# Multiplication Operators between Lipschitz-Type Spaces on a Tree

Authors: Robert F. Allen; Flavia Colonna; Glenn R. Easley
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

Letℒ be the space of complex-valued functions f on the set of vertices T of an infinite tree rooted at o such that the difference of the values of f at neighboring vertices remains bounded throughout the tree, and let ℒw be the set of functions f∈ℒ such that |f(v)-f(v-)|=O(|v|-1), where |v| is the distance between o and v and v- is the neighbor of v closest to o. In this paper, we characterize the bounded and the compact multiplication operators between ℒ and ℒw and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between ℒw and the space L∞ 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.


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## Body

## 1. Introduction

Let𝒳 and 𝒴 be complex Banach spaces of functions defined on a set Ω. For a complex-valued function ψ defined on Ω, the multiplication operator with symbolψfrom𝒳to𝒴 is defined as
(1.1)Mψf=ψf,∀f∈X.
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.WhenΩ is taken to be the open unit disk 𝔻 in the complex plane, an important space of functions to study is the Bloch space, defined as the set ℬ of analytic functions f:𝔻→ℂ for which
(1.2)βf=sup⁡z∈D(1-|z|2)|f′(z)|<∞.The Bloch space can also be described as the set consisting of the Lipschitz functions between metric spaces from𝔻 endowed with the Poincaré distance ρ to ℂ endowed with the Euclidean distance, a fact that was proved by the second author in [1] (see also [2]). In fact, f∈ℬ if and only if there exist β>0 such that for all z,w∈𝔻(1.3)|f(z)-f(w)|≤βρ(z,w),βf=sup⁡z≠w|f(z)-f(w)|ρ(z,w).More recently, considerable research has been carried out in the field of operator theory when the setΩ is taken to be a discrete structure, such as a discrete group or a graph. In this paper, we consider the case when Ω is taken to be an infinite tree.By atreeT we mean a locally finite, connected, and simplyconnected graph, which, as a set, we identify with the collection of its vertices. Two vertices u and v are called neighbors if there is an edge connecting them, and we use the notation u~v. A vertex is called terminal if it has a unique neighbor. A path is a finite or infinite sequence of vertices [v0,v1,…] such that vk~vk+1 and vk-1≠vk+1, for all k.Given a treeT rooted at o and a vertex u∈T, a vertex v is called a descendant of u if u lies in the unique path from o to v. The vertex u is then called an ancestor of v. Given a vertex v≠o, we denote by v- the unique neighbor which is an ancestor of v. For v∈T, the set Sv consisting of v and all its descendants is called the sector determined by v.Define thelength of a finite path [u=u0,u1,…,v=un] (with uk~uk+1 for k=0,…,n) to be the number n of edges connecting u to v. The distance, d(u,v), between vertices u and v is the length of the path connecting u to v. The tree T is a metric space under the distance d. Fixing o as the root of the tree, we define the length of a vertex v by |v|=d(o,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 vertexo and without terminal vertices (and hence infinite).Infinite trees are discrete structures which exhibit significant geometric and potential theoretic characteristics that are present in the Poincaré disk𝔻. 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 is [3]. In [4] it 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 𝔻 with the same hyperbolic length and the set of cluster points of the vertices is the entire unit circle.In [5], the last two authors defined the Lipschitz spaceℒ on a tree T as the set consisting of the functions f:T→ℂ which are Lipschitz with respect to the distance d on T and the Euclidean distance on ℂ. For this reason, the Lipschitz space ℒ can be viewed as a discrete analogue of the Bloch space ℬ. It was also shown that the Lipschitz functions on T are precisely the functions for which
(1.4)‖Df‖∞=sup⁡v∈T*Df(v)<∞,
where Df(v)=|f(v)-f(v-)| and T*=T∖{o}. Under the norm
(1.5)‖f‖L=|f(o)|+‖Df‖∞,ℒ is a Banach space containing the space L∞ of the bounded functions on T. Furthermore, for f∈L∞, ∥f∥ℒ≤2∥f∥∞.Thelittle Lipschitz space is defined as
(1.6)L0={f∈L:lim⁡|v|→∞Df(v)=0}
and was proven to be a separable closed subspace of ℒ. We state the following results that will be useful in the present paper.Lemma 1.1 (see [5, Lemma  3.4]).
(a) Iff∈ℒ and v∈T, then
(1.7)|f(v)|≤|f(o)|+|v|‖Df‖∞.
In particular, if ∥f∥ℒ≤1, then |f(v)|≤|v| for each v∈T*.
(b) Iff∈ℒ0, then
(1.8)lim⁡|v|→∞f(v)|v|=0.Lemma 1.2 (see [5, Proposition  2.4]).
Let{fn} be a sequence of functions in ℒ0 converging to 0 pointwise in T such that {∥fn∥ℒ} is bounded. Then fn→0 weakly in ℒ0.In [6], we introduced the weighted Lipschitz space on a tree T as the set ℒw of the functions f:T→ℂ such that
(1.9)sup⁡v∈T*|v|Df(v)<∞.
The interest in this space is due to its connection to the bounded multiplication operators on ℒ. Specifically, it was shown in [5] that the bounded multiplication operators on ℒ are precisely those operators Mψ whose symbol ψ is a bounded function in ℒw. The space ℒw was shown to be a Banach space under the norm
(1.10)‖f‖w=|f(o)|+sup⁡v∈T*|v|Df(v).Thelittle weighted Lipschitz space was defined as
(1.11)Lw,0={f∈Lw:lim⁡|v|→∞|v|Df(v)=0}
and was shown to be a closed separable subspace of ℒw.In this paper, we will make repeated use of the following results proved in [6].Lemma 1.3 (see [6, Propositions  2.1 and  2.6]).
(a) Iff∈ℒw, and v∈T*, then
(1.12)|f(v)|≤(1+log⁡|v|)‖f‖w.
(b) Iff∈ℒw,0, then
(1.13)lim⁡|v|→∞f(v)log⁡|v|=0.Lemma 1.4 (see [6, Proposition   2.7]).
Let{fn} be a sequence of functions in  ℒw,0 converging to 0 pointwise in T such that  {∥fn∥w} is bounded. Then fn→0 weakly in  ℒw,0.In this paper, we consider the multiplication operators betweenℒ and ℒw, as well as between ℒw and L∞. The multiplication operators between ℒ and L∞ were studied by the last two authors in [7].


