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

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

2_{Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA} 3_{System Planning Corporation, Arlington, VA 22209, USA}

Correspondence should be addressed to Flavia Colonna,[email protected]

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

Mψfψf, ∀f∈ X. 1.1

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

thatvk∼vk 1andvk−1/vk 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∼uk 1fork0, . . . , 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

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, iffL≤1, then|fv| ≤ |v|for eachv∈T∗.

bIff∈ L0, then

lim

|v| → ∞

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 setLwof 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 Lw. The spaceLw

was shown to be a Banach space under the norm

_f

wfo sup

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∈ Lw, andv∈T∗, then

_fv≤

1 log|v| f_w. 1.12

bIff∈ Lw,0, then

lim

|v| → ∞

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. Thenfn → 0 weakly inLw,0.

In this paper, we consider the multiplication operators betweenLandLw, 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 betweenLwandL. 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

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 quantitiesτψ and

σψ.

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

aMψ :Lw → Lis bounded.

bMψ :Lw,0 → L0is bounded.

cτψ andσψare finite.

Furthermore, under these conditions, we have

maxτψ, σψ

≤ Mψ ≤τψ σψ. 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

Dψf v Dψvfv ψv− Dfv, 2.4

one has

Dψvfv≤Dψf v ψv− Dfv

≤Mψf_L σψ|v|Dfv≤Mψ σψ.

2.5

Henceτψ <∞.

c⇒aAssumeτψandσψ are finite. Then, byLemma 1.3, forf∈ Lwandv∈T∗, we

have

Dψf v≤Dψvfv ψv− Dfv

≤Dψv1 log|v| f_w |v|σψDfv

≤τψf_w σψf_w−fo .

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 :Lw,0 → L0is bounded. The finiteness ofσψfollows again from

the fact thatψ Mψ1 ∈ L0and fromLemma 1.1. To prove thatτψ < ∞, let 0< α < 1 and,

forv ∈T, definefαv log1 |v|α. Thenfαo 0 and|v|Dfav → 0 as|v| → ∞; so

fα ∈ Lw,0. Since for 0 < α < 1, the functionx→ x−xαis increasing forx ≥1, the function

Dfαvis increasing inα, andDfαv≤Dfvforv∈T∗, wherefv log1 |v|, forv∈T.

Thus,fαw≤ fw1. Therefore, forv∈T∗, we have

Dψvfαv≤D

ψfα v ψ

v− Dfαv

≤Mψfα σψ|v|Dfαv≤Mψ σψ.

2.8

Lettingα → 1, we obtain

Dψvlog1 |v|≤Mψ σψ. 2.9

Henceτψ <∞.

c⇒bAssumeσψandτψare finite, and letf∈ Lw,0. Then, byLemma 1.3, forv∈T∗,

we have

Dψf v≤Dψvfv ψv− 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 ofMψ and the estimateMψfL≤τψ σψ can

be shown as in the proof ofc⇒a.

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

Forv∈ T∗, letfv 1/|v| 1χv, whereχvdenotes the characteristic function of{v}. Then

fvw1 and

ψfv_L

_ψv

|v| 1. 2.11

Furthermore, lettingfo 1/2χo, we see thatfo_w1 andψfoL |ψo|. Therefore, we

Next, fixv∈T∗and forw∈T, define

gvw ⎧ ⎨ ⎩

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

log1 |v| if|w| ≥ |v|.

2.12

Then,gv ∈ Lwand

lim

|v| → ∞gvw|v| → ∞lim |v|

log1 |v|−log|v|1. _2.13

Observe that, forw∈T∗, we have

Dψgv 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/gvw. Thenfvw1 and

Mψ≥Mψfv_L

Dψgv _∞

_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 operatorsMψ 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 ψL− |ψo| 0, which

implies thatψis a constant of modulus 1. Yet, forv∈T∗, lettingfv 1/|v| 1χv, we see

that

1fv_wMψfv_{L |} 1

v| 1, 2.17

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

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 fromLworLw,0toL.

Lemma 2.3. A bounded multiplication operatorMψ 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 Lw 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|ψfnk−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 inLwwithgn_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−gnkv<

ε

2|v|, gnk

v− −gv− < ε

2|v|. 2.20

We deduce

|v|Dgv≤ |v|gv−gnkv gnk

v− −gv− |v|Dgnkv

≤ |v|gv−gnkv |v|gnk

v− −gv− |v|Dgnkv

< ε |v|Dgnkv≤ε 1,

2.21

for allksufficiently large. Sog ∈ Lw. The sequence defined byfk gnk −g is bounded in

Lwand converges to 0 pointwise. Thus by hypothesis, we obtainψfk_L → 0 ask → ∞. It

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

fromLw,0toL0.

