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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 132541, 15 pages
http://dx.doi.org/10.1155/2011/132541
Research Article

Isometric and Closed-Range Composition Operators between Bloch-Type Spaces

Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2

Received 16 November 2010; Accepted 30 March 2011

Academic Editor: Asao Arai

Copyright © 2011 Nina Zorboska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present an overview of the known results describing the isometric and closed-range composition operators on different types of holomorphic function spaces. We add new results and give a complete characterization of the isometric univalently induced composition operators acting between Bloch-type spaces. We also add few results on the closed-range determination of composition operators on Bloch-type spaces and present the problems that are still open.

1. Introduction

A topic of interest in the paper is the description of isometric and, more generally, of closed-range composition operators on the Bloch-type spaces, in terms of the specific behaviour of the inducing function. The goal of the paper is to present an overview of the known results by emphasizing the intuitive idea and geometrical aspects of the corresponding conditions, to contribute to the classification with few new results and to list a number of open questions related to this topic.

One of the earliest results on isometric composition operators, acting on spaces of functions analytic on the open unit disk, is Nordgren's result [1] from 1968: if is inner, then is an isometry on if and only if . Martín and Vukotić have generalized recently in [2] that, indeed, is an isometry on for all if and only if is inner and .

Since rotations induce isometric composition operators on most of the spaces, it is of particular interest to determine the function spaces on which these are the only kind of isometric composition operators. Such are, for example, all of the weighted Bergman spaces, as shown by Martín and Vukotić in [2]. They have also classified the isometric composition operators on the Dirichlet space and, under the univalence condition of the inducing function, on some of the other Besov spaces (see [2, 3]). The isometric composition operators on the BMOA space have been determined by Laitila (see [4]).

As for the classification of the larger class of closed-range composition operators on spaces of functions analytic on the unit disk, the known results include some of the weighted Bergman spaces (see [5, 6]), and some of the Bloch-type spaces, which we will state and refer to in the next few sections.

In most of the cases, the general rule is that a composition operator is either isometric or has a closed range, whenever the image of the unit disc under the inducing function covers a significant (in some sense) part of . As we will see below, that stays true in the case of the Bloch-type spaces, with a specific description of what the “significant part” means in this context. Note that this represents a logical contrast to the description of the compact composition operators on all of these spaces, where the image of the unit disc under the inducing function must stay away significantly (again, in some sense) from the unit circle.

2. Definitions, Few Basic Notions, and Overview of the Existing Results

For a nonconstant analytic function that maps the unit disk into itself, the composition operator on the Banach space is defined by

with in , where is the space of functions analytic on . We will say that is the inducing function for .

Depending on the space , one gets various conditions on the inducing function under which the corresponding composition operator satisfies a certain operator theoretic property such as, for example, being bounded, compact, invertible, normal, subnormal, isometric, closed range, Fredholm, and many others. For general results and references on composition operators acting on various spaces of analytic functions, see, for example, [7, 8].

For , the - Bloch spaces (also referred to as Bloch-type spaces) are spaces of functions in such that

Each is a Banach space with a norm given by

The family of Bloch-type spaces includes the classical Bloch space . The spaces with are the analytic Lipschitz spaces . Thus, for , , where is the disk algebra. In general, for , we have that and so the -Bloch spaces form an increasing, uniform family of function spaces. Note also that the Bloch space contains and is included in all of the Bergman spaces , , while for large , such as , includes the Bergman space .

For further references and details on these general facts about the Bloch-type spaces stated above and more, see [9, 10].

The boundedness and compactness of composition operators acting between Bloch-type spaces has been established in a series of papers [1115]. We state the most general form of these results and use the following notation: for and being an analytic self map of , let

We write whenever , and if .

Theorem A (see [12, 15]). For and an analytic selfmap of , the composition operator is a bounded operator from into if and only if and is a compact operator from into if and only if

Here are a few simple consequences of Theorem A and of some basic complex analysis: recall the Schwarz-Pick lemma which states that for a self-map of the unit disk

Thus, when , , and we get from Theorem A that every composition operator is bounded on . Moreover, the Schwarz-Pick lemma and Theorem A imply that if , then maps boundedly into , since then . If furthermore , then from into must also be compact. Note also that if and is bounded, then is compact.

All of the spaces include the identity function, and so a necessary condition for to be bounded from into is that belongs to . Thus, for , every analytic self-map of that is in induces an unbounded composition operator from to .