### 1.1. Organization of the Paper

In Sections2 and 3, we study the multiplication operators between ℒw and ℒ. 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.In Section4, we characterize the bounded operators and the compact operators from ℒw to L∞ and determine their operator norm and their essential norm. As was the case in Sections 2 and 3, we show that no isometries exist amongst such operators. In addition, we characterize the multiplication operators that are bounded from below.Finally, in Section5, we characterize the bounded and the compact multiplication operators from L∞ to ℒw. 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.


## 1.1. Organization of the Paper

In Sections2 and 3, we study the multiplication operators between ℒw and ℒ. 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.In Section4, we characterize the bounded operators and the compact operators from ℒw to L∞ and determine their operator norm and their essential norm. As was the case in Sections 2 and 3, we show that no isometries exist amongst such operators. In addition, we characterize the multiplication operators that are bounded from below.Finally, in Section5, we characterize the bounded and the compact multiplication operators from L∞ to ℒw. 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 fromℒw to ℒ

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


### 2.1. Boundedness and Operator Norm Estimates

Letψ be a function on the tree T. Define
(2.1)τψ=sup⁡v∈T*Dψ(v)log⁡(1+|v|),σψ=sup⁡v∈T|ψ(v)||v|+1.
In the following theorem, we give a boundedness criterion in terms of the quantities τψ and σψ.Theorem 2.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:ℒw→ℒ is bounded.
(b)Mψ:ℒw,0→ℒ0 is bounded.
(c)τψ and σψ are finite.
Furthermore, under these conditions, we have(2.2)max⁡{τψ,σψ}≤∥Mψ∥≤τψ+σψ.Proof.
(a)⇒(c) Assume Mψ:ℒw→ℒ is bounded. Applying Mψ to the constant function 1, we have ψ∈ℒ, so that, by Lemma 1.1, we have σψ<∞. Next, consider the function f on T defined by f(v)=log(1+|v|). Then f(o)=0; for v≠o, a straightforward calculation shows that
(2.3)|v|Df(v)=|v|(log⁡(1+|v|)-log⁡|v|)≤1
and lim|v|→∞|v|Df(v)=1. Thus, ∥f∥w=1 and so ∥Mψf∥ℒ≤∥Mψ∥. Therefore, for v∈T*, noting that
