Isometric and Closed Range Composition Operators between Bloch Type Spaces

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Volume 2011, Article ID 132541,15pages doi:10.1155/2011/132541

Research Article

Isometric and Closed-Range Composition

Operators between Bloch-Type Spaces

Nina Zorboska

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

Correspondence should be addressed to Nina Zorboska,[email protected]

Received 16 November 2010; Accepted 30 March 2011

Academic Editor: Asao Arai

Copyrightq2011 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.

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, thenCφ is an isometry onH2 if and only ifφ0 0. Mart´ın and Vukoti´c have generalized

recently in2that, indeed,Cφ is an isometry onHpfor allp≥1 if and only ifφis inner and

φ0 0.

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on some of the other Besov spacessee2,3. The isometric composition operators on the BMOA space have been determined by Laitilasee4.

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 spacessee5,6, 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 significantlyagain, in some sensefrom the unit circle.

2. Definitions, Few Basic Notions, and Overview of

the Existing Results

For a nonconstant analytic function φthat maps the unit disk into itself, the composition

operatorCφon the Banach spaceX ⊆H is defined by

Cφf f◦φ 2.1

withf inX, whereH is the space of functions analytic on . We will say that φis the

inducing function forCφ.

Depending on the spaceX, 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.

Forα > 0, theα- Bloch spacesBα also referred to as Bloch-type spacesare spaces of functionsfinH such that

_f

Bα sup

z∈

_f_z₁_{− |z|}2α_<_∞.

2.2

EachBαis a Banach space with a norm given by

_f_B_α _f0f_Bα. 2.3

The family of Bloch-type spaces includes the classical Bloch spaceBB1_{. The spaces}_Bα_with

0 < α < 1 are the analytic Lipschitz spaces Lip₁₋_α. Thus, for 0 < α < 1,Bα ⊂ A ⊂ H∞, whereA is the disk algebra. In general, for 0 < α < β, we have thatBα ⊂ Bβand so the α-Bloch spacesBα _{form an increasing, uniform family of function spaces. Note also that the}

Bloch spaceBcontainsH∞and is included in all of the Bergman spacesLpa,p≥1, while for

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For further references and details on these general facts about the Bloch-type spaces stated above and more, see9,10.

The boundedness and compactness of composition operators acting between Bloch-type spaces has been established in a series of papers11–15. We state the most general form of these results and use the following notation: forα >0, β > 0 andφbeing an analytic self map of , let

τφ,α,βz

1− |z|2

_β

φ_z

1−φz2α

. 2.4

We writeτφ,αwheneverαβ, andτφifαβ1.

Theorem Asee12,15. Forα, β >0 andφan analytic selfmap of , the composition operatorCφ

is a bounded operator fromBαintoBβif and only if

sup

z∈ τφ,α,βz<∞, 2.5

andC_φis a compact operator fromBαintoBβif and only if

lim

|φz|→1τφ,α,βz 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 forφa self-map of the unit disk

1− |z|2φz

1−φz2 ≤1.

2.7

Thus, whenαβ1,τφz≤1, and we get from Theorem A that every composition operator

is bounded onB. Moreover, the Schwarz-Pick lemma and Theorem A imply that ifα≤1≤β, then Cφ maps Bα boundedly intoBβ, since thenτφ,α,βz ≤ 1− |φz|21−α. If furthermore

α < 1 ≤ β, then Cφ fromBα intoBβmust also be compact. Note also that ifφ∞ < 1 and

Cφ:Bα → Bβis bounded, thenCφis compact.

All of the spacesBαinclude the identity function, and so a necessary condition forCφ

to be bounded fromBα intoBβ is that φbelongs toBβ. Thus, for 0 < β < 1, every analytic self-map of that is inH∞\ Bβinduces an unbounded composition operator fromBαtoBβ. 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 operatorCφ :

Bα _{→ B}β_{is isometric whenever}

_C_φ_f

Bβ f_Bα, ∀f∈ Bα. 2.8

Since every nonconstantφis an open map, the composition operatorCφis always one

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if it is bounded below. Thus,Cφhas a closed range if and only if it is bounded below, namely,

if and only if there existsM >0 such that

_C_φ_f

Bβ ≥Mf_Bα, ∀f ∈ Bα. 2.9

In particular, every isometricCφhas a closed range.

