Topological Structures of Derivative Weighted Composition Operators on the Bergman Space

Abstract

We characterize the difference of derivative weighted composition operators on the Bergman space in the unit disk and determine when linear-fractional derivative weighted composition operators belong to the same component of the space of derivative weighted composition operators on the Bergman space under the operator norm topology.

1. Introduction

Let be the unit disk in the complex plane. The algebra of all holomorphic functions on domain will be denoted by . Let be the set of analytic self-maps of . Every induces the composition operator defined by for .

Let be analytic, and the weighted composition operator, , is defined by for any and . When , it becomes a composition operator.

Derivative weighted composition (-composition) operator is defined by It is obvious that .

For and , we recall that a weighted Bergman space is the set of holomorphic functions on the unit disk for which where is the Lebesgue area measure on the unit disk and is the root of this integral. Moreover, the Bergman space is a Hilbert space with the inner product

A weighted Dirichlet space is the set of holomorphic functions on the unit disk for which with norm given by and the Dirichlet space is also a Hilbert space with the inner product It is easy to show that (Exercise 2.1.6 in [1])

Much effort has been expended on characterizing those analytic maps which induce bounded or compact composition operators. Readers interested in this topic can refer to the books [2] by Shapiro, [1] by Cowen and MacCluer, and [3, 4] by Zhu, which are excellent sources for the development of the theory of composition operators and function spaces, and the recent papers [5–8].

Another area of particular interest is the topological structure of the space of composition operators acting on a given function space. When is a Banach space of analytic functions, we write for the space of composition operators and for the space of derivative weighted composition operators on under the operator norm topology. The investigation of the topological structure of was initiated by Berkson [9] in 1981. Central problem focuses on the relations between the structure of and the compactness properties of its members.

Continuing the work, in 1989, MacCluer [10] showed that, on the weighted Bergman space for , the compact composition operators form an arcwise connected set in and gave necessary conditions for two composition operators to have compact difference. At about the same time, Shapiro and Sundberg [11] gave further results on compact difference and isolation and, among other things, posed the following fundamental question and conjectured that it had a positive answer:

() Do the compact composition operators form a connected component of the set ?

In 2008, Gallardo-Gutiérrez et al. [12] gave a negative answer to the question for a variety of spaces in addition to . In 2003, Bourdon [13] determined two linear-fractional self-maps of the disk having the same first-order data at a point on the boundary of the disk and different second derivatives at lie in the same component of , while the induced composition operators do not have compact difference. In 2005, Moorhouse [14] answers the question of compact difference for composition operators acting on , , and gave a partial answer to the component structure of . Later, Kriete and Moorhouse [15] extended their study to general linear combinations. Saukko [16, 17] obtained a complete characterization of bounded and compact differences between standard weighted Bergman spaces. Recently, Choe et al. [18, 19] extend Moorhouses characterization to the unit polydisk and unit ball in .

Topological structure of composition operators, which are from some analytic function spaces into the Bloch type spaces or the space of bounded analytic functions, has been studied intensively during the past decades. Interested readers can refer to [20, 21] and the references therein.

In 2004 Čučkovič and Zhao [5] characterized the bounded, compact, and Schatten class weighted composition operators on the Bergman space by generalized Berezin transforms. Derivative weighted composition operators are a special class of the weighted composition operators. The boundedness is characterized as follows.

Theorem 1 (Theorem   in [5]). Let be an analytic self-map of . Then the -weighted composition operator is bounded on if and only if

Building on those foundations, the present paper continues this line of research. The remainder is assembled as follows. In Section 2, we characterize the compact difference of two derivative weighted composition operators on the Bergman space in the unit disk and discuss the isolation and component structure of the derivative weighted composition operator in , and some similar results about composition operators on the Dirichlet space are also presented there. In Section 3, we show that the -composition operators form an arcwise connected set in . Finally, we determine when linear-fractional -composition operators belong to the same component on the Bergman space under the operator norm topology in Section 4.

2. Difference of Derivative Weighted Composition Operators

Our discussion begins with the characterization of the compact difference of -composition operators. When we accomplished the following theorem, we read a new paper [22], in which Allen et al. obtained the same estimation (Lemma 10). Nonetheless, we present our version in the following, since our method is totally different.

Theorem 2. Suppose has finite angular derivative at some point in the unit circle. Let and consider the difference of and acting on the Bergman space . Then we have if the symbol satisfies one of the following two conditions:(i) (as radial limits).(ii) but .

