Multiplication Operators between Lipschitz-Type Spaces on a Tree

Volume 2011, Article ID 472495,36pages doi:10.1155/2011/472495

Research Article

Multiplication Operators between Lipschitz-Type Spaces on a Tree

Robert F. Allen,

Flavia Colonna,

and Glenn R. Easley

1Department of Mathematics, University of Wisconsin-La Crosse, La Crosse, WI 54601, USA

2Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

3System Planning Corporation, Arlington, VA 22209, USA

Correspondence should be addressed to Flavia Colonna,fcolonna@gmu.edu Received 4 December 2010; Accepted 16 March 2011

Academic Editor: Ingo Witt

Copyrightq2011 Robert F. Allen et al. 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.

Let L be the space of complex-valued functionsf on the set of vertices T of an infinite tree rooted atosuch that the difference of the values off at neighboring vertices remains bounded throughout the tree, and letLwbe the set of functionsf∈ Lsuch that|fv−fv⁻|O|v|⁻¹, where|v|is the distance betweenoandvandv⁻is the neighbor ofvclosest too. In this paper, we characterize the bounded and the compact multiplication operators betweenLandLwand provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators betweenLw and the spaceL^∞ of bounded functions on T and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces.

1. Introduction

LetXandYbe complex Banach spaces of functions defined on a setΩ. For a complex-valued functionψdefined onΩ, the multiplication operator with symbolψfromXtoYis defined as

M_ψfψf, ∀f∈ X. 1.1

A fundamental objective in the study of the operators with symbol is to tie the properties of the operator to the function theoretic properties of the symbol.

WhenΩis taken to be the open unit disk in the complex plane, an important space of functions to study is the Bloch space, defined as the setBof analytic functionsf : → for which

βf sup

z∈

1− |z|²fz<∞. 1.2

The Bloch space can also be described as the set consisting of the Lipschitz functions between metric spaces from endowed with the Poincar´e distanceρto endowed with the Euclidean distance, a fact that was proved by the second author in1 see also2. In fact, f∈ Bif and only if there existβ >0 such that for allz, w∈

fz−fw≤βρz, w, βf sup

z /w

fz−fw

ρz, w . 1.3

More recently, considerable research has been carried out in the field of operator theory when the setΩis taken to be a discrete structure, such as a discrete group or a graph. In this paper, we consider the case whenΩis taken to be an infinite tree.

By a treeT we mean a locally finite, connected, and simplyconnected graph, which, as a set, we identify with the collection of its vertices. Two verticesuandvare called neighbors if there is an edge connecting them, and we use the notationu∼v. A vertex is called terminal if it has a unique neighbor. A path is a finite or infinite sequence of verticesv0, v₁, . . .such thatv_k∼v_{k 1}andv_k−1/v_{k 1}, for allk.

Given a treeT rooted atoand a vertexu∈T, a vertexvis called a descendant ofuifu lies in the unique path fromotov. The vertexuis then called an ancestor ofv. Given a vertex v /o, we denote byv⁻ the unique neighbor which is an ancestor ofv. Forv ∈T, the setSv

consisting ofvand all its descendants is called the sector determined byv.

Define the length of a finite pathuu0, u1, . . . , vun withuk∼u_{k 1}fork0, . . . , n to be the numbernof edges connectingutov. The distance,du, v, between verticesuandv is the length of the path connectingutov. The treeTis a metric space under the distanced.

Fixingoas the root of the tree, we define the length of a vertexvby|v|do, v. By a function on a tree we mean a complex-valued function on the set of its vertices.

In this paper, the tree will be assumed to be rooted at a vertexoand without terminal verticesand hence infinite.

Infinite trees are discrete structures which exhibit significant geometric and potential theoretic characteristics that are present in the Poincar´e disk . For instance, they have a boundary, which is defined as the set of equivalence classes of paths which differ by finitely many vertices. The union of the boundary with the tree yields a compact space. A useful resource for the potential theory on trees illustrating the commonalities with the disk is3.

In4it was shown that, if the tree has the property that all its vertices have the same number of neighbors, then there is a natural embedding of the tree in the unit disk such that the edges of the tree are arcs of geodesics in with the same hyperbolic length and the set of cluster points of the vertices is the entire unit circle.

In5, the last two authors defined the Lipschitz spaceLon a treeTas the set consisting of the functionsf :T → which are Lipschitz with respect to the distancedonT and the Euclidean distance on. For this reason, the Lipschitz spaceLcan be viewed as a discrete

analogue of the Bloch spaceB. It was also shown that the Lipschitz functions onT are pre- cisely the functions for which

Df

∞sup

v∈T^∗Dfv<∞, 1.4

whereDfv |fv−fv⁻|andT^∗T\ {o}. Under the norm f

Lfo Df

∞, 1.5

Lis a Banach space containing the spaceL^∞of the bounded functions onT. Furthermore, for f∈L^∞,fL≤2f∞.

The little Lipschitz space is defined as L0

f ∈ L: lim

|v| → ∞Dfv 0

1.6

and was proven to be a separable closed subspace ofL. We state the following results that will be useful in the present paper.