On the other hand, recall that the onlyclosedsubspaces of the range of a compact operator are the finite dimensional ones. Thus, since a composition operatorCφ never has a

finite rankbecauseφis an open map, a compactCφcan never have a closed range. Hence,

a compactCφ 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 papers16–19. We present the classification in the form that appears in18.

Theorem B see 18. Let φ be an analytic self-map of . The composition operator Cφ is an

isometry on the Bloch spaceB if and only ifφ0 0 and either φis a rotation, or for every ain there exists a sequence{zn}in such that|zn| → 1,φzn a, andτφzn → 1.

The result provides a large class of functions inducing isometric composition operators on the Bloch space. For example, ifφ is an almost thin infinite Blaschke product fixing the origin, that is a Blaschke product with a sequence of zeroes{zn}that includes 0 and is such

that lim sup_n_{→ ∞}|Bzn|1− |zn|2 1, thenCφ is an isometry onB. For more examples and

the proof of the theorem, see18.

For all of the other Bloch-type spaces, the result from20shows that the only isometric composition operators are the trivial ones.

Theorem Csee20. Let 0< α,α /1, and letφbe an analytic self-map of . Then the composition

operatorCφis an isometry onBαif and only ifφis 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 ifCφis an isometry onBα, then φ , which further implies that∂ ⊆φ .

Moreover, in the caseα /1,φmust be a rotation.

This is similar to many other cases of isometric composition operators acting on spaces of analytic functions. The general requirement is thatφ covers a significantin some sense part of , or even further, thatφhas to be a rotation. For example, ifCφ is an isometry either

on theHp spaces or on the weighted Bergman spacesAp_w with 1 ≤ p < ∞, then φ has to be either an inner function or a rotation, respectively, as shown in2. IfCφ is an isometry

on the Dirichlet space, thenφhas to be an univalent full map, that is, one to one and such that the area measure of \φ is zero,see3. The complete determination of isometric composition operators in all of this cases is given by adding the conditionφ0 0.

A specific isometric requirement in the case of the Bloch space is that whenφis not univalent, it has to be of infinite multiplicity and such that the functionτφ stays close to 1

over some of the preimages of each point inφ . Whenφis univalent,φhas to be a rotation, and thusτφz 1, for allz∈ .

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is not too surprising that similar requirements onφ andτφ,αplay a role in determining the

closed range composition operators onBα.

The following Theorem D below classifies the closed range composition operators on the Bloch spacesBα, α ≥ 1. The results are contained in a series of papers21–23, and we present their combined version. We need a few more definitions before we can state the theorem.

We say thatG⊆ is sampling forBαif∃S >0 such that for all f∈ Bα,

sup

z∈G

_f_z₁_{− |z|}2α_≥_S_f

Bα. 2.10

Letρz, w |ψzw|denote the pseudohyperbolic distance on , whereψzis a disc

automorphism of , that is,

ψzw z−w

1−zw. 2.11

We say thatG ⊆ is anr −netfor for somer ∈ 0,1if for allz ∈ ,∃w ∈ Gsuch that ρz, w< r.

Forc >0, let

Ωc,α,βz∈ :τφ,α,βz≥c, 2.12

and letGc,α,β φΩc,α,β. Ifαβ, we use the notationΩc,αandGc,α, and if furtherαβ1,

we writeΩcandGc.

Theorem Dsee21,23, 24. Let φ be a self-map of and letα ≥ 1. Then, the following are

equivalent.

iCφhas a closed range onBα.

iiThere existsc >0 such that the setGc,αis sampling forBα. iiiThere existc, r >0 withr <1 such thatGc,αis anr-net for .

Moreover, ifα1, then (i) to (iii) are also equivalent to the following.

ivThere existsk >0 such thatCφψaB≥kψaB, for alla∈ .

Note, for example, that whenα1 andφis a rotation,τφz 1, for allz∈ , and so

G1 . Thus,G1is trivially sampling forBand anr-net for for anyr > 0. Therefore,ii

andiiiare true. Partivof the theorem also holds true if one takesk1.

