm{Lip}\alpha}\leq 1\) and (1.1), we obtain

$$Q_{1}=\sup_{|\varphi(z)|>r_{N}}(1-|z|^{2})|(f^{\prime}(\varphi(z))||g(z)\varphi^{\prime}(z)|\frac{(1-|\varphi(z)|^{2})^{1-\alpha}}{(j-\alpha)\overline{\varphi(z)}}\frac{(j-\alpha)\overline{\varphi(z)}}{(1-|\varphi(z)|^{2})^{1-\alpha}}$$
$${}\preceq\frac{(j-\alpha)||f||_{\textrm{Lip}_{\alpha}}}{r_{N}}\sup_{|\varphi(z)|>r_{N}}(1-|z|^{2})|g(z)\varphi^{\prime}(z)|\frac{(j-\alpha)\overline{\varphi(z)}}{(1-|\varphi(z)|^{2})^{1-\alpha}}$$
$${}\preceq\sup_{|\varphi(z)|>r_{N}}(1-|z|^{2})|g(z)\varphi^{\prime}(z)|\frac{(j-\alpha)\overline{\varphi(z)}}{(1-|\varphi(z)|^{2})^{1-\alpha}}\preceq\sup_{|a|>r_{N}}||C_{\varphi}^{g}(f_{a,j})||,\quad j=1,2.$$

Taking the limit as \(N\to\infty\), we obtain

$$\limsup_{j\to\infty}Q_{1}\leq\limsup_{|a|\to\infty}||C_{\varphi}^{g}(f_{a,j})||_{\mathcal{Z}}.$$

Similarly,

$$\limsup_{j\to\infty}Q_{2}\leq\limsup_{|a|\to\infty}||C_{\varphi}^{g}(f_{a,j})||_{\mathcal{Z}}.$$

Hence, we get \(M_{2}\preceq\max\{A_{1},A_{2}\}\). Similarly, it can be shown that \(M_{4}\preceq\max\{A_{1},A_{2}\}\). This completes the proof of the theorem. \(\Box\)