Lemma 2.4. A bounded multiplication operatorMψ 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 Lw to L and from Lw,0 to L0

Theorem 2.6.

Lemma 2.5. Forf∈ Lwand v∈T

fv≤fo 2 log1 |v|swf , 2.22

whereswf 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− fv−

≤ _|1

v|sw

f fo 2 log|v|swf

fo

1

|v| 2 log|v|

swf .

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:

aMψ :Lw → Lis compact.

bMψ :Lw,0 → L0is compact.

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 limn→ ∞Dψvnlog|vn| 0. Let{vn} be such a sequence, and for eachn ∈ , define

fn 1/|vn| 1χvn. Thenfno 0,fn → 0 pointwise asn → ∞, and fnw 1. By

Lemma 2.3, it follows thatψfnL → 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| 1

2

|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

{fn}is bounded inLwconverging to 0 pointwise, thenψfnL → 0 asn → ∞. Let{fn}be

such a sequence,ssup_n∈fnw, and fixε >0. Note that

lim

|v| → ∞Dψvlog1 |v| |v| → ∞lim 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

Dψfn v≤Dψvfnv Dfn

v− ψv−

≤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

Dψfn v≤Dψvfnv ψ

v− Dfnv

≤ψ_Lfnv max

|w|≤MψwDfnv−→0,

2.34

uniformly on BM. 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 functionsfnandgndefined 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

Forψa function onT, define the quantities

Aψ lim

n→ ∞_|v|≥nsup

_ψv

|v| 1,

Bψ lim

n→ ∞_|v|≥nsupDψvlog|v|.

2.36

Theorem 2.7. LetMψ be a bounded multiplication operator fromLwtoL. Then

_Mψ

e≥max

Aψ , Bψ . 2.37

Proof. For eachn∈, definefn 1/n 1χn, whereχndenotes 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

Mψfn_Lsup

|v|n

_ψv

n 1 . 2.39

Hence

_Mψ

e≥infMψ−K:K∈ K

≥lim sup

n→ ∞

Mψfn_L

lim

n→ ∞sup_|v|≥n

_ψv

|v| 1

Aψ .

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

lim

n→ ∞Dψvnlog|vn|B

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| ≤

hnwsn≤

log|vn| −1|vn|

log|vn|

<2. 2.45

By lettinggn hn/hnw, we havegn ∈ Lw,0,gnw 1, andgn → 0 pointwise. By

Lemma 1.4, the sequence{gn}converges to 0 weakly inLw,0. ThusKgnL → 0 asn → ∞.

Therefore,

Mψ−K≥lim sup

n→ ∞

Mψ−K gn_L≥lim sup

n→ ∞

ψgn_L. _2.46

For eachn∈, we havegnvn gnv −

n log|vn|/sn. So

Dψgn vn 1

snDψvnlog|vn|.

2.47

Since limn→ ∞sn1, we have

_Mψ

e≥infMψ−K:K∈ K

≥lim sup

n→ ∞ supv∈T∗D

ψgn v

≥ lim

n→ ∞

1

sn

Dψvnlog|vn|

Bψ .

2.48

We now derive an upper estimate on the essential norm.

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

Mψ_e≤A

ψ Bψ . 2.49

Proof. Forn∈, define the operatorKnonLwby

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 ∈ Lw,0. LetBn {v ∈ T : |v| ≤ n}, and note that Knf attains finitely many values,

whose number does not exceed the cardinality ofBn. 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{Kngk}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 toginLwas well, proving thatKnis a compact operator onLw. SinceMψ is bounded as an

operator fromLwtoL, it follows thatMψKn: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|DJnf v |v|Jnf v−

Jnf v− ≤ |v|Dfv≤f_w. 2.51

By partaofLemma 1.3, we see that

_{Jnf v≤}

Using2.51and2.52, we obtain

Mψ−MψKn f_LψJnf_L

sup

|v|>n

_ψv

Jnf v−ψ

v− Jnf v−

≤sup

|v|>n

_{Jnf v}Dψv ψv− DJnf 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|Dψv

1 logn

logn sup_|v|≥n

ψv

|v| 1

_f

w.