Ifφis not a rotation andCφis an isometry onB, then wheneverc1−εwith 0< ε <1,

we have from Theorem B thatG1−ε . Hence, as before,G1−εis trivially sampling forBand

anr-net for , for anyr >0. Again, partivof the theorem is true withk1. Thus, we get a geometric description of closed-rangeor isometriccomposition operators on these Bloch spaces as composition operators for which the inducing functionφ has aτφ,α function that

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The geometrical aspects of the restrictions on the τφ function are particularly

interesting in the case whenφis univalent, since then, as a consequence of the K ¨oebe one-quarter theorem, we have that

τφz≈ dist

φz, ∂φ

1−φz . 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 ifφ is the Riemann mapping from onto the slit disk \ 0,1, then Cφ has a closed range on B. Or if φ 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 operatorCφ by controlling the size and the placement of the

pseudohyperbolic disks. For more details on these examples, see23.

It is not too hard to see that ifCφhas a closed range onBαfor someα >1, thenCφhas a

closed range onBαfor allα≥1. On the other hand, the slit disk example from above provides an example of a univalent mapφ such thatCφ has a closed range onBα only forα 1see 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 operatorsCφ : Bα → Bβ. As Theorem A from the previous section states, not

every such composition operator is bounded. Thus, the boundedness condition onτφ,α,βplays

a natural role in the characterization of the isometric and closed-range composition operators fromBαintoBβ.

In general, depending on the choices ofαandβ, 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. Letφz λz, with|λ|1. ThenCφ :Bα → Bβis

aan isometry, wheneverαβ,

ba compact operator ifα < β,

can unbounded operator ifα > β.

Note thataholds since|φz||λ|1, and we have that

_C_φ_f

Bα f◦φ_Bα f0sup

z∈

_f_λz₁_{− |λz|}2α_f

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The other two cases follow easily from Theorem A, since

τφ,α,β

1− |z|2βφz

1−φz2α

1− |z|2β−α. 3.2

As for many other spaces, a condition thatφmust satisfy wheneverCφis an isometry

fromBαintoBβ, regardless of the choices ofαandβ, is thatφ0 0.

Proposition 3.2. IfCφis an isometry fromBαintoBβ, thenφ0 0 and

sup

z∈ τφ,α,βz 1. 3.3

Proof. Note first that the identity functionez zbelongs to each of the Bloch-type spaces and has norm one. Thus, sinceCφ is an isometry, it must be thatφBβC_φe_Bβ e_Bα 1.

Letφ0 a. Using the functionfaz 1−az, we see that

_C_φ_f_a

Bβ fa◦φ_Bβ faasup

z∈

_f

aφz φz1− |z|2

β

1− |a|2|a|sup

z∈

_φ_z₁_{− |z|}2β₁_{− |a|}2_|a|₁_{− |a|}

1|a| −2|a|2.

3.4

But since Cφ is an isometry andfaBα ₁|a|sup_z_∈ ₁− |z|2α ₁|a|, it must be that

|a|2_{0. Thus}_φ₀ _a_0.

To show that sup_z_∈ τφ,α,βz 1, we use another type of test functions. Fora ∈ ,

let ϕ_az be the function such that ϕ_a0 0 andϕ_az ψ_azα, whereψ_a is the disc automorphism of defined by

ψaz ₁a₋−_azz. 3.5

Using that1− |z|2_|ψ

wz|1− |ψwz|2, we get

_ϕ_a

Bα sup

z∈

_ψ

azα

1− |z|2

_α

sup

z∈

1−ψaz2

α

1.

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Hence, sinceCφis an isometry fromBαintoBβ, we have that for anyz0in

1sup

a∈

_C_φ_ϕ_a

Bβsup

a∈ supz∈ τφ,α,βz

1−ψaφz 2

α

≥sup

a∈ τφ,α,βz0

1−ψaφz0 2

_α

≥τφ,α,βz0,

3.7

where the last inequality follows by choosinga φz0, and the fact thatψaa 0. Thus

τφ,α,βz≤1, for allz∈ . On the other hand, for anya∈

1Cφϕa_Bβ sup

z∈ τφ,α,βz

1−ψaz2

α

≤sup

z∈ τφ,α,βz, 3.8

and thus, sup_z_∈ τ_φ,α,βz 1.

Note that in the more general case whenCφ :Bα → Bβhas a closed range,φ0does

not necessarily have to be zero. Still, without loss of generality, we will consider only the case φ0 0. This is possible since the disk automorphisms induce invertibleand thus closed rangecomposition operators on every Bloch-type space. Namely, ifφ0 a, we have that φaz ψa◦φzis such thatφa0 0 andCφaCφCψa. Moreover,

τψa,αz ψaz

1−α_≤1|a|

1− |a|

|1−α|

, 3.9

and by Theorem A we have thatCψa : Bα → Bα is always a boundedand an invertible

operator. ThusC_φ : Bα → Bβhas a closed range if and only ifC_φ_a : Bα → Bβhas a closed range.