Lemma 1.1see5, Lemma 3.4. aIff ∈ Landv∈T, then fv≤fo |v|Df

∞. 1.7

In particular, iff_L≤1, then|fv| ≤ |v|for eachv∈T^∗. bIff∈ L0, then

|v| → ∞lim fv

|v| 0. 1.8

Lemma 1.2see5, Proposition 2.4. Let{fn}be a sequence of functions inL0 converging to 0 pointwise inT such that{fnL}is bounded. Thenfn → 0 weakly inL0.

In6, we introduced the weighted Lipschitz space on a treeTas the setL_wof the func- tionsf :T → such that

sup

v∈T^∗|v|Dfv<∞. 1.9

The interest in this space is due to its connection to the bounded multiplication operators onL. Specifically, it was shown in 5 that the bounded multiplication operators onLare precisely those operatorsMψ whose symbolψ is a bounded function in L_w. The spaceL_w was shown to be a Banach space under the norm

f

wfo sup

v∈T^∗|v|Dfv. 1.10

The little weighted Lipschitz space was defined as

Lw,0

f∈ Lw: lim

|v| → ∞|v|Dfv 0

1.11

and was shown to be a closed separable subspace ofLw.

In this paper, we will make repeated use of the following results proved in6.

Lemma 1.3see6, Propositions 2.1 and 2.6. aIff∈ L_w, andv∈T^∗, then fv≤

1 log|v| f

w. 1.12

bIff∈ Lw,0, then

|v| → ∞lim fv

log|v|0. 1.13

Lemma 1.4see6, Proposition 2.7. Let{fn}be a sequence of functions inLw,0converging to 0 pointwise inT such that{fnw}is bounded. Thenf_n → 0 weakly inLw,0.

In this paper, we consider the multiplication operators betweenLandL_w, as well as betweenLw andL^∞. The multiplication operators betweenLandL^∞ were studied by the last two authors in7.

1.1. Organization of the Paper

In Sections2and3, we study the multiplication operators betweenL_wandL. We characterize the bounded and the compact operators and give estimates on their operator norm and their essential norm. We also prove that no isometric multiplication operators exist between the respective spaces.

InSection 4, we characterize the bounded operators and the compact operators from Lw toL^∞and determine their operator norm and their essential norm. As was the case in Sections2and3, we show that no isometries exist amongst such operators. In addition, we characterize the multiplication operators that are bounded from below.

Finally, in Section 5, we characterize the bounded and the compact multiplication operators fromL^∞toLw. We also determine their operator norm and their essential norm.

As with all the other cases, we show that there are no isometries amongst such operators.

2. Multiplication Operators from L

to L

We begin the section with the study of the bounded multiplication operatorsMψ :Lw → L andMψ :L_w,0 → L0.

2.1. Boundedness and Operator Norm Estimates

Letψbe a function on the treeT. Define τψ sup

v∈T^∗Dψvlog1 |v|, σψ sup

v∈T

ψv

|v| 1.

2.1

In the following theorem, we give a boundedness criterion in terms of the quantitiesτ_ψ and σψ.

Theorem 2.1. For a functionψonT, the following statements are equivalent:

aMψ :Lw → Lis bounded.

bMψ :L_w,0 → L0is bounded.

cτ_ψ andσ_ψare finite.

Furthermore, under these conditions, we have max

τ_ψ, σ_ψ

≤ Mψ ≤τ_ψ σ_ψ. 2.2

Proof. a⇒cAssumeMψ :Lw → Lis bounded. ApplyingMψ to the constant function 1, we haveψ ∈ L, so that, byLemma 1.1, we haveσψ <∞. Next, consider the functionf onT defined byfv log1 |v|. Thenfo 0; forv /o, a straightforward calculation shows that

|v|Dfv |v|

log1 |v|−log|v| ≤1 2.3

and lim_{|v| → ∞}|v|Dfv 1. Thus,fw 1 and soMψfL ≤ Mψ. Therefore, forv ∈T^∗, noting that

D

ψf v Dψvfv ψ

v⁻ Dfv, 2.4

one has

Dψvfv≤D

ψf v ψ

v⁻ Dfv

≤M_ψf

L σ_ψ|v|Dfv≤M_ψ σ_ψ. 2.5 Henceτ_ψ <∞.

c⇒aAssumeτψandσψ are finite. Then, byLemma 1.3, forf∈ L_wandv∈T^∗, we have

D

ψf v≤Dψvfv ψ

v⁻ Dfv

≤Dψv

1 log|v| f

w |v|σψDfv

≤τ_ψf_w σ_ψf_w−fo .

