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

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 𝜙 is inner, then 𝐶_𝜙 is an isometry on 𝐻² if and only if 𝜙(0)=0. Martín and Vukotić have generalized recently in [2] that, indeed, 𝐶_𝜙 is an isometry on 𝐻^𝑝 for all 𝑝≥1 if and only if 𝜙 is inner and 𝜙(0)=0.

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 𝔻 under the inducing function covers a significant (in some sense) part of 𝔻. 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 𝜙 that maps the unit disk 𝔻 into itself, the composition operator 𝐶_𝜙 on the Banach space 𝑋⊆𝐻(𝔻) is defined by 𝐶_𝜙𝑓=𝑓∘𝜙 (2.1)

with 𝑓 in 𝑋, where 𝐻(𝔻) is the space of functions analytic on 𝔻. We will say that 𝜙 is the inducing function for 𝐶_𝜙.

Depending on the space 𝑋, 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 𝛼>0, the 𝛼- Bloch spaces ℬ^𝛼 (also referred to as Bloch-type spaces) are spaces of functions 𝑓 in 𝐻(𝔻) such that 𝑓_ℬ𝛼=sup_𝑧∈𝔻𝑓^′(𝑧)1−|𝑧|^2𝛼<∞. (2.2)

Each ℬ^𝛼 is a Banach space with a norm given by 𝑓_ℬ𝛼=𝑓(0)+𝑓_ℬ𝛼. (2.3)

The family of Bloch-type spaces includes the classical Bloch space ℬ=ℬ¹. The spaces ℬ^𝛼 with 0<𝛼<1 are the analytic Lipschitz spaces Lip_1−𝛼. Thus, for 0<𝛼<1, ℬ^𝛼⊂𝐴(𝔻)⊂𝐻^∞, where 𝐴(𝔻) is the disk algebra. In general, for 0<𝛼<𝛽, we have that ℬ^𝛼⊂ℬ^𝛽 and so the 𝛼-Bloch spaces ℬ^𝛼 form an increasing, uniform family of function spaces. Note also that the Bloch space ℬ contains 𝐻^∞ and is included in all of the Bergman spaces 𝐿^𝑝_𝑎, 𝑝≥1, while for large 𝛼, such as 𝛼≥2, ℬ^𝛼 includes the Bergman space 𝐿²_𝑎.

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 [11–15]. We state the most general form of these results and use the following notation: for 𝛼>0,𝛽>0 and 𝜙 being an analytic self map of 𝔻, let𝜏_{𝜙,𝛼,𝛽}(𝑧)=1−|𝑧|^2𝛽𝜙′(𝑧)1−𝜙(𝑧)^2𝛼. (2.4)

We write 𝜏_𝜙,𝛼 whenever 𝛼=𝛽, and 𝜏_𝜙 if 𝛼=𝛽=1.

Theorem A (see [12, 15]). For 𝛼,𝛽>0 and 𝜙 an analytic selfmap of 𝔻, the composition operator 𝐶_𝜙 is a bounded operator from ℬ^𝛼 into ℬ^𝛽 if and only if sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)<∞, (2.5) and 𝐶_𝜙 is a compact operator from ℬ^𝛼 into ℬ^𝛽 if and only if lim_{𝜙(𝑧)→1}𝜏_{𝜙,𝛼,𝛽}(𝑧)=0. (2.6)

Here are a few simple consequences of Theorem A and of some basic complex analysis: recall the Schwarz-Pick lemma which states that for 𝜙 a self-map of the unit disk 1−|z|²𝜙′(z)1−𝜙(z)²≤1. (2.7)

Thus, when 𝛼=𝛽=1, 𝜏_𝜙(z)≤1, and we get from Theorem A that every composition operator is bounded on ℬ. Moreover, the Schwarz-Pick lemma and Theorem A imply that if 𝛼≤1≤𝛽, then C_𝜙 maps ℬ^𝛼 boundedly into ℬ^𝛽, since then 𝜏_{𝜙,𝛼,𝛽}(z)≤(1−|𝜙(z)|²)^1−𝛼. If furthermore 𝛼<1≤𝛽, then C_𝜙 from ℬ^𝛼 into ℬ^𝛽 must also be compact. Note also that if ‖𝜙‖_∞<1 and C_𝜙∶ℬ^𝛼→ℬ^𝛽 is bounded, then C_𝜙 is compact.

All of the spaces ℬ^𝛼 include the identity function, and so a necessary condition for C_𝜙 to be bounded from ℬ^𝛼 into B^𝛽 is that 𝜙 belongs to ℬ^𝛽. Thus, for 0<𝛽<1, every analytic self-map of 𝔻 that is in H^∞⧵ℬ^𝛽 induces an unbounded composition operator from ℬ^𝛼 to ℬ^𝛽.