Theorem 3.3. Letφbe a selfmap of , letα, β >0, and letCφ :Bα → Bβbe bounded. Then

iCφ :Bα → Bβhas a closed range if and only if there existsc >0 such that the setGc,α,βis

sampling forBα;

iiifCφ :Bα → Bβhas a closed range, then there existc, r >0 withr <1, such thatGc,α,βis

anr-net for ;

iiiif there existc, r >0 withr <1 such thatGc,α,βcontains an open annulus centered at the

origin and with outer radius 1, thenCφhas a closed range.

Proof. The proof ofiis similar to the proof of Theorem 1 from23. SinceCφ :Bα → Bβis

bounded, by Theorem A, we have that∃K >0 such thatτφ,α,βz≤K. Thus, ifCφis bounded

below byM, and ifcM/2, we will show that setGc,α,βis sampling forBαwith a sampling

constantSM/K. We have that for anyf ∈ Bα,

_C_φ_f

Bβ sup

z∈ τφ,α,βz

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and since

sup

z/∈Ωc,α,βτφ,α,βz

1−φz2αfφz < cf_Bα M

2 fBα, 3.11

it must be that

Mf_Bα ≤sup

z∈ τφ,α,βz

1−φz2

_α

f_φ_z

sup

z∈Ωc,α,βτφ,α,βz

1−φz2αfφz

≤K sup

w∈Gc,α,β

1− |w|2αfw.

3.12

Thus sup_w_∈_G_c,α,β1−|w|2α_|f_w_{| ≥}_M/Kf

Bα, that is,G_c,α,βis sampling forBαwith sampling

constantS M/K. The other direction of the proof is fairly similar, and we leave it to the reader.

ii Letw ∈ and letϕwbe as in the proof of Proposition 3.2, that is, the function

defined byϕw0 0 andϕw ψw α. As shown before,ϕwBα 1, and

_ϕ_w_◦_φ

Bβ sup

z∈ τφ,α,βz

1−ψwφz 2

α

. _3.13

Furthermore, assuming thatCφis bounded and has a closed range, there existM, K >0 such

that sup_z_∈ τφ,α,βz Kand

K≥sup

z∈ τφ,α,βz

1−ψwφz 2

α

≥M. _3.14

But since 1− |ψwφz|2≤1, there existszw∈ such that

τφ,α,βzw≥ M₂ ,

1−ψwφzw 2

α

≥ _2KM. 3.15

Thus, forc M/2 andr

1−M/2K1/α, we have that for allw ∈ ; there existszw ∈ Ωc,α,βsuch thatρw, φzw< r, and soGc,α,βis anr-net for .

iiiLetCφ:Bα → Bβbe bounded and assume thatGc,α,βcontains the annulusA{z:

r0<|z|<1}. Suppose thatCφdoes not have a closed range, that is, there exists a sequence of

functions{fn}withfnBα 1 and such thatC_φf_n_Bβ → 0. Since sup_z_∈ 1− |z|2α|f_nz|1,

there exists a sequence{an}in such that for alln,

1− |a_n|2αf_na_n≥ 1

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For anyε >0, letNεbe such that for alln > Nε, and we haveCφfnBβ < cε. Then

sup

w∈Gc,α,β

1− |w|2αf_nw

≤ 1_c sup

z∈Ωc,α,βτφ,α,βz

1−φz2

_α

f

nφz

≤ 1 csup_z_∈

1− |z|2βf_nφz φz 1

cCφfnBβ<

cε c ε.

3.17

Considering furtherε <1/2, we get that eachanwithn > Nεbelongs to the complement of

Gc,α,β. Thus|an| ≤r0<1 andan → awith|a| ≤r0.

On the other hand, since fnBα 1, a normal families argument implies that there

exists a subsequence {fnm} that converges uniformly on compact subsets of to some

function f ∈ Bα. But then {f_n_m} converges to f uniformly on compact subsets of , and since sup_w_∈_G_c,α,β1− |w|2α_|f

nw| → 0 asn → ∞ andGc,α,β contains an infinite compact

subset of , we get thatf≡0. This contradicts the fact that1− |a|2α_|f_a_{| ≥}_{1/2 and so}_C_φ

must be bounded below. Hence,Cφ has a closed range.