2.6

Since|ψo| ≤σψ, we obtain Mψf

L≤ψofo τψf

w σψf

w−fo

τψ σψ f

w ψo−σψ fo

≤

τψ σψ f

w,

2.7

proving the boundedness ofMψ :Lw → Land the upper estimate.

b⇒cSupposeMψ :L_w,0 → L0is bounded. The finiteness ofσψfollows again from the fact thatψ M_ψ1 ∈ L0and fromLemma 1.1. To prove thatτ_ψ < ∞, let 0< α < 1 and, forv ∈T, definefαv log1 |v|^α. Thenfαo 0 and|v|Dfav → 0 as|v| → ∞; so fα ∈ L_w,0. Since for 0 < α < 1, the functionx→ x−x^αis increasing forx ≥1, the function Df_αvis increasing inα, andDf_αv≤Dfvforv∈T^∗, wherefv log1 |v|, forv∈T. Thus,fαw≤ fw1. Therefore, forv∈T^∗, we have

Dψvfαv≤D

ψfα v ψ

v⁻ Dfαv

≤M_ψf_α σ_ψ|v|Dfαv≤M_ψ σ_ψ. 2.8 Lettingα → 1, we obtain

Dψvlog1 |v|≤Mψ σψ. 2.9

Henceτψ <∞.

c⇒bAssumeσ_ψandτ_ψare finite, and letf∈ Lw,0. Then, byLemma 1.3, forv∈T^∗, we have

D

ψf v≤Dψvfv ψ

v⁻ Dfv

≤Dψvlog1 |v| fv log1 |v|

ψv⁻

|v| |v|Dfv

≤τψ

fv

log1 |v| σψ|v|Dfv−→0

2.10

as|v| → ∞. Thus,ψf ∈ L0. The boundedness ofM_ψ and the estimateMψf_L≤τ_ψ σ_ψ can be shown as in the proof ofc⇒a.

Finally we show that, under boundedness assumptions onMψ,Mψ ≥max{τψ, σψ}.

Forv∈ T^∗, letf_v 1/|v| 1χv, whereχ_vdenotes the characteristic function of{v}. Then fvw1 and

ψf_v

L ψv

|v| 1. 2.11

Furthermore, lettingfo 1/2χo, we see thatfo_w1 andψfoL |ψo|. Therefore, we deduce thatMψ ≥σψ.

Next, fixv∈T^∗and forw∈T, define

g_vw

⎧⎨

⎩

log1 |w| if|w|<|v|,

log1 |v| if|w| ≥ |v|. 2.12

Then,g_v ∈ Lwand

|v| → ∞lim gv

w lim

|v| → ∞|v|

log1 |v|−log|v|

1. 2.13

Observe that, forw∈T^∗, we have

D

ψgv w

⎧⎨

⎩

ψwlog1 |w|−ψw⁻log|w| if|w|<|v|,

Dψwlog1 |v| if|w| ≥ |v|. 2.14

Hence

sup

w∈T^∗D

ψgv w≥ sup

|w|≥|v|Dψwlog1 |v|≥Dψvlog1 |v|. 2.15

Definefvgv/gv_w. Thenfv_w1 and M_ψ≥M_ψf_v

L D

ψg_{v ∞} gv

w

≥ Dψvlog1 |v|

gv

w

. 2.16

Taking the limit as|v| → ∞, we obtainMψ ≥τψ. Therefore,Mψ ≥max{τψ, σψ}.

2.2. Isometries

In this section, we show there are no isometric multiplication operatorsM_ψ from the spaces LworLw,0to the spacesLorL0, respectively.

AssumeMψ :Lw → Lis an isometry. ThenψL Mψ1L 1. On the other hand,

|ψo| 1/2Mψχo_L 1/2χo_w 1. Thus sup_v∈T∗Dψv ψ_L− |ψo| 0, which implies thatψis a constant of modulus 1. Yet, forv∈T^∗, lettingf_v 1/|v| 1χv, we see that

1fv

wMψfv

L 1

|v| 1, 2.17

which yields a contradiction. Therefore, we obtain the following result.

Theorem 2.2. There are no isometriesMψ fromL_wtoLorL_w,0toL0, respectively.

2.3. Compactness and Essential Norm Estimates

In this section, we characterize the compact multiplication operators. As with many classical spaces, the characterization of the compact operators is a “little-oh” condition corresponding the the “big-oh” condition for boundedness. We first collect some useful results about com- pact operators fromL_worL_w,0toL.

Lemma 2.3. A bounded multiplication operatorM_ψ fromLwtoLis compact if and only if for every bounded sequence{fn}inLwconverging to 0 pointwise, the sequence{ψfnL} → 0 asn → ∞.

Proof. AssumeMψ is compact, and let {fn}be a bounded sequence in L_w converging to 0 pointwise. Without loss of generality, we may assume fnw ≤ 1 for alln ∈ . Then the sequence{Mψfn}{ψfn}has a subsequence{ψfnk}which converges in theL-norm to some functionf∈ L. Clearlyψofnko → ψofo, and by partaofLemma 1.1, forv∈T^∗, we have

ψvfnkv−fv≤ψofnko−fo |v|D

ψfnk−f _∞

≤1 |v|ψf_nk−f_L. 2.18

Thus,ψfnk → fpointwise onT. Sincefn → 0 pointwise, it follows thatfmust be identically 0, which implies thatψfnk_L → 0. With 0 being the only limit point of{ψfn}inL, it follows thatψfn_L → 0 asn → ∞.

