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We characterize the boundedness and compactness of differences of the composition operators followed by differentiation between weighted Banach spaces of holomorphic functions in the unit disk. As their corollaries, some related results on the differences of composition operators acting from weighted Banach spaces to weighted Bloch type spaces are also obtained.

Let

The weighted Bloch space

The

For a weight

We say that a weight

Let

By differentiation we are led to the linear operator

Recently, there has been an increasing interest in studying the compact difference of composition operators acting on different spaces of holomorphic functions. Some related results on differences of the composition operators or weighted composition operators on weighted Banach spaces of analytic functions, Bloch-type spaces, and weighted Bergman spaces can be found, for example [

For each

Throughout this paper, we will use the symbol

Now let us state a couple of lemmas, which are used in the proofs of the main results in the next sections. The first lemma is taken from [

Let

In order to handle the differences, we need the pseudohyperbolic metric. Recall that for any point

Let

For

From Lemma

The following result is well known (see, e.g., [

Assume that

Here and below we use the abbreviated notation

The following lemma is the crucial criterion for compactness, and its proof is an easy modification of that of Proposition 3.11 of [

Suppose that

In this section we will characterize the boundedness of

Suppose that

The conditions (

The conditions (

First, we prove the implication (i)

Next prove that

Next we prove (

(ii)

(iii)

Suppose that

The conditions (

The conditions (

In this section, we turn our attention to the question of compact difference. Here we consider the following conditions:

Suppose that

First we suppose that

Now, let

We divide the argument into a few cases.

On the other hand, it follows from Remark

The desired result follows by an argument analogous to that in the proof of Case 2. Thus, together with the above cases, we conclude that

For the converse direction, we suppose that

Let

By the compactness of

Letting

Now we need only to show the condition (

By Lemma

Suppose that

In this final section we give an example of function

In this example we will show that there exist weight

Let

Since for

Now we will show that

This work was supported in part by the National Natural Science Foundation of China (Grant nos. 11371276, 11301373, and 11201331).