Abstract and Applied Analysis

1. Introduction and Preliminaries

Let be the upper half-plane, a domain in or , and the space of all holomorphic functions on . Let , and let be a holomorphic self-map of . Then by is defined a linear operator on which is called weighted composition operator. If , then becomes composition operator and is denoted by , and if , then becomes multiplication operator and is denoted by .

During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in or (see, e.g., [1–22] and the references therein). There are many reasons for this interest, for example, it is well known that the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see [23, 24]). For some other operators related to weighted composition operators, see [25–30] and the references therein.

While there is a vast literature on composition and weighted composition operators between spaces of holomorphic functions on the unit disk , there are few papers on these and related operators on spaces of functions holomorphic in the upper half-plane (see, e.g., [2, 3, 5, 7–9, 11, 12, 16–18, 31] and the references therein). For related results in the setting of the complex plane see also papers [19–21].

The behaviour of composition operators on spaces of functions holomorphic in the upper half-plane is considerably different from the behaviour of composition operators on spaces of functions holomorphic in the unit disk . For example, there are holomorphic self-maps of which do not induce composition operators on Hardy and Bergman spaces on the upper half-plane, whereas it is a well-known consequence of the Littlewood subordination principle that every holomorphic self-map of induces a bounded composition operator on the Hardy and weighted Bergman spaces on . Also, Hardy and Bergman spaces on the upper half-plane do not support compact composition operators (see [3, 5]).

For and , let denote the collection of all Lebesgue -integrable functions such that where , and .

Let . For is a Banach space with the norm defined by With this norm becomes a Banach space when , while for it is a Fréchet space with the translation invariant metric

Recall that for every the following estimate holds: where is a positive constant independent of .

Let . The weighted-type space (or growth space) on the upper half-plane consists of all such that It is easy to check that is a Banach space with the norm defined above. For weighted-type spaces on the unit disk, polydisk, or the unit ball see, for example, papers [10, 32, 33] and the references therein.

Given two Banach spaces and , we recall that a linear map is bounded if is bounded for every bounded subset of . In addition, we say that is compact if is relatively compact for every bounded set .

In this paper, we consider the boundedness and compactness of weighted composition operators acting from to the weighted-type space . Related results on the unit disk and the unit ball can be found, for example, in [6, 13, 15].

Throughout this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that . Moreover, if both and hold, then one says that .

2. Main Results

The boundedness and compactness of the weighted composition operator are characterized in this section.

Theorem 2.1. Let , and let be a holomorphic self-map of . Then is bounded if and only if Moreover, if the operator is bounded then the following asymptotic relationship holds:

Proof. First suppose that (2.1) holds. Then for any and , by (1.6) we have and so by (2.1), is bounded and moreover Conversely suppose is bounded. Consider the function Then and moreover (see, e.g., Lemma 1 in [18]).
Thus the boundedness of implies that for every . In particular, if is fixed then for , we get Since is arbitrary, (2.1) follows and moreover If is bounded then from (2.4) and (2.8) asymptotic relationship (2.2) follows.

Corollary 2.2. Let , and be such that and . Then is bounded if and only if , where

Example 2.3. Let and be such that and . Let be a holomorphic map of defined as For and in , we have Thus if . Similarly if . By Corollary 2.2, it follows that is bounded.

Corollary 2.4. Let , and let be a holomorphic self-map of . Then is bounded if and only if

Corollary 2.5. Let be the linear fractional map Then necessary and sufficient condition that is bounded is that and .

Proof. Assume that is bounded. Then which is finite only if and .
Conversely, if and , then from (2.13) we get , and by some calculation Hence is bounded.

Corollary 2.6. Let , and be such that . Let be a holomorphic self-map of and . Then the weighted composition operator acts boundedly from to .

Proof. By Theorem 2.1, is bounded if and only if By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map of and all where the equality holds when is a Möbius transformation given by (2.13). From (2.17), condition (2.16) follows and consequently the boundedness of the operator .

Corollary 2.6 enables us to show that there exist , and holomorphic maps and of the upper half-plane such that neither nor is bounded, but is bounded.

Example 2.7. Let , and be such that . Let , and . Then by Corollary 2.5, is not bounded. On the other hand, if then and so by Corollary 2.2, is not bounded. However, by Corollary 2.6, we have that is bounded.
The next Schwartz-type lemma characterizes compact weighted composition operators and it follows from standard arguments ([4]).

Lemma 2.8. Let , and let be a holomorphic self-map of  . Then is compact if and only if, for any bounded sequence converging to zero on compacts of  , one has

Theorem 2.9. Let and be a holomorphic self-map of   . If is compact, then

Proof. Suppose is compact and (2.20) does not hold. Then there is a and a sequence such that and for all . Let , , and Then is a norm bounded sequence and on compacts of as . By Lemma 2.8 it follows that On the other hand, which is a contradiction. Hence (2.20) must hold, as claimed.

Before we formulate and prove a converse of Theorem 2.9, we define, for every such that , the following subset of :

Theorem 2.10. Let , and let be a holomorphic self-map of and be bounded. Suppose that and as within for all and . Then is compact if condition (2.20) holds.

Proof. Assume (2.20) holds. Then for each , there is an such that Let be a sequence in such that and uniformly on compact subsets of as . Thus for such that and each , we have
From estimate (1.6) we have Thus there is an such that whenever . Hence for such that and each we have If , then by the assumption there is an such that , whenever . Therefore, for each we have whenever and .
If and , then there exists some such that for all , and so Combining (2.27)–(2.32), we have that for and some independent of . Since is an arbitrary positive number, by Lemma 2.8, it follows that is compact.

Example 2.11. Let , and be such that . Let and , then and . It is easy to see that . Beside this, for , we have
Also and the set is empty. Thus and satisfy all the assumptions of Theorem 2.10, and so is compact.

Acknowledgments

The work is partially supported by the Serbian Ministry of Science, projects III 41025 and III 44006. The work of the second author is a part of the research project sponsored by National Board of Higher Mathematics (NBHM)/DAE, India (Grant no. 48/4/2009/R&D-II/426).