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 C_𝜙∶ℬ^𝛼→ℬ^𝛽 is isometric whenever C_𝜙f_ℬ𝛽=‖f‖_ℬ^𝛼,∀f∈ℬ^𝛼. (2.8)

Since every nonconstant 𝜙 is an open map, the composition operator C_𝜙 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, C_𝜙 has a closed range if and only if it is bounded below, namely, if and only if there exists M>0 such that C_𝜙f_ℬ𝛽≥M‖f‖_ℬ^𝛼,∀f∈ℬ^𝛼. (2.9)

In particular, every isometric C_𝜙 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 C_𝜙 never has a finite rank (because 𝜙 is an open map), a compact C_𝜙 can never have a closed range. Hence, a compact C_𝜙 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 [16–19]. We present the classification in the form that appears in [18].

Theorem B (see [18]). Let 𝜙 be an analytic self-map of 𝔻. The composition operator 𝐶_𝜙 is an isometry on the Bloch space ℬ if and only if 𝜙(0)=0 and either 𝜙 is a rotation, or for every 𝑎 in 𝔻 there exists a sequence {𝑧_𝑛} in 𝔻 such that |𝑧_𝑛|→1, 𝜙(𝑧_𝑛)=𝑎, and 𝜏_𝜙(𝑧_𝑛)→1.

The result provides a large class of functions inducing isometric composition operators on the Bloch space. For example, if 𝜙 is an almost thin infinite Blaschke product fixing the origin, that is a Blaschke product with a sequence of zeroes {𝑧_𝑛} that includes 0 and is such that limsup_𝑛→∞|𝐵′(𝑧_𝑛)|(1−|𝑧_𝑛|²)=1, then 𝐶_𝜙 is an isometry on ℬ. 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 0<𝛼, 𝛼≠1, and let 𝜙 be an analytic self-map of 𝔻. Then the composition operator 𝐶_𝜙 is an isometry on ℬ^𝛼 if and only if 𝜙 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 𝐶_𝜙 is an isometry on ℬ^𝛼, then 𝔻=𝜙(𝔻), which further implies that 𝜕𝔻⊆𝜙(𝔻). Moreover, in the case 𝛼≠1, 𝜙 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 𝜙(𝔻) covers a significant (in some sense) part of 𝔻, or even further, that 𝜙 has to be a rotation. For example, if 𝐶_𝜙 is an isometry either on the 𝐻^𝑝 spaces or on the weighted Bergman spaces 𝐴^𝑝_𝑤 with 1≤𝑝<∞, then 𝜙 has to be either an inner function or a rotation, respectively, as shown in [2]. If 𝐶_𝜙 is an isometry on the Dirichlet space, then 𝜙 has to be an univalent full map, that is, one to one and such that the area measure of 𝔻⧵𝜙(𝔻) is zero, (see [3]). The complete determination of isometric composition operators in all of this cases is given by adding the condition 𝜙(0)=0.
A specific isometric requirement in the case of the Bloch space is that when 𝜙 is not univalent, it has to be of infinite multiplicity and such that the function 𝜏_𝜙 stays close to 1 over some of the preimages of each point in 𝜙(𝔻). When 𝜙 is univalent, 𝜙 has to be a rotation, and thus 𝜏_𝜙(𝑧)=1, for all 𝑧∈𝔻.
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 𝜙(𝔻) and 𝜏_𝜙,𝛼 play a role in determining the closed range composition operators on ℬ^𝛼.
The following Theorem D below classifies the closed range composition operators on the Bloch spaces ℬ^𝛼,𝛼≥1. The results are contained in a series of papers [21–23], and we present their combined version. We need a few more definitions before we can state the theorem.
We say that 𝐺⊆𝔻 is sampling for ℬ^𝛼 if ∃𝑆>0 such that for all 𝑓∈ℬ^𝛼, sup_𝑧∈𝐺𝑓′(𝑧)1−|𝑧|^2𝛼≥𝑆𝑓_ℬ𝛼. (2.10)
Let 𝜌(𝑧,𝑤)=|𝜓_𝑧(𝑤)| denote the pseudohyperbolic distance on 𝔻, where 𝜓_𝑧 is a disc automorphism of 𝔻, that is, 𝜓_𝑧(𝑤)=𝑧−𝑤1−.𝑧𝑤 (2.11) We say that 𝐺⊆𝔻 is an 𝑟−𝑛𝑒𝑡 for 𝔻 for some 𝑟∈(0,1) if for all 𝑧∈𝔻, ∃𝑤∈𝐺 such that 𝜌(𝑧,𝑤)<𝑟.
For 𝑐>0, let Ω_{𝑐,𝛼,𝛽}=𝑧∈𝔻∶𝜏_{𝜙,𝛼,𝛽}(𝑧)≥𝑐, (2.12) and let 𝐺_{𝑐,𝛼,𝛽}=𝜙(Ω_{𝑐,𝛼,𝛽}). If 𝛼=𝛽, we use the notation Ω_𝑐,𝛼 and 𝐺_𝑐,𝛼, and if further 𝛼=𝛽=1, we write Ω_𝑐 and 𝐺_𝑐.