Conversely, assume every bounded sequence {fn}inLw converging to 0 pointwise has the property thatψfn_L → 0 asn → ∞. Let{gn}be a sequence inL_wwithgn_w≤1 for alln ∈ . Then|gno| ≤ 1 for alln ∈ , and by partaofLemma 1.2, forv ∈ T^∗, we obtain

gnv≤

1 log|v| gn

w≤1 log|v|. 2.19

Thus,{gn}is uniformly bounded on finite subsets ofT. So some subsequence{gnk}converges pointwise to some functiong. Fixv∈T^∗andε >0. Then forksufficiently large, we have

gv−g_nkv< ε

2|v|, g_nk

v⁻ −g

v⁻ < ε

2|v|. 2.20

We deduce

|v|Dgv≤ |v|gv−g_nkv g_nk

v⁻ −g

v⁻ |v|Dgnkv

≤ |v|gv−gnkv |v|gnk

v⁻ −g

v⁻ |v|Dgnkv

< ε |v|Dgnkv≤ε 1,

2.21

for allksufficiently large. Sog ∈ Lw. The sequence defined byf_k g_nk −g is bounded in Lwand converges to 0 pointwise. Thus by hypothesis, we obtainψfk_L → 0 ask → ∞. It follows thatMψgnk ψgnk → ψgin theL-norm, thus proving the compactness ofMψ.

By an analogous argument, we obtain the corresponding compactness criterion forMψ

fromL_w,0toL0.

Lemma 2.4. A bounded multiplication operatorM_ψ fromLw,0 toL0 is compact if and only if for every bounded sequence{fn}in Lw,0 converging to 0 pointwise, the sequence {ψfnL} → 0 as n → ∞.

The following result is a variant ofLemma 1.3a, which will be needed to prove a char-

acterization of the compact multiplication operators from L_w to L and from L_w,0 to L0

Theorem 2.6.

Lemma 2.5. Forf∈ Lwand v∈T

fv≤fo 2 log1 |v|sw

f , 2.22

wheres_wf sup_w∈T∗|w|Dfw.

Proof. Fixv∈Tand argue by induction onn|v|. Forn0, inequality2.22is obvious. So assume|v|n >0 and|fu| ≤ |fo| 2 log1 |u|swffor all verticesusuch that|u|< n.

Then

fv≤fv−f

v⁻ f v⁻

≤ 1

|v|s_w

f fo 2 log|v|sw f

fo 1

|v| 2 log|v|

s_w

f .

2.23

Next, observe that 1/|v| 1≤log|v| 1/|v|, so 1

|v| ≤ 2

|v| 1 ≤2 log

|v| 1

|v|

. 2.24

Hence

1

|v| 2 log|v| ≤2 log|v| 1. 2.25

Inequality2.22now follows immediately from2.23and2.25.

Theorem 2.6. LetMψbe a bounded multiplication operator fromLwtoL(or equivalently fromLw,0

toL0). Then the following statements are equivalent:

aM_ψ :Lw → Lis compact.

bMψ :Lw,0 → L0is compact.

clim_{|v| → ∞}|ψv|/|v| 1 0 and lim_{|v| → ∞}Dψvlog|v|0.

Proof. We first provea⇒c. AssumeMψ : Lw → Lis compact. It suffices to show that, for any sequence{vn}inT such that 2≤ |vn| → ∞, we have limn→ ∞|ψvn|/|vn| 1 0 and lim_{n→ ∞}Dψvnlog|vn| 0. Let{vn} be such a sequence, and for eachn ∈ , define f_n 1/|vn| 1χvn. Thenf_no 0,f_n → 0 pointwise asn → ∞, and fnw 1. By Lemma 2.3, it follows thatψfn_L → 0 asn → ∞. Furthermore

ψfn

Lsup

v∈T^∗

ψvfnv−ψv⁻fnv⁻ψvnfnvn ψvn

|vn| 1. 2.26

Thus limn→ ∞|ψvn|/|vn| 1 0.

Next, for eachn∈andv∈T, define

gnv

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0 if|v|<

|vn|, 2 log|v| −log|vn| if

|vn| ≤ |v|<|vn| −1, log|vn| if|v| ≥ |vn| −1.

2.27

ThenDgnv 0 if|v| ≤

|vn| or|v| > |vn| −1. In addition, if

|vn| < |v| ≤ |vn| −1, then

|v|Dgnv<4. Indeed, there are two possibilities. Either

|vn| ≤ |v| −1, in which case

|v|Dgnv 2|v|

log|v| −log|v| −1 ≤ 2|v|

|v| −1 ≤3, 2.28

or|v| −1<

|vn|<|v|, in which case

|v|Dgnv |v|

2 log|v| −log|vn|

≤

|vn| 1

log

|vn| 1₂

|vn|

≤ 2

|vn| 1 |vn| ≤2

1 1

√2

<4.

2.29

Thus{gnw}is bounded, and{gn}converges to 0 pointwise. ByLemma 2.3, it follows that ψgnL → 0 asn → ∞. Moreover

ψgn

L≥ψvngnvn−ψv⁻_ngnv⁻_nDψvnlog|vn|. 2.30 Therefore limn→ ∞Dψvnlog|vn|0.