2.53

Since

Mψ_e≤lim sup

n→ ∞

Mψ−MψKnlim sup

n→ ∞

sup

fw1

Mψ−MψKn f_L, _2.54

taking the limit asn → ∞, we obtain

Mψ_e ≤B

ψ Aψ , 2.55

as desired.

3. Multiplication Operators from

L

to

L

We begin this section with a boundedness criterion for the multiplication operators fromMψ :

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|

2_Dψv,

ωψ sup

v∈T|v| 1

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

aMψ :L → Lwis bounded.

bMψ :L0 → Lw,0is bounded.

cθψ andωψare finite.

Furthermore, under the above conditions, one has

maxθψ, ωψ

≤Mψ≤θψ ωψ. 3.2

Proof. a⇒cAssumeMψ is bounded fromLtoLw. The functionfo 1/2χo ∈ Land

fo_L1. Thus

ψoψfow≤ Mψ. 3.3

Next, fixv∈T∗. Thenχv ∈ LandχvL1; so

|v| 1ψvψχv_w≤Mψ. 3.4

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

ωψ ≤Mψ. 3.5

Withv∈T∗, we now define

fvw ⎧ ⎨ ⎩

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

|v| if |w| ≥ |v|.

3.6

Thenfv∈ L,fvo 0 andfv_L1. By the boundedness ofMψ we obtain

_Mψ≥Mψfv

w

≥ sup

1≤|w|≤|v||w|

_{ψw|w| −ψ}

w− |w| −1

≥ sup

1≤|w|≤|v||w|

2_{Dψw− sup}

1≤|w|≤|v||w|

_ψ

w− .

Therefore,

|v|2Dψv≤ sup

1≤|w|≤|v||w|

2_Dψw≤M

ψ ωψ. _3.8

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

lower estimate

Mψ≥max

θψ, ωψ

. 3.9

c⇒aAssumeθψandωψare finite. Then,ψ∈ Lw, and byLemma 1.1, forf ∈ Lwith

fL 1 andv∈T∗, we have

|v|Dψf v≤ |v|Dψvfv |v|ψv− Dfv

≤ |v|Dψvfo |v|2DψvDf_∞ ωψDf_∞

≤ |v|Dψvfo θψ ωψ Df_∞.

3.10

Thus,ψf∈ Lw. Note that|fo| Df∞1 and

_ψ

wψo sup

v∈T∗|v|Dψv≤ωψ sup_v∈T∗|v|

2_{Dψv ω}

ψ θψ. _3.11

From this, we have

_ψf

w≤ψwfo

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⇒bAssumeθψ andωψare finite, and letf∈ L0. Then, byLemma 1.1, forv∈T∗,

we have

|v|Dψf v≤ |v|Dψvfv |v|ψv− Dfv

≤ |v|2Dψvfv

|v| |v|ψ

v− Dfv

≤θψ

_fv

|v| ωψDfv−→0

3.14

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

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

2ψχow 1

2χoL1. 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. Sinceχv ∈ L0for allv∈T, ifMψ :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 fromLtoLwor fromL0toLw,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 fromLtoLwor fromL0toLw,0.

Lemma 3.3. A bounded multiplication operatorMψ fromLtoLw(L0 toLw,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. SupposeMψis compact fromLtoLwand{fn}is a bounded sequence inLconverging

to 0 pointwise. Without loss of generality, we may assumefn_L ≤1 for alln∈. SinceMψ

is compact, the sequence{ψfn}has a subsequence{ψfnk}that converges in theLw-norm to

some functionf∈ Lw.

ByLemma 1.3, forv∈T∗we have

_{ψvfnkv−fv≤}

1 log|v| ψfnk−f_w. 3.17

Thus, ψfnk → 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

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−gnkv gnk

v− −gv− Dgnkv< ε Dgnkv. 3.18

Thusg ∈ L. The sequencefnk gnk −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 → Lw,0is similar.

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

toLw,0). Then the following are equivalent:

aMψ :L → Lwis compact.

bMψ :L0 → Lw,0is compact.

clim|v| → ∞|v|2Dψv 0 and lim|v| → ∞|v| 1|ψv|0.

Proof. a⇒c Suppose Mψ : L → Lw is compact. We need to show that if {vn} is

a sequence in T such that 2 ≤ |vn|increasing unboundedly, then limn→ ∞|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 limn→ ∞|vn| 1|ψvn|0.