Proof. Since the Bergman space is rotationally invariant, we may assume , , and . We first estimate by considering the adjoint of , acting on reproducing kernel functions . Sincethe formulas for the kernel functions and their norms show thatfor all in .
By the Julia-Carathéodory Theorem, has nontangential limit as goes to , so the first term of the right-hand side in the above expression has nontangential limit as goes to .
To deal with the third term of the right-hand side in (10) we distinguish two cases.
Case 1. Consider . We can find approaching nontangentially so that .
From Schwarz-Pick Theorem we have that for every .
It follows that has limit as tends to infinity since Combining with the above discussions and by (10), we can conclude that .
Case 2. Consider but . We first considerLet , the boundary of a nontangential approach region at . As approaches along , the Julia-Carathéodory Theorem shows that and eitherorIt is easy to show that . For any given positive number , we may, by choosing large enough, find a sequence approaching along so that for large enough It follows from (13) that for large enough. So, for arbitrary positive , choose such that ; we have from which and (10), we deduce that for arbitrary positive , so by letting .
It is standard to check the lower bound of the essential norm :This completes the proof of the theorem.

As a direct consequence, we can obtain the following corollary.

Corollary 3. If has finite angular derivative on a set of positive measures, then is isolated in . For any , .

Definition 4. Let and be two self-mappings of the unit disk, and we say that and have the same first-order boundary data or satisfy SBD, provided that whenever either or , then and .

Theorem 5. Suppose is in the component of containing . Then and must have the same first-order boundary data, where has finite angular derivative.

Proof. Suppose either or at in the unit circle. If or but , by Theorem 2, , is not in the component of containing . Hence and ; that is, and must have the same first-order boundary data.

-composition operators on the weighted Bergman space are highly connected with composition operators on weighted Dirichlet space. In the following, we will use the symbol to denote a finite positive number which is not necessarily the same at each occurrence.

Lemma 6. For and , let be such that and . Then is bounded (compact) from into if and only if is bounded (compact) from into .

Proof. We just verify the condition of boundedness. Suppose is bounded from into . Then there exists a constant such that for all . Take and let the function be such that and . Then we have Thus is bounded.
Conversely, assume is bounded. Take such that . Then we have

Theorem 7. Suppose is an analytic self-map of unit disk and has finite angular derivative at some point in the unit circle. Let be another analytic self-map of unit disk and consider and acting on . Then unless both(1),(2),we have , where the constant depends only on the spaces .

Proof. Note that and . Setting and in Lemma 6, from the proof of Lemma 6, we know there exist two constants and such that The desired result now follows from Theorem 2.

Corollary 8. If has finite angular derivative on a set of positive measures, then is isolated in . For any , , where constant depends only on the .

Corollary 9. Suppose is in the component of containing . Then and must have the same first-order boundary data, where has finite angular derivative.

3. Compact Composition Operators on Dirichlet Space

The proof of the following lemma is routine, which is omitted here.

Lemma 10. Let be an analytic self-map of such that the induced composition operator is bounded on . Then acts on compactly if and only if whenever is a bounded sequence in converging to zero uniformly on compact subsets of , then .

It is well known (e.g., Proposition  9.9 in [1]) that the compact composition operators on form an arcwise connected set in for all . By Lemma 10, we find the result is also true on the Dirichlet if we note that if , then is compact on the Dirichlet space.

Theorem 11. The compact composition operators on form an arcwise connected set in .

The proof of this theorem is almost the same as the proof of Proposition  9.9 in [1], so we just give an outline here.

Proof. Given and compact on we first construct a continuous map of into taking to and to , the composition operator whose symbol is the constant map . The path , where , will satisfy our demand ( implying the compactness of ).
Secondly, we construct continuous arc joining.
to . Let be the line segment in from to and then map into by , where . It is easy to verify that this map is continuous.

It follows from Lemma 6 and Theorem 11 that we have following corollary.

Corollary 12. The compact -composition operators on form an arcwise connected set in .

From Theorem 11, we find that the compact composition operators on are in the same component, but we do not know whether there is any noncompact composition operator belonging to the component generated by compact ones on . The answers to those similar questions on different analytic function spaces appear somewhat different. One has positive answers when the spaces are classical Bergman space, Bloch space, and , while the answer is negative on classical Hardy space.