(2.4)D(ψf)(v)=Dψ(v)f(v)+ψ(v-)Df(v),
one has
(2.5)Dψ(v)|f(v)|≤D(ψf)(v)+|ψ(v-)|Df(v)≤‖Mψf‖L+σψ|v|Df(v)≤‖Mψ‖+σψ.
Hence τψ<∞.
(c)⇒(a) Assume τψ and σψ are finite. Then, by Lemma 1.3, for f∈ℒw and v∈T*, we have
(2.6)D(ψf)(v)≤Dψ(v)|f(v)|+|ψ(v-)|Df(v)≤Dψ(v)(1+log⁡|v|)‖f‖w+|v|σψDf(v)≤τψ‖f‖w+σψ(‖f‖w-|f(o)|).
Since |ψ(o)|≤σψ, we obtain
(2.7)‖Mψf‖L≤|ψ(o)||f(o)|+τψ‖f‖w+σψ(‖f‖w-|f(o)|)=(τψ+σψ)‖f‖w+(|ψ(o)|-σψ)|f(o)|≤(τψ+σψ)‖f‖w,
proving the boundedness of Mψ:ℒw→ℒ and the upper estimate.
(b)⇒(c) Suppose Mψ:ℒw,0→ℒ0 is bounded. The finiteness of σψ follows again from the fact that ψ=Mψ1∈ℒ0 and from Lemma 1.1. To prove that τψ<∞, let 0<α<1 and, for v∈T, define fα(v)=(log(1+|v|))α. Then fα(o)=0 and |v|Dfa(v)→0 as |v|→∞; so fα∈ℒw,0. Since for 0<α<1, the function x↦x-xα is increasing for x≥1, the function Dfα(v) is increasing in α, and Dfα(v)≤Df(v) for v∈T*, where f(v)=log(1+|v|), for v∈T. Thus, ∥fα∥w≤∥f∥w=1. Therefore, for v∈T*, we have
(2.8)Dψ(v)|fα(v)|≤D(ψfα)(v)+|ψ(v-)|Dfα(v)≤‖Mψfα‖+σψ|v|Dfα(v)≤‖Mψ‖+σψ.
Letting α→1, we obtain
(2.9)Dψ(v)log⁡(1+|v|)≤‖Mψ‖+σψ.
Hence τψ<∞.
(c)⇒(b) Assume σψ and τψ are finite, and let f∈ℒw,0. Then, by Lemma 1.3, for v∈T*, we have
(2.10)D(ψf)(v)≤Dψ(v)|f(v)|+|ψ(v-)|Df(v)≤Dψ(v)log⁡(1+|v|)|f(v)|log⁡(1+|v|)+|ψ(v-)||v||v|Df(v)≤τψ|f(v)|log⁡(1+|v|)+σψ|v|Df(v)⟶0
as |v|→∞. Thus, ψf∈ℒ0. The boundedness of Mψ and the estimate ∥Mψf∥ℒ≤τψ+σψ can be shown as in the proof of (c)⇒(a).
Finally we show that, under boundedness assumptions onMψ, ∥Mψ∥≥max{τψ,σψ}. For v∈T*, let fv=1/(|v|+1)χv, where χv denotes the characteristic function of {v}. Then ∥fv∥w=1 and
(2.11)‖ψfv‖L=|ψ(v)||v|+1.
Furthermore, letting fo=(1/2)χo, we see that ∥fo∥w=1 and ∥ψfo∥ℒ=|ψ(o)|. Therefore, we deduce that ∥Mψ∥≥σψ.
Next, fixv∈T* and for w∈T, define
(2.12)gv(w)={log⁡(1+|w|)if|w|<|v|,log⁡(1+|v|)if|w|≥|v|.
Then, gv∈ℒw and
(2.13)lim⁡|v|→∞‖gv‖w=lim⁡|v|→∞|v|[log⁡(1+|v|)-log⁡|v|]=1.
Observe that, for w∈T*, we have
(2.14)D(ψgv)(w)={|ψ(w)log⁡(1+|w|)-ψ(w-)log⁡|w||if|w|<|v|,Dψ(w)log⁡(1+|v|)if|w|≥|v|.
Hence
(2.15)sup⁡w∈T*D(ψgv)(w)≥sup⁡|w|≥|v|Dψ(w)log⁡(1+|v|)≥Dψ(v)log⁡(1+|v|).
Define fv=gv/∥gv∥w. Then ∥fv∥w=1 and
(2.16)‖Mψ‖≥‖Mψfv‖L=‖D(ψgv)‖∞‖gv‖w≥Dψ(v)log⁡(1+|v|)‖gv‖w.
Taking the limit as |v|→∞, we obtain ∥Mψ∥≥τψ. Therefore, ∥Mψ∥≥max{τψ,σψ}.