In this paper, we are particularly interested in the closed-range composition operators and, even more specific, in the isometric composition operators. Recall that the operator is isometric whenever

Since every nonconstant is an open map, the composition operator is always one to one. By a basic operator theory result, a one-to-one operator has a closed range if and only if it is bounded below. Thus, has a closed range if and only if it is bounded below, namely, if and only if there exists such that

In particular, every isometric has a closed range.

On the other hand, recall that the only (closed) subspaces of the range of a compact operator are the finite dimensional ones. Thus, since a composition operator never has a finite rank (because is an open map), a compact can never have a closed range. Hence, a compact can never be an isometry.

The first few classification results on isometric composition operators acting on the Bloch-type spaces were done for the classical Bloch space in a series of papers [1619]. We present the classification in the form that appears in [18].

Theorem B (see [18]). Let be an analytic self-map of . The composition operator is an isometry on the Bloch space if and only if and either is a rotation, or for every in there exists a sequence in such that , , and .

The result provides a large class of functions inducing isometric composition operators on the Bloch space. For example, if is an almost thin infinite Blaschke product fixing the origin, that is a Blaschke product with a sequence of zeroes that includes 0 and is such that , then is an isometry on . For more examples and the proof of the theorem, see [18].

For all of the other Bloch-type spaces, the result from [20] shows that the only isometric composition operators are the trivial ones.

Theorem C (see [20]). Let , , and let be an analytic self-map of . Then the composition operator is an isometry on if and only if is a rotation.

Remarks 2.1. As already mentioned in the introduction, one notices a general behavior of the functions inducing an isometric composition operator. From the two previous theorems, we can see that if is an isometry on , then , which further implies that . Moreover, in the case , must be a rotation.
This is similar to many other cases of isometric composition operators acting on spaces of analytic functions. The general requirement is that covers a significant (in some sense) part of , or even further, that has to be a rotation. For example, if is an isometry either on the spaces or on the weighted Bergman spaces with , then has to be either an inner function or a rotation, respectively, as shown in [2]. If is an isometry on the Dirichlet space, then has to be an univalent full map, that is, one to one and such that the area measure of is zero, (see [3]). The complete determination of isometric composition operators in all of this cases is given by adding the condition .
A specific isometric requirement in the case of the Bloch space is that when is not univalent, it has to be of infinite multiplicity and such that the function stays close to 1 over some of the preimages of each point in . When is univalent, has to be a rotation, and thus , for all .
Since every isometric composition operator has a closed range, and since the closed-range composition operators are semi-Fredholm, that is, in some sense close to invertible, it is not too surprising that similar requirements on and play a role in determining the closed range composition operators on .
The following Theorem D below classifies the closed range composition operators on the Bloch spaces . The results are contained in a series of papers [2123], and we present their combined version. We need a few more definitions before we can state the theorem.
We say that is sampling for if such that for all ,
Let denote the pseudohyperbolic distance on , where is a disc automorphism of , that is, We say that is an for for some if for all , such that .
For , let and let . If , we use the notation and , and if further , we write and .

Theorem D (see [21, 23, 24]). Let be a self-map of and let . Then, the following are equivalent. (i) has a closed range on .(ii)There exists such that the set is sampling for .(iii)There exist with such that is an -net for .
Moreover, if , then (i) to (iii) are also equivalent to the following. There exists such that , for all .

Note, for example, that when and is a rotation, , for all , and so . Thus, is trivially sampling for and an -net for for any . Therefore, (ii) and (iii) are true. Part (iv) of the theorem also holds true if one takes .

If is not a rotation and is an isometry on , then whenever with , we have from Theorem B that . Hence, as before, is trivially sampling for and an -net for , for any . Again, part (iv) of the theorem is true with . Thus, we get a geometric description of closed-range (or isometric) composition operators on these Bloch spaces as composition operators for which the inducing function has a function that stays away from zero (or is close enough to one) over a set with large enough image.

The geometrical aspects of the restrictions on the function are particularly interesting in the case when is univalent, since then, as a consequence of the Köebe one-quarter theorem, we have that

For the isometry case, this implies that has to be a rotation, and for the closed-range case, it gives a deeper insight on the boundary behavior of , providing a number of interesting examples and counterexamples.