Remark 13. Let be a linear-fractional self-map of the unit disk with , where both and are unimodular. Then the induced composition operator cannot be in the component generated by compact composition operators on Dirichlet space. Suppose did. The linear-fractional self-map must have finite angular derivative at . And the composition operator induced by is compact on . By Theorem 11 compact composition operators must be in the same component and and are in the same component. By Corollary 9   and must satisfy SBD. That is impossible.

Lemma 14 (Corollary   in [5]). Let be an analytic function on and let be an analytic self-map of . Then the weighted composition operator is compact on if and only if , where is the normalized reproducing kernel of at .

There is a practical corollary of Lemma 14.

Lemma 15 (Proposition   in [5]). Let be an analytic function on and . If the weighted composition operator is compact on , then

From this lemma, in next section we will see that some linear-fractional composition operators cannot be compact on Dirichlet space, and we will discuss that component in through the characterization of the component in .

4. Linear-Fractional -Composition Operators

Throughout this section, and will denote linear-fractional self-mappings of . To determine when linear-fractional -composition operators belong to the same component of , we need to state some lemmas.

Lemma 16 (Theorem   in [3]). Suppose and is analytic in ; then if and only if is in .

If and are linear-fractional self-maps of the unit disk with the same boundary data, , where and are in the unit circle. Then it is trivial that and are in the same path component in if and only if and are in the same path component in .

So we can just discuss the situations when and . The following lemma follows Lemma  3.2 in [13] and its proof.

Lemma 17. Suppose that is a nonautomorphic, linear-fractional self-mapping of such that and . Denote for ; then there is a complex constant with positive real part such that

Conversely if we put , it is clear that is a linear-fractional self-mapping of such that and . By Lemma 15 and direct computation, it is easy to check all are noncompact on . So all are also noncompact on equivalently by Lemma 6.

Lemma 18. Let be defined by so that is the straight-line path in the open right half-plane joining and . Let be the self-map of given byThen is continuous from the unit interval to , placing both and in the same path component.

Proof. Let and be distinct points in with . This lemma will be proved if we proved that there is constant such that We require some preliminary estimates. Fix in the closure of , let , and calculate the derivative of with respect to : For any and , note that ; we haveSince , it follows from (30) thatCombining (29) and (32), there is a constant such thatfor all and . Calculate the derivative of with respect to : Similarly we havefor a constant independent of and .
An easy calculation shows that Thus, applying the triangle inequality, we see that, for every , Because is a proper subdisk of internally tangent to the unit circle at , we may set where ; that is, . It is easy to show that and this implies thatfor any .
It follows from (27) that . And the image of the circleunder is the line , where in right half-plane, and we put back into (41). We have that Consequently,For any , we have that This produces the crucial estimate: where the constant denotes upper bound of multiplied by an absolute constant: where and last inequality follows by using Lemma 16. Thus we have where () and depend only on . This completes the proof of the lemma.

Theorem 19. Suppose that and are linear-fractional self-mappings of . The following are equivalent:(a) and lie in the same component of .(b) and satisfy SBD.(c) and are joined by a continuous path in .

Proof. Clearly implies . That implies is proved in Theorem 5. Thus to complete the proof of Theorem 19, we establish implies .
If is an automorphism, then, for any in the unit circle, . That and satisfy SBD implies for any in the unit circle. It turns out that by Maximum Principle, and the conclusion is obvious.
Suppose that is not an automorphism; then is a proper subdisk of ; there are exactly two possibilities:(i) for any .(ii)There exists a point such that Suppose that applies and and satisfy SBD. Then also must map the closure of into . In this situation, both and are compact operators on . By Lemma 17, we have both and are compact on . Compact -composition operators are shown to belong to the same arcwise connected subset in Corollary 12. Thus, to complete the proof of the theorem, we need to handle case .
We may suppose that . Our setting condition is that share the same first-order boundary data with so and we denote . By Lemma 17, there are two constants and satisfying and , such that Denote described by Lemma 18; then is continuous from the unit interval to , placing both and in the same path component by Lemma 18. This ends the proof of the theorem.

Next corollary is an obvious consequence of Theorem 19 and gives a partial description of component in .

Corollary 20. Suppose that and are linear-fractional self-mappings of . The following are equivalent:(a) and lie in the same component of .(b) and satisfy SBD.(c) and are joined by a continuous path in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported in part by National Science Foundation of China (Grant nos. 11301132; 11371276; 11501415; and 11171087) and Natural Science Foundation of Hebei Province (Grant no. A2013202265) and Tianjin Advanced Education Development Fund (Grant no. 20111005).