Theorem D (see [21, 23, 24]). Let 𝜙 be a self-map of 𝔻 and let 𝛼≥1. Then, the following are equivalent. (i)𝐶_𝜙 has a closed range on ℬ^𝛼.(ii)There exists 𝑐>0 such that the set 𝐺_𝑐,𝛼 is sampling for ℬ^𝛼.(iii)There exist 𝑐,𝑟>0 with 𝑟<1 such that 𝐺_𝑐,𝛼 is an 𝑟-net for 𝔻.
Moreover, if 𝛼=1, then (i) to (iii) are also equivalent to the following.(iv) There exists 𝑘>0 such that ‖𝐶_𝜙𝜓_𝑎‖_ℬ≥𝑘‖𝜓_𝑎‖_ℬ, for all 𝑎∈𝔻.

Note, for example, that when 𝛼=1 and 𝜙 is a rotation, 𝜏_𝜙(𝑧)=1, for all 𝑧∈𝔻, and so 𝐺₁=𝔻. Thus, 𝐺₁ is trivially sampling for 𝐵 and an 𝑟-net for 𝔻 for any 𝑟>0. Therefore, (ii) and (iii) are true. Part (iv) of the theorem also holds true if one takes 𝑘=1.

If 𝜙 is not a rotation and 𝐶_𝜙 is an isometry on ℬ, then whenever 𝑐=1−𝜀 with 0<𝜀<1, we have from Theorem B that 𝐺_1−𝜀=𝔻. Hence, as before, 𝐺_1−𝜀 is trivially sampling for ℬ and an 𝑟-net for 𝔻, for any 𝑟>0. Again, part (iv) of the theorem is true with 𝑘=1. 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 𝜙 has a 𝜏_𝜙,𝛼 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 𝜏_𝜙 function are particularly interesting in the case when 𝜙 is univalent, since then, as a consequence of the Köebe one-quarter theorem, we have that 𝜏_𝜙(𝑧)≈dist𝜙(𝑧),𝜕𝜙(𝔻)𝜙(𝑧.1−) (2.13)

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

For example, one gets that if 𝜙 is the Riemann mapping from 𝔻 onto the slit disk 𝔻⧵[0,1], then 𝐶_𝜙 has a closed range on ℬ. Or if 𝜙 is the Riemann mapping onto the simply connected region in 𝔻 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 𝐶_𝜙 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 𝐶_𝜙 has a closed range on 𝐵^𝛼 for some 𝛼>1, then 𝐶_𝜙 has a closed range on ℬ^𝛼 for all 𝛼≥1. On the other hand, the slit disk example from above provides an example of a univalent map 𝜙 such that 𝐶_𝜙 has a closed range on ℬ^𝛼 only for 𝛼=1 (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 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽. As Theorem A from the previous section states, not every such composition operator is bounded. Thus, the boundedness condition on 𝜏_{𝜙,𝛼,𝛽} plays a natural role in the characterization of the isometric and closed-range composition operators from ℬ^𝛼 into ℬ^𝛽.

In general, depending on the choices of 𝛼 and 𝛽, 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 𝜙(𝑧)=𝜆𝑧, with |𝜆|=1. Then 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 is (a)an isometry, whenever 𝛼=𝛽, (b)a compact operator if 𝛼<𝛽, (c)an unbounded operator if 𝛼>𝛽. Note that (a) holds since |𝜙′(𝑧)|=|𝜆|=1, and we have that 𝐶_𝜙𝑓_ℬ𝛼=𝑓∘𝜙_ℬ𝛼=𝑓(0)+sup_𝑧∈𝔻𝑓^′(𝜆𝑧)1−|𝜆𝑧|^2𝛼=𝑓_ℬ𝛼. (3.1) The other two cases follow easily from Theorem A, since 𝜏_{𝜙,𝛼,𝛽}=1−|𝑧|^2𝛽𝜙^′(𝑧)1−𝜙(𝑧)^2𝛼=1−|𝑧|^2𝛽−𝛼. (3.2)
As for many other spaces, a condition that 𝜙 must satisfy whenever 𝐶_𝜙 is an isometry from ℬ^𝛼 into ℬ^𝛽, regardless of the choices of 𝛼 and 𝛽, is that 𝜙(0)=0.

Proposition 3.2. If 𝐶_𝜙 is an isometry from ℬ^𝛼 into ℬ^𝛽, then 𝜙(0)=0 and sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)=1. (3.3)