For example, one gets that if is the Riemann mapping from onto the slit disk , then has a closed range on . Or if is the Riemann mapping onto the simply connected region in created by taking away an infinite countable number of slits and pseudo-hyperbolic disks connected to the slits, one can get either a closed-range or a nonclosed-range composition operator by controlling the size and the placement of the pseudohyperbolic disks. For more details on these examples, see [23].

It is not too hard to see that if has a closed range on for some , then has a closed range on for all . On the other hand, the slit disk example from above provides an example of a univalent map such that has a closed range on only for (see [21, 23]).

3. Further Results on Closed-Range and Isometric Composition Operators on the Bloch-Type Spaces

In this section, we present our results on the classification of the isometric and closed-range composition operators . As Theorem A from the previous section states, not every such composition operator is bounded. Thus, the boundedness condition on plays a natural role in the characterization of the isometric and closed-range composition operators from into .

In general, depending on the choices of and , a composition operator induced by a specific, fixed function can behave very differently. We illustrate this with the following example, in which the inducing function is one of the nicest univalent selfmaps of the unit disk, namely, a rotation.

Example 3.1. Let , with . Then is (a)an isometry, whenever , (b)a compact operator if , (c)an unbounded operator if . Note that (a) holds since , and we have that The other two cases follow easily from Theorem A, since
As for many other spaces, a condition that must satisfy whenever is an isometry from into , regardless of the choices of and , is that .

Proposition 3.2. If is an isometry from into , then and

Proof. Note first that the identity function belongs to each of the Bloch-type spaces and has norm one. Thus, since is an isometry, it must be that .
Let . Using the function , we see that But since is an isometry and , it must be that . Thus .
To show that , we use another type of test functions. For , let be the function such that and , where is the disc automorphism of defined by Using that , we get Hence, since is an isometry from into , we have that for any in where the last inequality follows by choosing , and the fact that . Thus , for all . On the other hand, for any and thus, .

Note that in the more general case when has a closed range, does not necessarily have to be zero. Still, without loss of generality, we will consider only the case . This is possible since the disk automorphisms induce invertible (and thus closed range) composition operators on every Bloch-type space. Namely, if , we have that is such that and . Moreover,

and by Theorem A we have that is always a bounded (and an invertible) operator. Thus has a closed range if and only if has a closed range.

Theorem 3.3. Let be a selfmap of , let , and let be bounded. Then(i) has a closed range if and only if there exists such that the set is sampling for ; (ii)if has a closed range, then there exist with , such that is an -net for ; (iii)if there exist with such that contains an open annulus centered at the origin and with outer radius 1, then has a closed range.

Proof. The proof of (i) is similar to the proof of Theorem  1 from [23]. Since is bounded, by Theorem A, we have that such that . Thus, if is bounded below by , and if , we will show that set is sampling for with a sampling constant . We have that for any , and since it must be that Thus , that is, is sampling for with sampling constant . The other direction of the proof is fairly similar, and we leave it to the reader.
(ii) Let and let be as in the proof of Proposition 3.2, that is, the function defined by and . As shown before, , and Furthermore, assuming that is bounded and has a closed range, there exist such that and But since , there exists such that Thus, for and , we have that for all ; there exists such that , and so is an -net for .
(iii) Let be bounded and assume that contains the annulus . Suppose that does not have a closed range, that is, there exists a sequence of functions with and such that . Since , there exists a sequence in such that for all , For any , let be such that for all , and we have . Then Considering further , we get that each with belongs to the complement of . Thus and with .
On the other hand, since , a normal families argument implies that there exists a subsequence that converges uniformly on compact subsets of to some function . But then converges to uniformly on compact subsets of , and since as and contains an infinite compact subset of , we get that . This contradicts the fact that and so must be bounded below. Hence, has a closed range.

Example 3.4. Let , and let . Then For with , we have that . Thus(i)if , we have that as , and so is compact;(ii)if , we have that as , and so is not bounded;(iii)if , we have that , that is, and so is bounded below and has a closed range. Recall that cannot be an isometry on any of the spaces.
Note that the same conclusions as in Example 3.4 hold if we choose , , or even further, if we choose other particularly nice functions, such as, for example, finite Blaschke products. The sufficient condition, as we will see later, is the boundedness of the derivative over points that are mapped close to the unit circle.
As for the boundedness of the composition operator from into when in general, we mention the following useful condition which might be known, but we could not find it in the literature.