Example 3.4. Letφz z2_{, and let}_C

φ :Bα → Bβ. Then

τφ,α,βz

2|z|1− |z|2β

1− |z|4α

1− |z|2β−α 2|z|

1|z|2α

. 3.18

Forz∈ with|z|>1/2, we have that1/2α1− |z|2β−α_{< τ}

φ,α,βz≤21− |z|2β−α. Thus

iifα < β, we have thatτφ,α,βz → 0 as|z| → 1, and soCφis compact; iiifα > β, we have thatτφ,α,βz → ∞as|z| → 1, and soCφ is not bounded;

iiiifαβ, we have that1/2α< τφ,αz≤2, that is,A{z:1/2<|z|<1} ⊂G1/2α_,α

and soCφ is bounded below and has a closed range. Recall thatCφ cannot be an

isometry on any of theBαspaces.

Note that the same conclusions as inExample 3.4hold if we chooseφz zk,k∈, 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 fromBα intoBβwhenα > βin 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 letφbe an analytic selfmap of . If the composition operatorCφ

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Proof. Recall that ifφhas an angular derivative at a pointζ∈∂ , then|φζ|1. Thus, if the setEis empty, we are done. If not, recall further that ifφhas an angular derivativeφζatζ, then for any sequence{zn}in that converges nontangentially toζ, we have that

lim

n→ ∞

1−φzn

1− |zn|

_φ_ζ _lim

n→ ∞φ

_z

n. 3.19

Also, ifφ0 0, it must be that|φζ| ≥1. For more details on angular derivatives, see, for example,8.

Without loss of generality, assume thatφ0 0 and suppose thatφ has an angular derivative at some ζ ∈ ∂ . Hence, whenever{zn} in converges nontangentially toζ, we

have that for large enoughn,

τφ,α,βzn

1− |zn|2

_β

φ_z_n

1−φzn2

α ≥ 1₂

⎛ ⎜ ⎝

1− |zn|2

1−φzn2

⎞ ⎟ ⎠ β 1

1−φzn2

_α₋_β → ∞ 3.20

asn → ∞, since1− |zn|2/1− |φzn|2 → |φζ|−βand|φzn| → 1. By Theorem A,Cφ

cannot be bounded fromBαintoBβ.

Furthermore, by a result from 14, Corollary, page 71, if the set E has a positive measure in∂ , thenφmust have an angular derivative at some point of∂ , and soCφ can

not be bounded fromBαintoBβ.

Recall that Theorems B, C, and D consider the closed-range and isometry classifica-tions of composition operators acting between the same spaceBα. The following theorem looks at the determination of closed-range and isometric composition operators acting between diﬀerent Bloch-type spaces, namely, the case whenα /β.

Theorem 3.6. Letα, β >0, α /β, and letCφfromBαintoBβbe bounded. Then

iCφcan not have a closed range, except possibly when 0< α < β <1, or 1≤β < α; iiCφcan not be an isometry, except possibly when 0< α < β <1.

Proof. i

Case 1. Letα > β. We will show that ifα > βandβ <1, thenCφbounded implies thatφ∞<1

and so, by Theorem A,Cφis compact. ThusCφcan not have a closed range.

This follows from the more general fact that ifCφ : Bα → Bβ is bounded, then for

0< ε < αwe have thatCφ:Bα−ε → Bβmust be compact, since

1− |z|2

_β

φ_z

1−φz2α−ε

≤1−φz2εsup

z∈

1− |z|2

_β

φ_z

1−φz2α

, 3.21

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So, since 0< εα−β < α, we have that ifCφ :Bα → Bβis bounded, thenCφ :Bβ → Bβ

is compact, and sinceβ <1, it must be thatφ∞<1 as shown, for example, in14.

Case 2. Letα < β.

aIfα≥1, then eachCφis compact and thus can not have a closed range. This follows

from the fact that

τφ,α,βz

1− |z|2βφz

1−φz2α ≤

1− |z|2φz

1−φz2

1− |z|2β−α≤1− |z|2β−α. 3.22

The first inequality is true sinceφ0 0 and so

1

1−φz2α−1

≤ 1

1− |z|2α−1

, _3.23

while in the second inequality one uses the Schwarz-Pick lemma. Again, since φ0 0, we have that 1−|z|2_≤₁_−|φ_z_|2_{, and by Theorem A,}_C

φmust be compact. bWhenα < 1 ≤β, then again, eachCφ is compact and so,Cφ can not have a closed

range. This follows similarly from the Schwarz-Pick lemma and Theorem A:

1− |z|2βφz

1−φz2α

⎛ ⎜ ⎝

1− |z|2φz

1−φz2

⎞ ⎟ ⎠

α

1− |z|21−αφz1−α1− |z|2β−1

≤1−φz21−α1− |z|2β−1 → 0

3.24

as|φz| → 1.

iiSince every isometry has a closed range, by parti, the leftover case to consider when determining the isometric composition operatorsCφfromBαintoBβis when 1≤β < α.