Proof. Note first that the identity function 𝑒(𝑧)=𝑧 belongs to each of the Bloch-type spaces and has norm one. Thus, since 𝐶_𝜙 is an isometry, it must be that ‖𝜙‖_ℬ^𝛽=‖𝐶_𝜙𝑒‖_ℬ^𝛽=‖𝑒‖_ℬ^𝛼=1.
Let 𝜙(0)=𝑎. Using the function 𝑓_𝑎(𝑧)=1−𝑎𝑧, we see that 𝐶_𝜙𝑓_𝑎ℬ𝛽=𝑓_𝑎∘𝜙_ℬ𝛽=𝑓_𝑎(𝑎)+sup_𝑧∈𝔻𝑓^′_𝑎𝜙(𝑧)𝜙^′(𝑧)1−|𝑧|^2𝛽=1−|𝑎|²+|𝑎|sup_𝑧∈𝔻𝜙′(𝑧)1−|𝑧|^2𝛽=1−|𝑎|²+|𝑎|𝑎|𝑎||𝑎||(1−|)=1+−2². (3.4) But since 𝐶_𝜙 is an isometry and ‖𝑓_𝑎‖_ℬ^𝛼=1+|𝑎|sup_𝑧∈𝔻(1−|𝑧|²)^𝛼=1+|𝑎|, it must be that |𝑎|²=0. Thus 𝜙(0)=𝑎=0.
To show that sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)=1, we use another type of test functions. For 𝑎∈𝔻, let 𝜑_𝑎(𝑧) be the function such that 𝜑_𝑎(0)=0 and 𝜑^′_𝑎(𝑧)=(𝜓^′_𝑎(𝑧))^𝛼, where 𝜓_𝑎 is the disc automorphism of 𝔻 defined by 𝜓_𝑎(𝑧)=𝑎−𝑧1−.𝑎𝑧 (3.5) Using that (1−|𝑧|²)|𝜓^′_𝑤(𝑧)|=1−|𝜓_𝑤(𝑧)|², we get 𝜑_𝑎ℬ𝛼=sup_𝑧∈𝔻𝜓^′_𝑎(𝑧)^𝛼1−|𝑧|^2𝛼=sup_𝑧∈𝔻1−𝜓_𝑎(𝑧)^2𝛼=1. (3.6)Hence, since 𝐶_𝜙 is an isometry from ℬ^𝛼 into ℬ^𝛽, we have that for any 𝑧₀ in 𝔻1=sup_𝑎∈𝔻𝐶_𝜙𝜑_𝑎ℬ𝛽=sup_𝑎∈𝔻sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜓_𝑎𝜙(𝑧)^2𝛼≥sup_𝑎∈𝔻𝜏_{𝜙,𝛼,𝛽}𝑧₀1−𝜓_𝑎𝜙𝑧₀^2𝛼≥𝜏_{𝜙,𝛼,𝛽}𝑧₀, (3.7) where the last inequality follows by choosing 𝑎=𝜙(𝑧₀), and the fact that 𝜓_𝑎(𝑎)=0. Thus 𝜏_{𝜙,𝛼,𝛽}(𝑧)≤1, for all 𝑧∈𝔻. On the other hand, for any 𝑎∈𝔻1=𝐶_𝜙𝜑_𝑎ℬ𝛽=sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜓_𝑎(𝑧)^2𝛼≤sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧), (3.8) and thus, sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)=1.

Note that in the more general case when 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 has a closed range, 𝜙(0) does not necessarily have to be zero. Still, without loss of generality, we will consider only the case 𝜙(0)=0. This is possible since the disk automorphisms induce invertible (and thus closed range) composition operators on every Bloch-type space. Namely, if 𝜙(0)=𝑎, we have that 𝜙_𝑎(𝑧)=𝜓_𝑎∘𝜙(𝑧) is such that 𝜙_𝑎(0)=0 and 𝐶_𝜙𝑎=𝐶_𝜙𝐶_𝜓𝑎. Moreover, 𝜏_{𝜓𝑎,𝛼}(𝑧)=𝜓^′_𝑎(𝑧)^1−𝛼≤1+|𝑎|1−|𝑎|^|1−𝛼|, (3.9)

and by Theorem A we have that 𝐶_𝜓𝑎∶ℬ^𝛼→ℬ^𝛼 is always a bounded (and an invertible) operator. Thus 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 has a closed range if and only if 𝐶_𝜙𝑎∶ℬ^𝛼→ℬ^𝛽 has a closed range.

Theorem 3.3. Let 𝜙 be a selfmap of 𝔻, let 𝛼,𝛽>0, and let 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 be bounded. Then(i)𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 has a closed range if and only if there exists 𝑐>0 such that the set 𝐺_{𝑐,𝛼,𝛽} is sampling for ℬ^𝛼; (ii)if 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 has a closed range, then there exist c,𝑟>0 with 𝑟<1, such that 𝐺_{𝑐,𝛼,𝛽} is an 𝑟-net for 𝔻; (iii)if there exist 𝑐,𝑟>0 with 𝑟<1 such that 𝐺_{𝑐,𝛼,𝛽} contains an open annulus centered at the origin and with outer radius 1, then 𝐶_𝜙 has a closed range.