### 2.2. Isometries

In this section, we show there are no isometric multiplication operatorsMψ from the spaces ℒw or ℒw,0 to the spaces ℒ or ℒ0, respectively.AssumeMψ:ℒw→ℒ is an isometry. Then ∥ψ∥ℒ=∥Mψ1∥ℒ=1. On the other hand, |ψ(o)|=(1/2)∥Mψχo∥ℒ=(1/2)∥χo∥w=1. Thus supv∈T*Dψ(v)=∥ψ∥ℒ-|ψ(o)|=0, which implies that ψ is a constant of modulus 1. Yet, for v∈T*, letting fv=(1/(|v|+1))χv, we see that
(2.17)1=‖fv‖w=‖Mψfv‖L=1|v|+1,
which yields a contradiction. Therefore, we obtain the following result.Theorem 2.2.
There are no isometriesMψ from  ℒw to ℒ or  ℒw,0 to ℒ0, 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 compact operators fromℒw or ℒw,0 to ℒ.Lemma 2.3.
A bounded multiplication operatorMψ from  ℒw to ℒ is compact if and only if for every bounded sequence {fn} in ℒw converging to 0 pointwise, the sequence {∥ψfn∥ℒ}→0 as n→∞.Proof.
AssumeMψ is compact, and let {fn} be a bounded sequence in ℒw converging to 0 pointwise. Without loss of generality, we may assume ∥fn∥w≤1 for all n∈ℕ. Then the sequence {Mψfn}={ψfn} has a subsequence {ψfnk} which converges in the ℒ-norm to some function f∈ℒ. Clearly ψ(o)fnk(o)→ψ(o)f(o), and by part (a) of Lemma 1.1, for v∈T*, we have
(2.18)|ψ(v)fnk(v)-f(v)|≤|ψ(o)fnk(o)-f(o)|+|v|‖D(ψfnk-f)‖∞≤(1+|v|)‖ψfnk-f‖L.
Thus, ψfnk→f pointwise on T. Since fn→0 pointwise, it follows that f must be identically 0, which implies that ∥ψfnk∥ℒ→0. With 0 being the only limit point of {ψfn} in ℒ, it follows that ∥ψfn∥ℒ→0 as n→∞.
Conversely, assume every bounded sequence{fn} in ℒw converging to 0 pointwise has the property that ∥ψfn∥ℒ→0 as n→∞. Let {gn} be a sequence in ℒw with ∥gn∥w≤1 for all n∈ℕ. Then |gn(o)|≤1 for all n∈ℕ, and by part (a) of Lemma 1.2, for v∈T*, we obtain
(2.19)|gn(v)|≤(1+log⁡|v|)‖gn‖w≤1+log⁡|v|.
Thus, {gn} is uniformly bounded on finite subsets of T. So some subsequence {gnk} converges pointwise to some function g. Fix v∈T* and ɛ>0. Then for k sufficiently large, we have
(2.20)|g(v)-gnk(v)|<ɛ2|v|,|gnk(v-)-g(v-)|<ɛ2|v|.
We deduce
(2.21)|v|Dg(v)≤|v||g(v)-gnk(v)+gnk(v-)-g(v-)|+|v|Dgnk(v)≤|v||g(v)-gnk(v)|+|v||gnk(v-)-g(v-)|+|v|Dgnk(v)<ɛ+|v|Dgnk(v)≤ɛ+1,
for all k sufficiently large. So g∈ℒw. The sequence defined by fk=gnk-g is bounded in ℒw and converges to 0 pointwise. Thus by hypothesis, we obtain ∥ψfk∥ℒ→0 as k→∞. It follows that Mψgnk=ψgnk→ψg in the ℒ-norm, thus proving the compactness of Mψ.By an analogous argument, we obtain the corresponding compactness criterion forMψ from  ℒw,0 to ℒ0.Lemma 2.4.