To prove the implication c⇒a, suppose lim|v| → ∞Dψvlog|v| 0 and lim_{|v| → ∞}|ψv|/|v| 1 0. Clearly, if ψ is identically 0, then Mψ is compact. So assume Mψ :L_w → Lis bounded withψnot identically 0. ByLemma 2.3, it suffices to show that if

{fn}is bounded inLwconverging to 0 pointwise, thenψfnL → 0 asn → ∞. Let{fn}be such a sequence,ssup_n∈fn_w, and fixε >0. Note that

|v| → ∞lim Dψvlog1 |v| lim

|v| → ∞Dψvlog|v|log1 |v|

log|v| 0. 2.31

Thus there exists anM∈such that fno< ε

3sψ

L

, Dψvlog1 |v|< ε

6s, ψv

|v| 1 < ε

3s, 2.32

for|v| ≥M. UsingLemma 2.5, for|v|> M, we have

D

ψfn v≤Dψvfnv Dfn

v⁻ ψ v⁻

≤Dψvfno 2 log|v| 1 fn

w fn

w

ψv⁻

|v|

≤ ψ

Lfno 2Dψvlog|v| 1 ψv⁻

|v|

fn

w

< ε.

2.33

On the other hand, on the setBM{v∈T :|v| ≤M},{fn}converges to 0 uniformly, and thusDfndoes as well. Moreover

D

ψfn v≤Dψvfnv ψ

v⁻ Dfnv

≤ψ

Lfnv max

|w|≤MψwDfnv−→0, 2.34

uniformly on B_M. Therefore Dψfn → 0 uniformly on T. Furthermore, the sequence {ψfno}converges to 0 asn → ∞. HenceψfnL → 0 asn → ∞, proving thatMψ is compact.

Finally, note that the functionsf_nandg_ndefined in the proof ofa⇒care inLw,0. So the equivalence ofbandcis proved analogously.

Recall the essential norm of a bounded operatorSbetween Banach spacesXandYis defined as

S_einf

S−K:K is compact fromXtoY

. 2.35

Forψa function onT, define the quantities

A

ψ lim

n→ ∞sup

|v|≥n

ψv

|v| 1, B

ψ lim

n→ ∞sup

|v|≥nDψvlog|v|.

2.36

Theorem 2.7. LetM_ψ be a bounded multiplication operator fromLwtoL. Then Mψ

e≥max A

ψ , B

ψ . 2.37

Proof. For eachn∈, definef_n 1/n 1χn, whereχ_ndenotes the characteristic function of the set {v ∈ T : |v| n}. Thenfn ∈ Lw,0,fnw 1, andfn → 0 pointwise. Thus, by Lemma 1.4,{fn}converges to 0 weakly inLw,0. LetKbe the set of compact operators from Lw,0 toL0, and letK ∈ K. ThenK is completely continuous 8, and soKfnL → 0 as n → ∞. Thus

Mψ −K ≥lim sup

n→ ∞

Mψ−K fn

L≥lim sup

n→ ∞

Mψfn

L. 2.38

Now note that

M_ψf_n

Lsup

|v|n

ψv

n 1 . 2.39

Hence

Mψ

e≥infMψ−K:K∈ K

≥lim sup

n→ ∞

M_ψf_n

L

lim

n→ ∞sup

|v|≥n

ψv

|v| 1 A

ψ .

2.40

We will now show thatMψ_e ≥ Bψ. This estimate is clearly true ifBψ 0. So assume {vn}is a sequence inT such that 2≤ |vn| → ∞asn → ∞and

nlim→ ∞Dψvnlog|vn|B

ψ . 2.41

Forn∈ andv∈T, define

hnv

⎧⎪

⎨

⎪⎩

log|v| 12

log|vn| if 0≤ |v|<|vn|, log|vn| if |v| ≥ |vn|.

2.42

Thenhno 0,hnvn hnv⁻_n log|vn|, and

|v|Dhnv

⎧⎪

⎨

⎪⎩

|v|

log|vn|log

|v| 1

|v|

log|v||v| 1 if 1≤ |v|<|vn|,

0 if|v| ≥ |vn|. 2.43

The supremum of|v|Dhnvis attained at the vertices of length|vn| −1 and is given by snsup

v∈T^∗|v|Dhnv |vn| −1log |vn|

|vn| −1

log|vn| −1|vn|

log|vn| . 2.44

Since|vn| −1log|vn|/|vn| −1≤1, we have log 2 ²

log|vn| ≤ h_nws_n≤ log|vn| −1|vn|

log|vn| <2. 2.45

By lettinggn hn/hn_w, we havegn ∈ L_w,0,gn_w 1, andgn → 0 pointwise. By Lemma 1.4, the sequence{gn}converges to 0 weakly inL_w,0. ThusKgn_L → 0 asn → ∞.

Therefore,

M_ψ−K≥lim sup

n→ ∞

M_ψ−K g_n

L≥lim sup

n→ ∞

ψg_nL. 2.46

For eachn∈, we haveg_nvn g_nv⁻_n log|vn|/sn. So D

ψgn vn 1

snDψvnlog|vn|. 2.47

Since lim_{n→ ∞}s_n1, we have Mψ

e≥infMψ−K:K∈ K

≥lim sup

n→ ∞ sup

v∈T^∗D

ψgn v

≥ lim

n→ ∞

1

s_nDψvnlog|vn| B

ψ .

2.48

Therefore,Mψe≥max{Aψ, Bψ}.

We now derive an upper estimate on the essential norm.