Proof. The proof of (i) is similar to the proof of Theorem  1 from [23]. Since 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 is bounded, by Theorem A, we have that ∃𝐾>0 such that 𝜏_{𝜙,𝛼,𝛽}(𝑧)≤𝐾. Thus, if 𝐶_𝜙 is bounded below by 𝑀, and if 𝑐=𝑀/2, we will show that set 𝐺_{𝑐,𝛼,𝛽} is sampling for ℬ^𝛼 with a sampling constant 𝑆=𝑀/𝐾. We have that for any 𝑓∈ℬ^𝛼, 𝐶_𝜙𝑓_ℬ𝛽=sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜙(𝑧)^2𝛼𝑓′𝜙(𝑧)≥𝑀𝑓_ℬ𝛼, (3.10) and since sup_{𝑧∉Ω𝑐,𝛼,𝛽}𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜙(𝑧)^2𝛼𝑓′𝜙(𝑧)<𝑐𝑓_ℬ𝛼=𝑀2𝑓_ℬ𝛼, (3.11) it must be that 𝑀𝑓_ℬ𝛼≤sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜙(𝑧)^2𝛼𝑓′𝜙(𝑧)=sup_{𝑧∈Ω𝑐,𝛼,𝛽}𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜙(𝑧)^2𝛼𝑓′𝜙(𝑧)≤𝐾sup_{𝑤∈𝐺𝑐,𝛼,𝛽}1−|𝑤|^2𝛼𝑓′(𝑤). (3.12)Thus sup_{𝑤∈𝐺𝑐,𝛼,𝛽}(1−|𝑤|²)^𝛼|𝑓′(𝑤)|≥𝑀/𝐾‖𝑓‖_ℬ^𝛼, that is, 𝐺_{𝑐,𝛼,𝛽} is sampling for ℬ^𝛼 with sampling constant 𝑆=𝑀/𝐾. The other direction of the proof is fairly similar, and we leave it to the reader.
(ii) Let 𝑤∈𝔻 and let 𝜑_𝑤 be as in the proof of Proposition 3.2, that is, the function defined by 𝜑_𝑤(0)=0 and 𝜑^′_𝑤=(𝜓^′_𝑤)^𝛼. As shown before, ‖𝜑_𝑤‖_ℬ^𝛼=1, and 𝜑_𝑤∘𝜙_ℬ𝛽=sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜓_𝑤𝜙(𝑧)^2𝛼. (3.13) Furthermore, assuming that 𝐶_𝜙 is bounded and has a closed range, there exist 𝑀,𝐾>0 such that sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)=𝐾 and 𝐾≥sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜓_𝑤𝜙(𝑧)^2𝛼≥𝑀. (3.14) But since 1−|𝜓_𝑤(𝜙(𝑧))|²≤1, there exists 𝑧_𝑤∈𝔻 such that 𝜏_{𝜙,𝛼,𝛽}𝑧_𝑤≥𝑀2,1−𝜓_𝑤𝜙𝑧_𝑤^2𝛼≥𝑀.2𝐾 (3.15) Thus, for 𝑐=𝑀/2 and 𝑟=1−(𝑀/2𝐾)^1/𝛼, we have that for all 𝑤∈𝔻; there exists 𝑧_𝑤∈Ω_{𝑐,𝛼,𝛽} such that 𝜌(𝑤,𝜙(𝑧_𝑤))<𝑟, and so 𝐺_{𝑐,𝛼,𝛽} is an 𝑟-net for 𝔻.
(iii) Let 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 be bounded and assume that 𝐺_{𝑐,𝛼,𝛽} contains the annulus 𝐴={𝑧∶𝑟₀<|𝑧|<1}. Suppose that 𝐶_𝜙 does not have a closed range, that is, there exists a sequence of functions {𝑓_𝑛} with ‖𝑓_𝑛‖_ℬ^𝛼=1 and such that ‖𝐶_𝜙𝑓_𝑛‖_ℬ^𝛽→0. Since sup_𝑧∈𝔻(1−|𝑧|²)^𝛼|𝑓^′_𝑛(𝑧)|=1, there exists a sequence {𝑎_𝑛} in 𝔻 such that for all 𝑛, 1−𝑎_𝑛^2𝛼𝑓^′_𝑛𝑎_𝑛≥12. (3.16) For any 𝜀>0, let 𝑁_𝜀 be such that for all 𝑛>𝑁_𝜀, and we have ‖𝐶_𝜙𝑓_𝑛‖_ℬ^𝛽<𝑐𝜀. Then sup_{𝑤∈𝐺𝑐,𝛼,𝛽}1−|𝑤|^2𝛼𝑓^′_𝑛(𝑤≤1)𝑐sup_{𝑧∈Ω𝑐,𝛼,𝛽}𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜙(𝑧)^2𝛼𝑓^′_𝑛𝜙(𝑧)≤1𝑐sup_𝑧∈𝔻1−|𝑧|^2𝛽𝑓^′_𝑛𝜙(𝑧)𝜙^′(𝑧=1)𝑐𝐶_𝜙𝑓_𝑛ℬ𝛽<𝑐𝜀𝑐=𝜀. (3.17) Considering further 𝜀<1/2, we get that each 𝑎_𝑛 with 𝑛>𝑁_𝜀 belongs to the complement of 𝐺_{𝑐,𝛼,𝛽}. Thus |𝑎_𝑛|≤𝑟₀<1 and 𝑎_𝑛→𝑎 with |𝑎|≤𝑟₀.
On the other hand, since ‖𝑓_𝑛‖_ℬ^𝛼=1, a normal families argument implies that there exists a subsequence {𝑓_𝑛𝑚} that converges uniformly on compact subsets of 𝔻 to some function 𝑓∈ℬ^𝛼. But then {𝑓_𝑛𝑚′} converges to 𝑓′ uniformly on compact subsets of 𝔻, and since sup_{𝑤∈𝐺𝑐,𝛼,𝛽}(1−|𝑤|²)^𝛼|𝑓^′_𝑛(𝑤)|→0 as 𝑛→∞ and 𝐺_{𝑐,𝛼,𝛽} contains an infinite compact subset of 𝔻, we get that 𝑓′≡0. This contradicts the fact that (1−|𝑎|²)^𝛼|𝑓′(𝑎)|≥1/2 and so 𝐶_𝜙 must be bounded below. Hence, 𝐶_𝜙 has a closed range.