We will show that in this case also, the composition operator can not be an isometry.

aLet 1< β < α. Then, ifCφ:Bα → Bβis an isometry, we have thatφBβ 1. Thus

sup

z∈

1− |z|2β−11− |z|2φz1, 3.25

where1−|z|2β−1_≤_{1 and}₁_−|z|2_|φ_z_{| ≤}₁_−|φ_z_|2_≤_{1. Therefore, the supremum}

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bIf 1β < αandCφ:Bα → Bis an isometry, then, as before,φB1. In the proof of

Theorem B in18, it is shown that this is enough to imply that eitherφis a rotation

but thenCφis not boundedor thatφis such that for everyain there exists{zn}

withzn → 1,φzn a, andτφzn → 1.

Let{ak}be a sequence of points in such that|ak| → 1, and let{zk}be such that

φzk akandτφzk≥1/2. Then, asak → 1,

τφ,α,1zk

1− |zk|2φzk

1−φzk2

α

τφzk

1− |ak|2

α−1 ≥

1

21− |ak|2

α−1 → ∞, 3.26

and soCφ :Bα → Bis not bounded.

Note that in the proof of the previous theorem, we have shown that in general, wheneverφ0 0,φ_Bβ1, and 1≤β < α, thenC_φ:Bα → Bβis 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 functionφsatisfies 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 onBα, α /1, has a closed range if and only ifφis a disk automorphism.

Theorem 3.7. Letα, β >0, and letCφ :Bα → Bβbe bounded.

iIfφis univalent and not a rotation, thenCφ is not an isometry.

iiIfα /βand if∃M, r >0,r <1 such that|φz| ≤M, whenever|φz|> r, thenCφdoes

not have a closed range.

Proof. iLetφ be univalent, not a rotation, and such thatCφ : Bα → Bβis an isometry. By Proposition 3.2, thenφ0 0 and sup_z_∈ τφ,α,βz 1. The function fz z is such that

fBα 1 and so

1Cφf_Bβ sup

z∈ τφ,α,βz

1−φz2

_α

. 3.27

Since both parts of the last product are less or equal to 1, there is a sequence{zn}in such

thatτφ,α,βzn → 1 and|φzn| → 0. Sinceφis univalent, by the K ¨oebe distortion theorem, it

can not be that|zn| → 1. Hence,∃z0 ∈ such thatzn → z0,φzn → φz0 0. Again,φis

univalent, and soz0 0. But thenτφ,α,β0 |φ0|1 and by Schwarz lemma,φhas to be a

rotation, which is a contradiction. Hence,Cφ is not an isometry.

iiLetα /βand let∃M, r > 0, r < 1 such that|φz| ≤M, whenever|φz|> r. If φ∞ <1, thenCφ is compact and does not have a closed range. So, letζ∈∂ be such that

|φζ|1.

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Ifα < β, then wheneverφzn → 1,

τφ,α,βzn

1− |zn|2

_β

φ_z_n

1−φzn2

_α ≤M1− |zn|2

β−α

→ 0. 3.28

ThusCφis compact and so it can not have a closed range.

Note that partiiof the previous theorem together with partiiiofTheorem 3.3show that, for example, ifφis a finite Blaschke product, thenCφ :Bα → Bβhas a closed range only

whenαβ.

We finish with the following two open questions.

Question 1. If the composition operatorC_φ :Bα → Bβis bounded, is the necessary closed-range condition in partiiofTheorem 3.3also suﬃcient for anyα, β >0? By Theorem D, this is the case whenαβ≥1.

Question 2. Are there any closed-range composition operatorsCφ :Bα → Bβwhenαandβ

are such that either 0< α < β <1 or such that 1≤β < α? ByTheorem 3.7partii, if there is such aCφ, thenφwill not be a finite Blaschke product.

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