Next, forn∈, define

gnv ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

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 gn → 0 pointwise, and

gn_L 2. Sincegnvn gnvn− |vn|, we have

_ψgn

w≥ |vn|ψvngnvn−ψ

v−_n gn

v−_n |vn|2Dψvn. 3.22

ByLemma 3.3we obtain limn→ ∞|vn|2Dψvn≤limn→ ∞ψgnw0.

c⇒aSuppose lim|v| → ∞|v|2Dψv 0 and lim|v| → ∞|v| 1|ψv|0. Assumeψis

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∈fnL, and fix

ε >0. There existsM∈ such that|v| 1|ψv|< ε/2sand|v|

2_{Dψv< ε/2sfor|v| ≥M.}

Forv∈T∗and byLemma 1.1, we have

|v|Dψfn v≤ |v|ψvDfnv |v|Dψvfn

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|Dψfn 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

Aψ lim

n→ ∞sup_|v|≥n|v|ψv,

Bψ lim

n→ ∞sup_|v|≥n|v|

2_Dψv.

3.25

Theorem 3.5. LetMψ be a bounded multiplication operator fromLtoLw. Then

_Mψ

e≥max

Aψ ,1

2B

ψ . 3.26

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

En,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

Therefore, ifK∈ K0, then

Mψ−K≥lim sup

n→ ∞

Mψ−K fn,k_w

≥lim sup

n→ ∞

_Mψfn,k

w

≥lim sup

n→ ∞ v∈En,ksup|v| 1

_ψv.

3.28

Similarly,

Mψ−K≥lim sup

n→ ∞

sup

v∈On,k|v| 1

ψv. _3.29

Therefore, combining3.28and3.29, we obtain

_Mψ

einfMψ−K:K∈ K0

≥lim sup

n→ ∞

sup

kn≥|v|≥n|v| 1

_ψv

≥lim sup

n→ ∞ kn≥|v|≥nsup |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

lim

n→ ∞|vn|

2_Dψv

n B

ψ . 3.31

Forn∈, define

hnv ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0 if vo,

|v| 12 |vn|

if 1≤ |v|<|vn|,

|vn| if |v| ≥ |vn|.

3.32

Clearly,hno 0, hnvn hnv−n |vn|, and

Dhnv ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 4

|vn| if |v|1,

2|v| 1 |vn|

if 1<|v|<|vn|,

0 if |v| ≥ |vn|.

The supremum ofDhnvis attained on the set{v∈T :|v||vn| −1}. Thushn_L 2|vn| −

1/|vn|<2. Definegnhn/hnL, and observe thatgn∈ L0,gnL1, andgn → 0 pointwise

onT. ByLemma 1.2,gn → 0 weakly inL0. ThusKgn_w → 0 asn → ∞for anyK∈ K0.

For eachn∈,gnvn gnv −

n |vn|2/2|vn| −1. Thus

|vn|D

ψgn vn |vn|ψvngnvn−ψ

v−_n gn

v−_n

|vn|

2|vn| −1|

vn|2Dψvn.

3.34

We deduce that

_Mψ

einfMψ−K:K∈ K0

≥lim sup

n→ ∞

_{Mψ −K gn}

w

≥lim sup

n→ ∞

_Mψgn

w

≥ lim

n→ ∞sup_v∈T∗|v|D

ψgn v

≥ lim

n→ ∞|vn|D

ψgn vn

lim

n→ ∞

|vn|

2|vn| −1|

vn|2Dψvn≥ 1

2B

ψ .

3.35

Therefore,

_Mψ

e≥max

Aψ ,1

2B

ψ . 3.36

We next derive an upper estimate on the essential norm.

Theorem 3.6. LetMψ be a bounded multiplication operator fromLtoLw. Then

_Mψ

e≤ A

ψ Bψ . 3.37

Proof. For eachn∈, consider the operatorKndefined by

Knf v ⎧ ⎨ ⎩

fv if |v| ≤n,

fvn if |v|> n,

3.38

forf ∈ L, wherevn is the ancestor ofvof lengthn. ThenKnfo fo, andKnf ∈ Lw,0.