Example 3.4. Let 𝜙(𝑧)=𝑧², and let 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽. Then 𝜏_{𝜙,𝛼,𝛽}(𝑧)=2|𝑧|1−|𝑧|^2𝛽1−|𝑧|^4𝛼=1−|𝑧|^2𝛽−𝛼2|𝑧|1+|𝑧|^2𝛼. (3.18) For 𝑧∈𝔻 with |𝑧|>1/2, we have that (1/2^𝛼)(1−|𝑧|²)^𝛽−𝛼<𝜏_{𝜙,𝛼,𝛽}(𝑧)≤2(1−|𝑧|²)^𝛽−𝛼. Thus(i)if 𝛼<𝛽, we have that 𝜏_{𝜙,𝛼,𝛽}(𝑧)→0 as |𝑧|→1, and so 𝐶_𝜙 is compact;(ii)if 𝛼>𝛽, we have that 𝜏_{𝜙,𝛼,𝛽}(𝑧)→∞ as |𝑧|→1, and so 𝐶_𝜙 is not bounded;(iii)if 𝛼=𝛽, we have that (1/2^𝛼)<𝜏_𝜙,𝛼(𝑧)≤2, that is, 𝐴={𝑧∶(1/2)<|𝑧|<1}⊂𝐺_1/2^𝛼_,𝛼 and so 𝐶_𝜙 is bounded below and has a closed range. Recall that 𝐶_𝜙 cannot be an isometry on any of the ℬ^𝛼 spaces.
Note that the same conclusions as in Example 3.4 hold if we choose 𝜙(𝑧)=𝑧^𝑘, 𝑘∈ℕ, 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 ℬ^𝛼 into ℬ^𝛽 when 𝛼>𝛽 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 0<𝛽<𝛼, and let 𝜙 be an analytic selfmap of 𝔻. If the composition operator 𝐶_𝜙 from ℬ^𝛼 into ℬ^𝛽 is bounded, then 𝜙 has no angular derivative on 𝜕𝔻, and if 𝐸={𝜁∈𝜕𝔻∶|𝜙(𝜁)|=1}, then the (linear) measure of 𝐸 must be zero.

Proof. Recall that if 𝜙 has an angular derivative at a point 𝜁∈𝜕𝔻, then |𝜙(𝜁)|=1. Thus, if the set 𝐸 is empty, we are done. If not, recall further that if 𝜙 has an angular derivative 𝜙′(𝜁) at 𝜁, then for any sequence {𝑧_𝑛} in 𝔻 that converges nontangentially to 𝜁, we have that lim_𝑛→∞1−𝜙𝑧_𝑛1−𝑧_𝑛=𝜙^′(𝜁)=lim_𝑛→∞𝜙^′𝑧_𝑛. (3.19) Also, if 𝜙(0)=0, it must be that |𝜙′(𝜁)|≥1. For more details on angular derivatives, see, for example, [8].
Without loss of generality, assume that 𝜙(0)=0 and suppose that 𝜙 has an angular derivative at some 𝜁∈𝜕𝔻. Hence, whenever {𝑧_𝑛} in 𝔻 converges nontangentially to 𝜁, we have that for large enough 𝑛, 𝜏_{𝜙,𝛼,𝛽}𝑧_𝑛=1−𝑧_𝑛^2𝛽𝜙^′𝑧_𝑛1−𝜙𝑧_𝑛^2𝛼≥121−𝑧_𝑛²1−𝜙𝑧_𝑛^2𝛽11−𝜙𝑧_𝑛^2𝛼−𝛽→∞ (3.20) as 𝑛→∞, since (1−|𝑧_𝑛|²)/(1−|𝜙(𝑧_𝑛)|²)→|𝜙′(𝜁)|^−𝛽 and |𝜙(𝑧_𝑛)|→1. By Theorem A, 𝐶_𝜙 cannot be bounded from ℬ^𝛼 into ℬ^𝛽.
Furthermore, by a result from [14, Corollary, page 71], if the set 𝐸 has a positive measure in 𝜕𝔻, then 𝜙 must have an angular derivative at some point of 𝜕𝔻, and so 𝐶_𝜙 can not be bounded from ℬ^𝛼 into ℬ^𝛽.