Theorem 2.8. LetMψ be a bounded multiplication operator fromL_wtoL. Then

M_ψ

e≤A

ψ B

ψ . 2.49

Proof. Forn∈, define the operatorKnonL_wby

Knf v

⎧⎨

⎩

fv if |v| ≤n,

fvn if |v|> n, 2.50

wheref ∈ Lwand vn is the ancestor of vof length n. Forf ∈ Lw,Knfo fo, and Knf ∈ L_w,0. LetBn {v ∈ T : |v| ≤ n}, and note that Knf attains finitely many values, whose number does not exceed the cardinality ofB_n. Let{gk}be a sequence inLwsuch that gkw≤1 for eachk∈. Thenasup_k∈|gko| ≤1, and|Kngko| ≤a. Furthermore, by part aofLemma 1.3, for eachv∈T^∗and for eachk∈, we have|Kngkv| ≤1 logn. Thus, some subsequence of{Kng_k}_k∈must converge to a functiong onT attaining constant values on the sectors determined by the vertices of lengthn. It follows that this subsequence converges toginL_was well, proving thatKnis a compact operator onL_w. SinceMψ is bounded as an operator fromLwtoL, it follows thatM_ψK_n:Lw → Lis compact for alln∈.

Define the operatorJn I−Kn, whereI denotes the identity operator onLw. Then Jnfo 0, and forv∈T^∗, we have

|v|D

Jnf v |v|Jnf v−

Jnf v⁻ ≤ |v|Dfv≤f

w. 2.51

By partaofLemma 1.3, we see that

Jnf v≤

1 log|v| f

w. 2.52

Using2.51and2.52, we obtain M_ψ−M_ψK_n f

LψJnf_L sup

|v|>n

ψv

Jnf v−ψ

v⁻ Jnf v⁻

≤sup

|v|>n

J_nf vDψv ψ v⁻ D

J_nf v

sup

|v|>n

Jnf vDψv ψv⁻

|v| |v|D

Jnf v

≤sup

|v|≥n

1 log|v| Dψv ψv

|v| 1 f

w

≤sup

|v|≥n

log|v|Dψv1 log|v|

log|v|

ψv

|v| 1 f_w

≤

sup

|v|≥nlog|v|Dψv1 logn logn sup

|v|≥n

ψv

|v| 1 f

w.

2.53

Since

M_ψ

e≤lim sup

n→ ∞

M_ψ−M_ψK_nlim sup

n→ ∞ sup

fw1

M_ψ−M_ψK_n f

L, 2.54

taking the limit asn → ∞, we obtain M_ψ

e ≤B

ψ A

ψ , 2.55

as desired.

3. Multiplication Operators from L to L

We begin this section with a boundedness criterion for the multiplication operators fromM_ψ : L → LwandMψ :L0 → Lw,0.

3.1. Boundedness and Operator Norm Estimates Letψbe a function on the treeT. Define the quantities

θψ sup

v∈T^∗|v|²Dψv, ωψ sup

v∈T|v| 1ψv.

3.1

Theorem 3.1. For a functionψonT, the following statements are equivalent:

aMψ :L → L_wis bounded.

bM_ψ :L0 → Lw,0is bounded.

cθ_ψ andω_ψare finite.

Furthermore, under the above conditions, one has

max θ_ψ, ω_ψ

≤M_ψ≤θ_ψ ω_ψ. 3.2

Proof. a⇒cAssumeM_ψ is bounded fromLtoLw. The functionf_o 1/2χo ∈ Land fo_L1. Thus

ψoψfow≤ Mψ. 3.3

Next, fixv∈T^∗. Thenχ_v ∈ LandχvL1; so

|v| 1ψvψχv

w≤Mψ. 3.4

Taking the supremum over allv∈T, from3.3and3.4we see thatωψ is finite and

ω_ψ ≤M_ψ. 3.5

Withv∈T^∗, we now define

fvw

⎧⎨

⎩

|w| if |w|<|v|,

|v| if |w| ≥ |v|. 3.6

Thenf_v∈ L,f_vo 0 andfv_L1. By the boundedness ofM_ψ we obtain Mψ≥Mψfv

w

≥ sup

1≤|w|≤|v||w|ψw|w| −ψ

w⁻ |w| −1

≥ sup

1≤|w|≤|v||w|²Dψw− sup

1≤|w|≤|v||w|ψ w⁻ .

3.7

Therefore,

|v|²Dψv≤ sup

1≤|w|≤|v||w|²Dψw≤Mψ ωψ. 3.8

Taking the supremum over allv∈T^∗, we obtainθψ <∞. From this and3.5, we deduce the lower estimate

M_ψ≥max θ_ψ, ω_ψ

. 3.9

c⇒aAssumeθψandωψare finite. Then,ψ∈ L_w, and byLemma 1.1, forf ∈ Lwith fL 1 andv∈T^∗, we have

|v|D

ψf v≤ |v|Dψvfv |v|ψ

v⁻ Dfv

≤ |v|Dψvfo |v|²DψvDf_∞ ω_ψDf_∞

≤ |v|Dψvfo

θψ ωψ Df

∞.