Arguing as in the proof ofTheorem 2.8, by the boundedness ofMψ, it follows thatMψKnis

Define the operatorJnI−Kn, whereIis the identity operatorIonL. Then,

DJnf v≤Dfv≤f_L. 3.39

SinceJnfv 0 for|v| ≤n, byLemma 1.1, we obtain

_{Jnf v≤ |v|f}

L. 3.40

From these two estimates, we arrive at

_MψJnf

wsup

|v|>n|v|

_ψv

Jnf v−ψ

v− Jnf v−

≤sup

|v|>n

|v|DψvJnf v |v|ψ

v− DJnf v

≤sup

|v|>n|v|

2_DψvJnf v

|v| sup_|v|>n|v|ψ

v− DJnf v

≤sup

|v|>n|v|

2_Dψvf

L sup

|v|>n|v|

_ψ

v− f_L.

3.41

Since

_Mψ

e≤lim sup

n→ ∞

_Mψ−MψKn

lim sup

n→ ∞

sup

fL1

_{Mψ−MψKn f}

w

lim sup

n→ ∞ fsupL1

_MψJnf

w,

3.42

from3.41, taking the limit asn → ∞, we obtain

_Mψ

e≤ B

ψ Aψ , 3.43

as desired.

4. Multiplication Operators from

L

or

L

to

L

In this section, we study the multiplication operatorsMψ from the weighted Lipschitz space

4.1. Boundedness and Operator Norm

For a functionψonT, define

γψ max

_ψo,sup

v∈T∗

1 log|v| ψv

. 4.1

Theorem 4.1. For a functionψonT, the following statements are equivalent:

aMψ :Lw → L∞is bounded.

bMψ :Lw,0 → L∞is bounded.

csup_v∈T∗log|vψv|is finite.

Furthermore, under the above conditions, one hasMψγψ.

Proof. The implicationa⇒bis obvious.

b⇒aWe begin by showing that, for eachf∈ Lw, the functionψfis bounded. Since

Mψ is bounded onLw,0,ψ Mψ1∈L∞. Thus, iffis constant, thenψf ∈L∞. Fixf ∈ Lw,f

nonconstant,v∈T, and setn|v|. Forw∈T, define

fnw ⎧ ⎨ ⎩

fw if|w| ≤n,

fwn if|w|> n,

4.2

wherewnis the ancestor ofwof lengthn. Thenfn∈ Lw,0, andfn_w≤ f_w. Thus,ψfn∈L∞

and

ψfn_∞≤Mψf_w. 4.3

So|ψvfv||ψvfnv| ≤ Mψf_w. Therefore,ψf∈L∞and

ψf_∞≤Mψf_w, 4.4

proving the boundedness ofMψ as an operator fromLwtoL∞.

a⇒cAssumeMψ :Lw → L∞is bounded. ThenψMψ1∈L∞, and

_Mψ≥ψ

∞≥ψo. 4.5

Forv ∈ T, definefv log1 |v|. Thenfo 0, and since forx ≥ 1 the functionx → xlogx 1/xis increasing and has limit 1 asx → ∞,f∈ Lwandf_w1. Thus

Mψ≥ψf_∞sup

v∈T∗log1 |v|

ψv, _4.6

provingc. Furthermore, from4.5and4.6, we obtain

c⇒a Assume sup_v∈T∗log|v||ψv| < ∞. Let f ∈ Lw such that f_w 1. Then

|ψofo| ≤ |ψo|, and byLemma 1.3, forv∈T∗, we have

ψvfv≤1 log|v| ψv≤γψ. 4.8

Thus,ψf∈L∞and

_ψf

∞≤γψ, 4.9

proving the boundedness ofMψas an operator fromLwtoL∞. Taking the supremum over all

functionsf ∈ Lwsuch thatf_w1, from4.9we obtainMψ ≤γψ. Therefore, from4.7

we conclude thatMψγψ.

4.2. Boundedness from Below

Recall that an operatorSfrom a Banach spaceXto a Banach spaceYis bounded below if there exists a constantC >0 such that for all x∈X

Sx ≥Cx. 4.10

Theorem 4.2. A bounded multiplication operatorMψ fromLw orLw,0 toL∞is bounded below if and only if

inf

v∈T

_ψv

|v| 1 >0. 4.11

Proof. AssumeMψ is bounded below, and, arguing by contradiction, assume there existsv∈

T such thatψv 0. ThenMψχv is identically 0. Since operators that are bounded below

are necessarily injective 8, it follows that Mψ is not bounded below. Therefore, if Mψ is

bounded below, thenψis nonvanishing.