Proposition 3.5. Let , and let be an analytic selfmap of . If the composition operator from into is bounded, then has no angular derivative on , and if , then the (linear) measure of must be zero.

Proof. Recall that if has an angular derivative at a point , then . Thus, if the set is empty, we are done. If not, recall further that if has an angular derivative at , then for any sequence in that converges nontangentially to , we have that Also, if , it must be that . For more details on angular derivatives, see, for example, [8].
Without loss of generality, assume that and suppose that has an angular derivative at some . Hence, whenever in converges nontangentially to , we have that for large enough , as , since and . By Theorem A, cannot be bounded from into .
Furthermore, by a result from [14, Corollary, page 71], if the set has a positive measure in , then must have an angular derivative at some point of , and so can not be bounded from into .

Recall that Theorems B, C, and D consider the closed-range and isometry classifications of composition operators acting between the same space . The following theorem looks at the determination of closed-range and isometric composition operators acting between different Bloch-type spaces, namely, the case when .

Theorem 3.6. Let , and let from into be bounded. Then(i) can not have a closed range, except possibly when , or ;(ii) can not be an isometry, except possibly when .

Proof. (i)
Case 1. Let . We will show that if and , then bounded implies that and so, by Theorem A, is compact. Thus can not have a closed range.
This follows from the more general fact that if is bounded, then for we have that must be compact, since which converges to 0, as .
So, since , we have that if is bounded, then is compact, and since , it must be that as shown, for example, in [14].
Case 2. Let .(a)If , then each is compact and thus can not have a closed range. This follows from the fact that The first inequality is true since and so while in the second inequality one uses the Schwarz-Pick lemma. Again, since , we have that , and by Theorem A, must be compact.(b)When , then again, each is compact and so, can not have a closed range. This follows similarly from the Schwarz-Pick lemma and Theorem A: as .
(ii) Since every isometry has a closed range, by part (i), the leftover case to consider when determining the isometric composition operators from into is when . We will show that in this case also, the composition operator can not be an isometry. (a)Let . Then, if is an isometry, we have that . Thus where and . Therefore, the supremum is 1 only when both parts of the product converge or are equal to 1. This implies that the supremum is attained at and that . But then, by Schwarz's lemma, must be a rotation, and rotations induce an unbounded composition operators whenever . Hence, a composition operator from is never an isometry in the case .(b)If and is an isometry, then, as before, . In the proof of Theorem B in [18], it is shown that this is enough to imply that either is a rotation (but then is not bounded) or that is such that for every in there exists with , , and .
Let be a sequence of points in such that , and let be such that and . Then, as , and so is not bounded.

Note that in the proof of the previous theorem, we have shown that in general, whenever , , and , then is not bounded.

We state one more result on the isometric and closed-range composition operators on Bloch-type spaces, which takes care of the classification in the case when the inducing function satisfies extra conditions. For example, we show that the only univalently induced isometric composition operators are the trivial ones, that is, the ones induced by rotations. This is similar to a result from [25], where we have shown that a univalently induced composition operator on , has a closed range if and only if is a disk automorphism.

Theorem 3.7. Let , and let be bounded. (i)If is univalent and not a rotation, then is not an isometry. (ii)If and if , such that , whenever , then does not have a closed range.

Proof. (i) Let be univalent, not a rotation, and such that is an isometry. By Proposition 3.2, then and . The function is such that and so Since both parts of the last product are less or equal to 1, there is a sequence in such that and . Since is univalent, by the Köebe distortion theorem, it can not be that . Hence, such that , . Again, is univalent, and so . But then and by Schwarz lemma, has to be a rotation, which is a contradiction. Hence, is not an isometry.
(ii) Let and let such that , whenever . If , then is compact and does not have a closed range. So, let be such that .
If , the boundedness of the derivative as implies that must have an angular derivative at , and by Proposition 3.5, is not bounded.
If , then whenever , Thus is compact and so it can not have a closed range.

Note that part (ii) of the previous theorem together with part (iii) of Theorem 3.3 show that, for example, if is a finite Blaschke product, then has a closed range only when .

We finish with the following two open questions.

Question 1. If the composition operator is bounded, is the necessary closed-range condition in part (ii) of Theorem 3.3 also sufficient for any ? By Theorem D, this is the case when .

Question 2. Are there any closed-range composition operators when and are such that either or such that ? By Theorem 3.7 part (ii), if there is such a , then will not be a finite Blaschke product.

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