A bounded multiplication operatorMψ from  ℒw,0 to ℒ0 is compact if and only if for every bounded sequence {fn} in  ℒw,0 converging to 0 pointwise, the sequence {∥ψfn∥ℒ}→0 as n→∞.The following result is a variant of Lemma1.3(a), which will be needed to prove a characterization of the compact multiplication operators from ℒw to ℒ and from ℒw,0 to ℒ0 (Theorem 2.6).Lemma 2.5.
Forf∈ℒw andv∈T(2.22)|f(v)|≤|f(o)|+2log⁡(1+|v|)sw(f),
where sw(f)=supw∈T*|w|Df(w).Proof.
Fixv∈T and argue by induction on n=|v|. For n=0, inequality (2.22) is obvious. So assume |v|=n>0 and |f(u)|≤|f(o)|+2log(1+|u|)sw(f) for all vertices u such that |u|<n. Then
(2.23)|f(v)|≤|f(v)-f(v-)|+|f(v-)|≤1|v|sw(f)+|f(o)|+2log⁡|v|sw(f)=|f(o)|+(1|v|+2log⁡|v|)sw(f).
Next, observe that 1/(|v|+1)≤log((|v|+1)/|v|), so
(2.24)1|v|≤2|v|+1≤2log⁡(|v|+1|v|).
Hence
(2.25)1|v|+2log⁡|v|≤2log⁡(|v|+1).
Inequality (2.22) now follows immediately from (2.23) and (2.25).Theorem 2.6.
LetMψ be a bounded multiplication operator from  ℒw to ℒ (or equivalently from  ℒw,0 to ℒ0). Then the following statements are equivalent:
(a)Mψ:ℒw→ℒ is compact.
(b)Mψ:ℒw,0→ℒ0 is compact.
(c)lim|v|→∞|ψ(v)|/(|v|+1)=0 and lim|v|→∞Dψ(v)log|v|=0.Proof.
We first prove (a)⇒(c). Assume Mψ:ℒw→ℒ is compact. It suffices to show that, for any sequence {vn} in T such that 2≤|vn|→∞, we have limn→∞|ψ(vn)|/(|vn|+1)=0 and limn→∞Dψ(vn)log|vn|=0. Let {vn} be such a sequence, and for each n∈ℕ, define fn=(1/(|vn|+1))χvn. Then fn(o)=0, fn→0 pointwise as n→∞, and ∥fn∥w=1. By Lemma 2.3, it follows that ∥ψfn∥ℒ→0 as n→∞. Furthermore
(2.26)‖ψfn‖L=sup⁡v∈T*|ψ(v)fn(v)-ψ(v-)fn(v-)|=|ψ(vn)fn(vn)|=|ψ(vn)||vn|+1.
Thus limn→∞|ψ(vn)|/(|vn|+1)=0.
Next, for eachn∈ℕ and v∈T, define
(2.27)gn(v)={0if|v|<|vn|,2log⁡|v|-log⁡|vn|if|vn|≤|v|<|vn|-1,log⁡|vn|if|v|≥|vn|-1.
Then Dgn(v)=0 if |v|≤|vn| or |v|>|vn|-1. In addition, if |vn|<|v|≤|vn|-1, then |v|Dgn(v)<4. Indeed, there are two possibilities. Either |vn|≤|v|-1, in which case
(2.28)|v|Dgn(v)=2|v|(log⁡|v|-log⁡(|v|-1))≤2|v||v|-1≤3,
or |v|-1<|vn|<|v|, in which case
(2.29)|v|Dgn(v)=|v|(2log⁡|v|-log⁡|vn|)≤(|vn|+1)log⁡(|vn|+1)2|vn|≤2(|vn|+1)|vn|≤2(1+12)<4.
Thus {∥gn∥w} is bounded, and {gn} converges to 0 pointwise. By Lemma 2.3, it follows that ∥ψgn∥ℒ→0 as n→∞. Moreover
(2.30)‖ψgn‖L≥|ψ(vn)gn(vn)-ψ(vn-)gn(vn-)|=Dψ(vn)log⁡|vn|.
Therefore limn→∞Dψ(vn)log|vn|=0.