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

Theorem 3.6. Let 𝛼,𝛽>0,𝛼≠𝛽, and let 𝐶_𝜙 from ℬ^𝛼 into ℬ^𝛽 be bounded. Then(i)𝐶_𝜙 can not have a closed range, except possibly when 0<𝛼<𝛽<1, or 1≤𝛽<𝛼;(ii)𝐶_𝜙 can not be an isometry, except possibly when 0<𝛼<𝛽<1.

Proof. (i)
Case 1. Let 𝛼>𝛽. We will show that if 𝛼>𝛽 and 𝛽<1, then 𝐶_𝜙 bounded implies that ‖𝜙‖_∞<1 and so, by Theorem A, 𝐶_𝜙 is compact. Thus 𝐶_𝜙 can not have a closed range.
This follows from the more general fact that if 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 is bounded, then for 0<𝜀<𝛼 we have that 𝐶_𝜙∶ℬ^𝛼−𝜀→ℬ^𝛽 must be compact, since 1−|𝑧|^2𝛽𝜙^′(𝑧)1−𝜙(𝑧)^2𝛼−𝜀≤1−𝜙(𝑧)^2𝜀sup_𝑧∈𝔻1−|𝑧|^2𝛽𝜙^′(𝑧)1−𝜙(𝑧)^2𝛼, (3.21) which converges to 0, as |𝜙(𝑧)|→1.
So, since 0<𝜀=𝛼−𝛽<𝛼, we have that if 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 is bounded, then 𝐶_𝜙∶ℬ^𝛽→ℬ^𝛽 is compact, and since 𝛽<1, it must be that ‖𝜙‖_∞<1 as shown, for example, in [14].Case 2. Let 𝛼<𝛽.(a)If 𝛼≥1, then each 𝐶_𝜙 is compact and thus can not have a closed range. This follows from the fact that 𝜏_{𝜙,𝛼,𝛽}(𝑧)=1−|𝑧|^2𝛽𝜙^′(𝑧)1−𝜙(𝑧)^2𝛼≤1−|𝑧|²𝜙^′(𝑧)1−𝜙(𝑧)²1−|𝑧|^2𝛽−𝛼≤1−|𝑧|^2𝛽−𝛼. (3.22) The first inequality is true since 𝜙(0)=0 and so 11−𝜙(𝑧)^2𝛼−1≤11−|𝑧|^2𝛼−1, (3.23) while in the second inequality one uses the Schwarz-Pick lemma. Again, since 𝜙(0)=0, we have that 1−|𝑧|²≤1−|𝜙(𝑧)|², and by Theorem A, 𝐶_𝜙 must be compact.(b)When 𝛼<1≤𝛽, then again, each 𝐶_𝜙 is compact and so, 𝐶_𝜙 can not have a closed range. This follows similarly from the Schwarz-Pick lemma and Theorem A: 1−|𝑧|^2𝛽𝜙^′(𝑧)1−𝜙(𝑧)^2𝛼=1−|𝑧|²𝜙^′(𝑧)1−𝜙(𝑧)^2𝛼1−|𝑧|^21−𝛼𝜙^′(𝑧)^1−𝛼1−|𝑧|^2𝛽−1≤1−𝜙(𝑧)^21−𝛼1−|𝑧|^2𝛽−1→0 (3.24) as |𝜙(𝑧)|→1.
(ii) Since every isometry has a closed range, by part (i), the leftover case to consider when determining the isometric composition operators 𝐶_𝜙 from ℬ^𝛼 into ℬ^𝛽 is when 1≤𝛽<𝛼. We will show that in this case also, the composition operator can not be an isometry. (a)Let 1<𝛽<𝛼. Then, if 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 is an isometry, we have that ‖𝜙‖_ℬ^𝛽=1. Thus sup_𝑧∈𝔻1−|𝑧|^2𝛽−11−|𝑧|²𝜙^′(𝑧)=1, (3.25) where (1−|𝑧|²)^𝛽−1≤1 and (1−|𝑧|²)|𝜙′(𝑧)|≤1−|𝜙(𝑧)|²≤1. 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 𝑧=0 and that |𝜙′(0)|=1. But then, by Schwarz's lemma, 𝜙 must be a rotation, and rotations induce an unbounded composition operators whenever 𝛽<𝛼. Hence, a composition operator from ℬ^𝛼→ℬ^𝛽 is never an isometry in the case 1<𝛽<𝛼.(b)If 1=𝛽<𝛼 and 𝐶_𝜙∶ℬ^𝛼→ℬ is an isometry, then, as before, ‖𝜙‖_ℬ=1. In the proof of Theorem B in [18], it is shown that this is enough to imply that either 𝜙 is a rotation (but then 𝐶_𝜙 is not bounded) or that 𝜙 is such that for every 𝑎 in 𝔻 there exists {𝑧_𝑛} with 𝑧_𝑛→1, 𝜙(𝑧_𝑛)=𝑎, and 𝜏_𝜙(𝑧_𝑛)→1.
Let {𝑎_𝑘} be a sequence of points in 𝔻 such that |𝑎_𝑘|→1, and let {𝑧_𝑘} be such that 𝜙(𝑧_𝑘)=𝑎_𝑘 and 𝜏_𝜙(𝑧_𝑘)≥1/2. Then, as 𝑎_𝑘→1, 𝜏_𝜙,𝛼,1𝑧_𝑘=1−𝑧_𝑘²𝜙^′𝑧_𝑘1−𝜙𝑧_𝑘^2𝛼=𝜏_𝜙𝑧_𝑘1−𝑎_𝑘^2𝛼−1≥121−𝑎_𝑘^2𝛼−1→∞, (3.26) and so 𝐶_𝜙∶ℬ^𝛼→ℬ is not bounded.