Next assume ψ is nonvanishing and infv∈T|ψv|/|v| 1 0. Then, there exists

a sequence{vn}inT with 1 ≤ |vn| → ∞, such that |ψvn|/|vn| 1→ 0 asn → ∞. For

n∈, definefn 1/|vn| 1χvn. Thenfnw1, but

ψfn∞

ψvn

|vn| 1 −→0.

4.12

Thus,Mψ is not bounded below.

Conversely, assume infv∈T|ψv|/|v| 1 c >0 and thatMψ is not bounded below.

Then, for eachn∈, there existsfn ∈ Lwsuch thatfnw 1 andψfn∞ <1/n. Then, for

eachv∈T, we have

c|v| 1fnv≤ψvfnv< 1

so that the sequence{gn}defined bygnv |v| 1fnvconverges to 0 uniformly.

On the other hand, forv∈T∗, we have

|v|Dfnv

_|v|_|v|₁gnv−gn

v−

≤gnv gn

v− −→0

4.14

uniformly asn → ∞. Since|ψofno|<1/n, yetfnw1, this yields a contradiction.

4.3. Isometries

In this section, we show there are no isometries among the multiplication operators from the spacesLworLw,0intoL∞.

SupposeMψ is an isometry fromLworLw,0toL∞. Then, forv∈T the functionfv

1/|v| 1χvis inLw,0,fv_w1, and

1

|v| 1ψvMψfv∞fvw1. 4.15

Thus,|ψv| |v| 1. On the other hand, sinceMψ is bounded, by Theorem 4.1, we have

sup_v∈T∗log|v||ψv|<∞; soψv → 0 as|v| → ∞, which yields a contradiction. Thus, we

proved the following result.

Theorem 4.3. There are no isometric multiplication operatorsMψfromLworLw,0toL∞.

4.4. Compactness and Essential Norm

We begin by giving a useful compactness criterion for the bounded operators fromLw or

Lw,0intoL∞.

Lemma 4.4. A bounded multiplication operatorMψfromLwtoL∞is compact if and only if for every

bounded sequence{fn}inLwconverging to 0 pointwise, the sequenceψfn∞approaches 0 asn →

∞.

Proof. AssumeMψ is compact onLw, and let{fn}be a bounded sequence inLwconverging

to 0 pointwise. By rescaling the sequence, if necessary, we may assumefnw≤1 for alln∈.

By the compactness ofMψ,{fn}has a subsequence{fnk}such that{ψfnk}converges in the

norm to some functionf ∈L∞. In particular,ψfnk → fpointwise. Since by assumption,fn →

0 pointwise, it follows that f must be identically 0. Thus, the only limit point of sequence {ψfn}inL∞is 0. Henceψfn∞ → 0.

Conversely, assume that for every bounded sequence {fn} in Lw converging to 0

pointwise, the sequence ψfn_∞ approaches 0 asn → ∞. Let {gn} be a sequence in Lw

with gnw ≤ 1. Fix w ∈ T, and, by replacing gn with gn −gnw, assume gnw 0

for all n ∈ . Then, for each v ∈ T, |gnv| |gnv−gnw| ≤ dv, w. Therefore, gn

pointwise to some function g onT. Fix ε > 0 and v ∈ T∗. Then, |go−gnko| < ε/2,

|gnkv−gv|< ε/2|v|, and|gnkv−−gv−|< ε/2|v|for allksufficiently large. Thus,

|v|Dgv≤ |v|gv−gv− −gnkv−gnk

v− |v|Dgnkv

< ε |v|Dgnkv,

4.16

forksufficiently large. Consequently,g ∈ Lw, we have

_g

wgo sup

v∈T∗|v|Dgv

≤go−gnko gnko ε sup

v∈T∗Dgnkv

<2ε gnk_w≤2ε 1.

4.17

Sinceεwas arbitrary, it follows thatg_w ≤1. Therefore, the sequence{fk}defined byfk

gnk−gis bounded inLwand converges to 0 pointwise, hence, by the hypothesis,ψfk_∞ → 0

asn → ∞. We conclude thatψgnk → ψginL∞, proving the compactness ofMψ.

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

Mψ :Lw,0 → L∞.

Lemma 4.5. A bounded multiplication operatorMψ fromLw,0 toL∞ is compact if and only if for

every bounded sequence{fn}inLw,0 converging to 0 pointwise, the sequenceψfn∞approaches 0

asn → ∞.

