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).