Note that in the proof of the previous theorem, we have shown that in general, whenever 𝜙(0)=0, ‖𝜙‖_ℬ^𝛽=1, and 1≤𝛽<𝛼, then 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 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 𝜙 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 ℬ^𝛼,𝛼≠1, has a closed range if and only if 𝜙 is a disk automorphism.

Theorem 3.7. Let 𝛼,𝛽>0, and let 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 be bounded. (i)If 𝜙 is univalent and not a rotation, then 𝐶_𝜙 is not an isometry. (ii)If 𝛼≠𝛽 and if ∃𝑀,𝑟>0, 𝑟<1 such that |𝜙′(𝑧)|≤𝑀, whenever |𝜙(𝑧)|>𝑟, then 𝐶_𝜙 does not have a closed range.

Proof. (i) Let 𝜙 be univalent, not a rotation, and such that 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 is an isometry. By Proposition 3.2, then 𝜙(0)=0 and sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)=1. The function 𝑓(𝑧)=𝑧 is such that ‖𝑓‖_ℬ^𝛼=1 and so 1=𝐶_𝜙𝑓_ℬ𝛽=sup_𝑧∈𝔻𝜏_{𝜙,𝛼,𝛽}(𝑧)1−𝜙(𝑧)^2𝛼. (3.27) Since both parts of the last product are less or equal to 1, there is a sequence {𝑧_𝑛} in 𝔻 such that 𝜏_{𝜙,𝛼,𝛽}(𝑧_𝑛)→1 and |𝜙(𝑧_𝑛)|→0. Since 𝜙 is univalent, by the Köebe distortion theorem, it can not be that |𝑧_𝑛|→1. Hence, ∃𝑧₀∈𝔻 such that 𝑧_𝑛→𝑧₀, 𝜙(𝑧_𝑛)→𝜙(𝑧₀)=0. Again, 𝜙 is univalent, and so 𝑧₀=0. But then 𝜏_{𝜙,𝛼,𝛽}(0)=|𝜙′(0)|=1 and by Schwarz lemma, 𝜙 has to be a rotation, which is a contradiction. Hence, 𝐶_𝜙 is not an isometry.
(ii) Let 𝛼≠𝛽 and let ∃𝑀,𝑟>0,𝑟<1 such that |𝜙′(𝑧)|≤𝑀, whenever |𝜙(𝑧)|>𝑟. If ‖𝜙‖_∞<1, then 𝐶_𝜙 is compact and does not have a closed range. So, let 𝜁∈𝜕𝔻 be such that |𝜙(𝜁)|=1.
If 𝛽<𝛼, the boundedness of the derivative as |𝜙(𝑧)|→1 implies that 𝜙 must have an angular derivative at 𝜁, and by Proposition 3.5, 𝐶_𝜙 is not bounded.
If 𝛼<𝛽, then whenever 𝜙(𝑧_𝑛)→1, 𝜏_{𝜙,𝛼,𝛽}𝑧_𝑛=1−𝑧_𝑛^2𝛽𝜙′𝑧_𝑛1−𝜙𝑧_𝑛^2𝛼≤𝑀1−𝑧_𝑛^2𝛽−𝛼→0. (3.28) Thus 𝐶_𝜙 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 𝜙 is a finite Blaschke product, then 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 has a closed range only when 𝛼=𝛽.

We finish with the following two open questions.

Question 1. If the composition operator 𝐶_𝜙∶ℬ^𝛼→ℬ^𝛽 is bounded, is the necessary closed-range condition in part (ii) of Theorem 3.3 also sufficient for any 𝛼,𝛽>0? By Theorem D, this is the case when 𝛼=𝛽≥1.

Question 2. Are there any closed-range composition operators 𝐶_𝜙∶𝐵^𝛼→𝐵^𝛽 when 𝛼 and 𝛽 are such that either 0<𝛼<𝛽<1 or such that 1≤𝛽<𝛼? By Theorem 3.7 part (ii), if there is such a 𝐶_𝜙, then 𝜙 will not be a finite Blaschke product.