Theorem 4.6. For a bounded operatorMψ fromLw toL∞ (or equivalently fromLw,0 toL∞) the following statements are equivalent:

aMψ :Lw → L∞is compact.

bMψ :Lw,0 → L∞is compact.

clim|v| → ∞log|v||ψv|0.

Proof. a⇒bis trivial.

b⇒c: Let{vn}be a sequence of vertices such that 1≤ |vn| → ∞asn → ∞. We need

to show that

lim

n→ ∞log|vn|ψvn0. 4.18

Forn∈ define

fnv ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0 ifvo,

log|v| 2

log|vn|

if 1≤ |v|<|vn|,

log|vn| if|v| ≥ |vn|.

Then{fn}converges to 0 pointwise. Using the fact that|v| −1log|v| −log|v| −1≤1 for

any choice ofvinT∗with|v|>1, we have

|v|Dfnv _| |v|

v| −1

|v| −1log|v| 2−log|v| −1 2

log|vn| ≤

2log|v| log|v| −1 log|vn| ≤4,

4.20

for 2 ≤ |v| ≤ |vn|. Moreover,|v|Dfnv 0 for|v| 1 and for|v| > |vn|. Thus,fn ∈ Lw,0,

and{fn_w}is bounded. By the compactness ofMψ as an operator fromLw,0 toL∞and by

Lemma 4.5, we deduce

log|vn|ψvn≤ψfn_∞−→0 4.21

asn → ∞.

c⇒aAssume{fn}is a sequence inLwconverging to 0 pointwise and such thata

sup_n∈fnw<∞. ByLemma 1.3, for allv∈T

∗_{and alln∈}

, we have

_ψvfnv≤a

1 log|v| ψv. 4.22

Fixε >0. There existsN ∈such thatN≥3, and for|v| ≥N, log|v||ψv|< ε/2a. Thus, for

|v| ≥Nand for alln∈,|ψvfnv| ≤2alog|v||ψv|< ε. On the other hand, sincefn → 0

pointwise, for each vertexvsuch that|v|< Nandψv/0, we obtain|fnv|< ε/|ψv|for all

nsufficiently large. Hence|ψvfnv|< εfor allv∈Tand allnsufficiently large. Therefore,

Mψfn_∞ → 0 asn → ∞, which, byLemma 4.4, proves the compactness ofMψ.

Next, we determine the essential norm of the bounded multiplication operatorsMψ

fromLworLw,0toL∞.

Theorem 4.7. LetMψ be a bounded multiplication operator fromLworLw,0toL∞. Then

Mψ_e lim

n→ ∞sup_|v|≥nlog|v|ψv. 4.23

Proof. DefineAψ limn→ ∞sup|v|≥nlog|v||ψv|. IfAψ 0, then byTheorem 4.6,Mψ is

compact, hence its essential norm is 0. So assumeAψ>0. We first show thatMψ_e ≥Aψ.

Let{vn}be a sequence inTsuch that 1≤ |vn| → ∞asn → ∞and

Aψ lim

n→ ∞log|vn|ψvn. 4.24

Fixp∈0,1, and for eachn∈, define

fn,pv ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0 ifvo,

log|v| p 1

log|vn|

p if 1≤ |v|<|vn|,

log|vn| if|v| ≥ |vn|.

Then{fn,p}converges to 0 pointwise,fn,p ∈ Lw,0,fn,pvn log|vn|, and

_fn,p

w sup

2≤|v|≤|vn|

|v|

log|vn| p

log|v| p 1−log|v| −1 p 1

|vn|

log|vn| p

log|vn| p 1

−log|vn| −1

p 1

≤p 1.

4.26

ByLemma 1.4,{fn,p}converges to 0 weakly inLw,0. LetKbe a compact operator fromLw,0

or equivalently, from Lw to L∞. Since compact operators are completely continuous, it

follows thatKfn,p_∞ → 0 asn → ∞. Thus,

_{Mψ−K≥lim sup}

n→ ∞

Mψ−K fn,p_∞

fn,pw

≥ 1

p 1lim sup_{n→ ∞} Mψfn,p∞

≥ 1

p 1lim supn→ ∞ log|vn|

_ψvn.

4.27

Taking the infimum over all such compact operators K and passing to the limit as p ap-proaches 0, we obtain