3.10

Thus,ψf∈ Lw. Note that|fo| Df∞1 and ψ

wψo sup

v∈T^∗|v|Dψv≤ω_ψ sup

v∈T^∗|v|²Dψv ω_ψ θ_ψ. 3.11 From this, we have

ψf

w≤ψ

wfo

θψ ωψ Df

∞≤θψ ωψ, 3.12

proving the boundedness ofMψ :L → Lwand the upper estimate

Mψ≤θψ ωψ. 3.13 b⇒cThe proof is the same as fora⇒c; since forv∈T^∗, the functionsχvandfv

used there belong toL0.

c⇒bAssumeθ_ψ andω_ψare finite, and letf∈ L0. Then, byLemma 1.1, forv∈T^∗, we have

|v|D

ψf v≤ |v|Dψvfv |v|ψ

v⁻ Dfv

≤ |v|²Dψvfv

|v| |v|ψ

v⁻ Dfv

≤θ_ψfv

|v| ω_ψDfv−→0

3.14

as |v| → ∞. Thus, ψf ∈ Lw,0. The proof of the boundedness ofMψ is similar to that in c⇒a.

3.2. Isometries

In this section, we show there are no isometric multiplication operatorsMψfrom the spaceL toLwor fromL0toLw,0.

SupposeMψ :L → Lwis an isometry. Thenψ_wMψ1_w1. On the other hand, ψo 1

2ψχ_o

w 1 2χ_o

L1. 3.15

Thus sup_v∈T∗|v|Dψv ψ_w− |ψo|0, which implies thatψis a constant of modulus 1.

Now observe that, forv∈T^∗, we have 1χv

LMψχ

w |v| 1ψv|v| 1, 3.16 which is a contradiction. Sinceχ_v ∈ L0for allv∈T, ifM_ψ :L0 → Lw,0is an isometry, then the above argument yields again a contradiction. Thus, we proved the following result.

Theorem 3.2. There are no isometriesMψ fromLtoL_wor fromL0toL_w,0.

3.3. Compactness and Essential Norm

We now characterize the compact multiplication operators, but first we give a useful com- pactness criterion for multiplication operators fromLtoL_wor fromL0toL_w,0.

Lemma 3.3. A bounded multiplication operatorM_ψ fromLtoL_w(L0 toL_w,0) is compact if and only if for every bounded sequence{fn}inL(L0) converging to 0 pointwise, the sequenceψfn_w converges to 0 asn → ∞.

Proof. SupposeM_ψis compact fromLtoLwand{fn}is a bounded sequence inLconverging to 0 pointwise. Without loss of generality, we may assumefn_L ≤1 for alln∈. SinceM_ψ is compact, the sequence{ψfn}has a subsequence{ψfnk}that converges in theL_w-norm to some functionf∈ Lw.

ByLemma 1.3, forv∈T^∗we have ψvfnkv−fv≤

1 log|v| ψfnk−f

w. 3.17

Thus, ψf_nk → f pointwise on T^∗. Furthermore, since |ψofnko−fo| ≤ ψfnk−f_w, ψ0fnk0 → f0ask → ∞. Thusψfnk → fpointwise onT. Since by assumption,fn → 0 pointwise, it follows thatf is identically 0, and thusψfnk_w → 0. Since 0 is the only limit point inLwof the sequence{ψfn}, we deduce thatψfn_w → 0 asn → ∞.

Conversely, suppose that every bounded sequence {fn} in L that converges to 0 pointwise has the property thatψfn_w → 0 asn → ∞. Let{gn}be a sequence inLsuch that gn_L ≤1 for alln∈. Then|gno| ≤ 1, and by partaofLemma 1.1, forv∈T^∗we have

|gnv| ≤ |v|. So{gn}is uniformly bounded on finite subsets ofT. Thus there is a subsequence {gnk}, which converges pointwise to some functiong.

Fixε >0 andv∈T^∗. Then|gnkv−gv|< ε/2 as well as|gnkv⁻−gv⁻|< ε/2 fork sufficiently large. Therefore, for allksufficiently large, we have

Dgv≤gv−g_nkv g_nk

v⁻ −g

v⁻ Dg_nkv< ε Dg_nkv. 3.18 Thusg ∈ L. The sequencef_nk g_nk −g is bounded inLand converges to 0 pointwise. So ψfnk_w → 0 ask → ∞. Thusψgnk → ψgin theLw-norm. Therefore,Mψis compact.

The proof for the case ofMψ :L0 → L_w,0is similar.

Theorem 3.4. LetM_ψ be a bounded multiplication operator fromLtoLw(or equivalently fromL0

toLw,0). Then the following are equivalent:

aMψ :L → L_wis compact.

bM_ψ :L0 → Lw,0is compact.

clim_{|v| → ∞}|v|²Dψv 0 and lim_{|v| → ∞}|v| 1|ψv|0.

Proof. a⇒c Suppose Mψ : L → L_w is compact. We need to show that if {vn} is a sequence in T such that 2 ≤ |vn|increasing unboundedly, then lim_{n→ ∞}|vn|²Dψvn 0 and limn→ ∞|vn| 1|ψvn| 0. Let {vn} be such a sequence, and for n ∈ define fn |vn| 1/|vn|χvn. Clearly fn → 0 pointwise, and fn_L ≤ 3/2. UsingLemma 3.3, we see that

ψfn

w−→0 asn−→ ∞. 3.19

On the other hand, sincefno 0 for alln∈, we have ψfn

wsup

v∈T^∗|v|D

ψfn v≥ |vn|

|vn| 1

|vn|

ψvn |vn| 1ψvn. 3.20

Hence lim_{n→ ∞}|vn| 1|ψvn|0.