To prove the implication (c)⇒(a), suppose lim|v|→∞Dψ(v)log|v|=0 and lim|v|→∞|ψ(v)|/(|v|+1)=0. Clearly, if ψ is identically 0, then Mψ is compact. So assume Mψ:ℒw→ℒ is bounded with ψ not identically 0. By Lemma 2.3, it suffices to show that if {fn} is bounded in ℒw converging to 0 pointwise, then ∥ψfn∥ℒ→0 as n→∞. Let {fn} be such a sequence, s=supn∈ℕ∥fn∥w, and fix ɛ>0. Note that
(2.31)lim⁡|v|→∞Dψ(v)log⁡(1+|v|)=lim⁡|v|→∞Dψ(v)log⁡|v|log⁡(1+|v|)log⁡|v|=0.
Thus there exists an M∈ℕ such that
(2.32)|fn(o)|<ɛ3s‖ψ‖L,Dψ(v)log⁡(1+|v|)<ɛ6s,|ψ(v)||v|+1<ɛ3s,
for |v|≥M. Using Lemma 2.5, for |v|>M, we have
(2.33)D(ψfn)(v)≤Dψ(v)|fn(v)|+Dfn(v-)|ψ(v-)|≤Dψ(v)(|fn(o)|+2log⁡(|v|+1))‖fn‖w+‖fn‖w|ψ(v-)||v|≤(‖ψ‖L|fn(o)|+2Dψ(v)log⁡(|v|+1)+|ψ(v-)||v|)‖fn‖w<ɛ.
On the other hand, on the setBM={v∈T:|v|≤M}, {fn} converges to 0 uniformly, and thus Dfn does as well. Moreover
(2.34)D(ψfn)(v)≤Dψ(v)|fn(v)|+|ψ(v-)|Dfn(v)≤‖ψ‖L|fn(v)|+max⁡|w|≤M|ψ(w)|Dfn(v)⟶0,
uniformly on BM. Therefore D(ψfn)→0 uniformly on T. Furthermore, the sequence {(ψfn)(o)} converges to 0 as n→∞. Hence ∥ψfn∥ℒ→0 as n→∞, proving that Mψ is compact.
Finally, note that the functionsfn and gn defined in the proof of (a)⇒(c) are in ℒw,0. So the equivalence of (b) and (c) is proved analogously.Recall theessential norm of a bounded operator S between Banach spaces 𝒳 and 𝒴 is defined as
(2.35)‖S‖e=inf⁡{‖S-K‖:KiscompactfromXtoY}.
For ψ a function on T, define the quantities
(2.36)A(ψ)=lim⁡n→∞sup⁡|v|≥n|ψ(v)||v|+1,B(ψ)=lim⁡n→∞sup⁡|v|≥nDψ(v)log⁡|v|.Theorem 2.7.
LetMψ be a bounded multiplication operator from  ℒw to ℒ. Then
(2.37)‖Mψ‖e≥max⁡{A(ψ),B(ψ)}.Proof.
For eachn∈ℕ, define fn=(1/(n+1))χn, where χn denotes the characteristic function of the set {v∈T:|v|=n}. Then fn∈ℒw,0, ∥fn∥w=1, and fn→0 pointwise. Thus, by Lemma 1.4, {fn} converges to 0 weakly in ℒw,0. Let 𝒦 be the set of compact operators from ℒw,0 to ℒ0, and let K∈𝒦. Then K is completely continuous [8], and so ∥Kfn∥ℒ→0 as n→∞. Thus
(2.38)∥Mψ-K∥≥limsup⁡n→∞‖(Mψ-K)fn‖L≥limsup⁡n→∞‖Mψfn‖L.
Now note that
(2.39)‖Mψfn‖L=sup⁡|v|=n|ψ(v)|n+1.
Hence
(2.40)‖Mψ‖e≥inf⁡{‖Mψ-K‖:K∈K}≥limsup⁡n→∞‖Mψfn‖L=lim⁡n→∞sup⁡|v|≥n|ψ(v)||v|+1=A(ψ).
We will now show that ∥Mψ∥e≥B(ψ). This estimate is clearly true if B(ψ)=0. So assume {vn} is a sequence in T such that 2≤|vn|→∞ as n→∞ and
(2.41)lim⁡n→∞Dψ(vn)log⁡|vn|=B(ψ).
For n∈ℕ and v∈T, define
(2.42)hn(v)={[log⁡(|v|+1)]2log⁡|vn|if0≤|v|<|vn|,log⁡|vn|if|v|≥|vn|.
Then hn(o)=0, hn(vn)=hn(vn-)=log|vn|, and
(2.43)|v|Dhn(v)={|v|log⁡|vn|log⁡(|v|+1|v|)log⁡[|v|(|v|+1)]if1≤|v|<|vn|,0if|v|≥|vn|.
The supremum of|v|Dhn(v) is attained at the vertices of length |vn|-1 and is given by
(2.44)sn=sup⁡v∈T*|v|Dhn(v)=(|vn|-1)log⁡(|vn||vn|-1)log⁡[(|vn|-1)|vn|]log⁡|vn|.
Since (|vn|-1)log(|vn|/(|vn|-1))≤1, we have
(2.45)(log⁡2)2log⁡|vn|≤‖hn‖w=sn≤log⁡[(|vn|-1)|vn|]log⁡|vn|<2.
By lettinggn=hn/∥hn∥w, we have gn∈ℒw,0, ∥gn∥w=1, and gn→0 pointwise. By Lemma 1.4, the sequence {gn} converges to 0 weakly in ℒw,0. Thus ∥Kgn∥ℒ→0 as n→∞. Therefore,
(2.46)‖Mψ-K‖≥limsup⁡n→∞‖(Mψ-K)gn‖L≥limsup⁡n→∞‖ψgn‖L.
For eachn∈ℕ, we have gn(vn)=gn(vn-)=log|vn|/sn. So
(2.47)D(ψgn)(vn)=1snDψ(vn)log⁡|vn|.
Since limn→∞sn=1, we have
(2.48)‖Mψ‖e≥inf⁡{‖Mψ-K‖:K∈K}≥limsup⁡n→∞sup⁡v∈T*D(ψgn)(v)≥lim⁡n→∞1snDψ(vn)log⁡|vn|=B(ψ).
Therefore, ∥Mψ∥e≥max{A(ψ),B(ψ)}.We now derive an upper estimate on the essential norm.Theorem 2.8.
LetMψ be a bounded multiplication operator from ℒw to ℒ. Then
(2.49)‖Mψ‖e≤A(ψ)+B(ψ).Proof.
Forn∈ℕ, define the operator Kn on ℒw by
(2.50)(Knf)(v)={f(v)if|v|≤n,f(vn)if|v|>n,
where f∈ℒw and vn is the ancestor of v of length n. For f∈ℒw, (Knf)(o)=f(o), and Knf∈ℒw,0. Let Bn={v∈T:|v|≤n}, and note that Knf attains finitely many values, whose number does not exceed the cardinality of Bn. Let {gk} be a sequence in ℒw such that ∥gk∥w≤1 for each k∈ℕ. Then a=supk∈ℕ|gk(o)|≤1, and |Kngk(o)|≤a. Furthermore, by part (a) of Lemma 1.3, for each v∈T* and for each k∈ℕ, we have |Kngk(v)|≤1+logn. Thus, some subsequence of {Kngk}k∈ℕ must converge to a function g on T attaining constant values on the sectors determined by the vertices of length n. It follows that this subsequence converges to g in ℒw as well, proving that Kn is a compact operator on ℒw. Since Mψ is bounded as an operator from ℒw to ℒ, it follows that MψKn:ℒw→ℒ is compact for all n∈ℕ.
Define the operatorJn=I-Kn, where I denotes the identity operator on ℒw. Then Jnf(o)=0, and for v∈T*, we have
(2.51)|v|D(Jnf)(v)=|v||(Jnf)(v)-(Jnf)(v-)|≤|v|Df(v)≤‖f‖w.
By part (a) of Lemma 1.3, we see that
(2.52)|(Jnf)(v)|≤(1+log⁡|v|)‖f‖w.
Using (2.51) and (2.52), we obtain
(2.53)‖(Mψ-MψKn)f‖L=‖(ψJnf)‖L=sup⁡|v|>n|ψ(v)(Jnf)(v)-ψ(v-)(Jnf)(v-)|≤sup⁡|v|>n[|(Jnf)(v)|Dψ(v)+|ψ(v-)|D(Jnf)(v)]=sup⁡|v|>n[|(Jnf)(v)|Dψ(v)+|ψ(v-)||v||v|D(Jnf)(v)]≤sup⁡|v|≥n[(1+log⁡|v|)Dψ(v)+|ψ(v)||v|+1]‖f‖w≤sup⁡|v|≥n[log⁡|v|Dψ(v)1+log⁡|v|log⁡|v|+|ψ(v)||v|+1]‖f‖w≤[sup⁡|v|≥nlog⁡|v|Dψ(v)1+log⁡nlog⁡n+sup⁡|v|≥n|ψ(v)||v|+1]‖f‖w.
Since
(2.54)‖Mψ‖e≤limsup⁡n→∞‖Mψ-MψKn‖=limsup⁡n→∞sup⁡∥f∥w=1‖(Mψ-MψKn)f‖L,
taking the limit as n→∞, we obtain
(2.55)‖Mψ‖e≤B(ψ)+A(ψ),
as desired.