Next, forn∈, define

g_nv

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

0 if|v|<

|vn| 2

, 2|v| − |vn| 2 if

|vn| 2

≤ |v|<|vn|,

|vn| if|v| ≥ |vn|,

3.21

wherexdenotes the largest integer less than or equal tox. Then g_n → 0 pointwise, and gn_L 2. Sincegnvn gnv_n⁻ |vn|, we have

ψgn

w≥ |vn|ψvngnvn−ψ v⁻_n gn

v⁻_n |vn|²Dψvn. 3.22

ByLemma 3.3we obtain lim_{n→ ∞}|vn|²Dψvn≤lim_{n→ ∞}ψgnw0.

c⇒aSuppose lim_{|v| → ∞}|v|²Dψv 0 and lim_{|v| → ∞}|v| 1|ψv|0. Assumeψis not identically zero, otherwiseMψ is trivially compact. ByLemma 3.3, to prove that Mψ is

compact, it suffices to show that if{fn}is a bounded sequence inLconverging to 0 pointwise, thenψfn_w → 0 asn → ∞. Let{fn}be such a bounded sequence,ssup_n∈fn_L, and fix ε >0. There existsM∈ such that|v| 1|ψv|< ε/2sand|v|²Dψv< ε/2sfor|v| ≥M.

Forv∈T^∗and byLemma 1.1, we have

|v|D

ψf_n v≤ |v|ψvDf_nv |v|Dψvf_n v⁻

≤ |v|ψvDfnv |v|Dψvfno |v|Dfn

∞

≤|v| 1ψvDfnv |v|²Dψvfno Dfn

∞

|v| 1ψvDfnv |v|²Dψvfn

L.

3.23

Sincefn → 0 uniformly on{v∈T :|v| ≤M}asn → ∞, so doesDfn. So, on the set{v∈T :

|v| ≤M},|v|Dψfnv → 0 asn → ∞. On the other hand, on{v∈T :|v| ≥M}, we have

|v|D

ψfn v≤|v| 1ψvDfnv |v|²Dψvfn

L< ε. 3.24 So|v|Dψfnv → 0 asn → ∞. Sincefn → 0 pointwise,ψofno → 0 asn → ∞. Thus ψfn_w → 0 asn → ∞. The compactness ofMψ follows at once fromLemma 3.3.

The proof of the equivalence ofbandcis analogous.

Forψa function onT, define A

ψ lim

n→ ∞sup

|v|≥n|v|ψv, B

ψ lim

n→ ∞sup

|v|≥n|v|²Dψv.

3.25

Theorem 3.5. LetMψ be a bounded multiplication operator fromLtoL_w. Then Mψ

e≥max

A ψ ,1

2B

ψ . 3.26

Proof. Fixk∈, and for eachn∈, consider the sets

E_n,k{v∈T :n≤ |v| ≤kn, |v|even},

On,k{v∈T :n≤ |v| ≤kn, |v|odd}. 3.27 Define the functionsfn,k χEn,k andgn,k χOn,k. Thenfn,k, gn,k ∈ L0,fn,k_L gn,k_L 1, andfn,kandgn,kapproach 0 pointwise asn → ∞. ByLemma 1.2, the sequences{fn,k}and {gn,k}approach 0 weakly inL0 asn → ∞. LetK0 be the set of compact operators fromL0

toLw,0, and note that every operator inK0is completely continuous. Thus, ifK ∈ K0, then Kfn,k_w → 0 andKgn,k_w → 0, asn → ∞.

Therefore, ifK∈ K0, then

M_ψ−K≥lim sup

n→ ∞

M_ψ−K f_n,k

w

≥lim sup

n→ ∞

Mψfn,k

w

≥lim sup

n→ ∞ sup

v∈En,k

|v| 1ψv.

3.28

Similarly,

M_ψ−K≥lim sup

n→ ∞ sup

v∈On,k

|v| 1ψv. 3.29

Therefore, combining3.28and3.29, we obtain Mψ

einfMψ−K:K∈ K0

≥lim sup

n→ ∞ sup

kn≥|v|≥n|v| 1ψv

≥lim sup

n→ ∞ sup

kn≥|v|≥n|v|ψv.

3.30

Lettingk → ∞, we obtainMψ_e≥ Aψ.

Next, we wish to show thatMψ_e ≥1/2Bψ. The result is clearly true ifBψ 0.

So assume there exists a sequence{vn}inTsuch that 2<|vn| → ∞asn → ∞and

nlim→ ∞|vn|²Dψvn B

ψ . 3.31

Forn∈, define

h_nv

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

0 if vo,

|v| 1²

|vn| if 1≤ |v|<|vn|,

|vn| if |v| ≥ |vn|.

3.32

Clearly,hno 0, hnvn hnv⁻_n |vn|, and

Dh_nv

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ 4

|vn| if |v|1, 2|v| 1

|vn| if 1<|v|<|vn|, 0 if |v| ≥ |vn|.

3.33