## 2.1. Boundedness and Operator Norm Estimates

Letψ be a function on the tree T. Define
(2.1)τψ=sup⁡v∈T*Dψ(v)log⁡(1+|v|),σψ=sup⁡v∈T|ψ(v)||v|+1.
In the following theorem, we give a boundedness criterion in terms of the quantities τψ and σψ.Theorem 2.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:ℒw→ℒ is bounded.
(b)Mψ:ℒw,0→ℒ0 is bounded.
(c)τψ and σψ are finite.
Furthermore, under these conditions, we have(2.2)max⁡{τψ,σψ}≤∥Mψ∥≤τψ+σψ.Proof.
(a)⇒(c) Assume Mψ:ℒw→ℒ is bounded. Applying Mψ to the constant function 1, we have ψ∈ℒ, so that, by Lemma 1.1, we have σψ<∞. Next, consider the function f on T defined by f(v)=log(1+|v|). Then f(o)=0; for v≠o, a straightforward calculation shows that
(2.3)|v|Df(v)=|v|(log⁡(1+|v|)-log⁡|v|)≤1
and lim|v|→∞|v|Df(v)=1. Thus, ∥f∥w=1 and so ∥Mψf∥ℒ≤∥Mψ∥. Therefore, for v∈T*, noting that
(2.4)D(ψf)(v)=Dψ(v)f(v)+ψ(v-)Df(v),
one has
(2.5)Dψ(v)|f(v)|≤D(ψf)(v)+|ψ(v-)|Df(v)≤‖Mψf‖L+σψ|v|Df(v)≤‖Mψ‖+σψ.
Hence τψ<∞.
(c)⇒(a) Assume τψ and σψ are finite. Then, by Lemma 1.3, for f∈ℒw and v∈T*, we have
(2.6)D(ψf)(v)≤Dψ(v)|f(v)|+|ψ(v-)|Df(v)≤Dψ(v)(1+log⁡|v|)‖f‖w+|v|σψDf(v)≤τψ‖f‖w+σψ(‖f‖w-|f(o)|).
Since |ψ(o)|≤σψ, we obtain
(2.7)‖Mψf‖L≤|ψ(o)||f(o)|+τψ‖f‖w+σψ(‖f‖w-|f(o)|)=(τψ+σψ)‖f‖w+(|ψ(o)|-σψ)|f(o)|≤(τψ+σψ)‖f‖w,
proving the boundedness of Mψ:ℒw→ℒ and the upper estimate.
(b)⇒(c) Suppose Mψ:ℒw,0→ℒ0 is bounded. The finiteness of σψ follows again from the fact that ψ=Mψ1∈ℒ0 and from Lemma 1.1. To prove that τψ<∞, let 0<α<1 and, for v∈T, define fα(v)=(log(1+|v|))α. Then fα(o)=0 and |v|Dfa(v)→0 as |v|→∞; so fα∈ℒw,0. Since for 0<α<1, the function x↦x-xα is increasing for x≥1, the function Dfα(v) is increasing in α, and Dfα(v)≤Df(v) for v∈T*, where f(v)=log(1+|v|), for v∈T. Thus, ∥fα∥w≤∥f∥w=1. Therefore, for v∈T*, we have
(2.8)Dψ(v)|fα(v)|≤D(ψfα)(v)+|ψ(v-)|Dfα(v)≤‖Mψfα‖+σψ|v|Dfα(v)≤‖Mψ‖+σψ.
Letting α→1, we obtain
(2.9)Dψ(v)log⁡(1+|v|)≤‖Mψ‖+σψ.
Hence τψ<∞.
(c)⇒(b) Assume σψ and τψ are finite, and let f∈ℒw,0. Then, by Lemma 1.3, for v∈T*, we have
(2.10)D(ψf)(v)≤Dψ(v)|f(v)|+|ψ(v-)|Df(v)≤Dψ(v)log⁡(1+|v|)|f(v)|log⁡(1+|v|)+|ψ(v-)||v||v|Df(v)≤τψ|f(v)|log⁡(1+|v|)+σψ|v|Df(v)⟶0
as |v|→∞. Thus, ψf∈ℒ0. The boundedness of Mψ and the estimate ∥Mψf∥ℒ≤τψ+σψ can be shown as in the proof of (c)⇒(a).
Finally we show that, under boundedness assumptions onMψ, ∥Mψ∥≥max{τψ,σψ}. For v∈T*, let fv=1/(|v|+1)χv, where χv denotes the characteristic function of {v}. Then ∥fv∥w=1 and
(2.11)‖ψfv‖L=|ψ(v)||v|+1.
Furthermore, letting fo=(1/2)χo, we see that ∥fo∥w=1 and ∥ψfo∥ℒ=|ψ(o)|. Therefore, we deduce that ∥Mψ∥≥σψ.
Next, fixv∈T* and for w∈T, define
(2.12)gv(w)={log⁡(1+|w|)if|w|<|v|,log⁡(1+|v|)if|w|≥|v|.
Then, gv∈ℒw and
(2.13)lim⁡|v|→∞‖gv‖w=lim⁡|v|→∞|v|[log⁡(1+|v|)-log⁡|v|]=1.
Observe that, for w∈T*, we have
(2.14)D(ψgv)(w)={|ψ(w)log⁡(1+|w|)-ψ(w-)log⁡|w||if|w|<|v|,Dψ(w)log⁡(1+|v|)if|w|≥|v|.
Hence
(2.15)sup⁡w∈T*D(ψgv)(w)≥sup⁡|w|≥|v|Dψ(w)log⁡(1+|v|)≥Dψ(v)log⁡(1+|v|).
Define fv=gv/∥gv∥w. Then ∥fv∥w=1 and
(2.16)‖Mψ‖≥‖Mψfv‖L=‖D(ψgv)‖∞‖gv‖w≥Dψ(v)log⁡(1+|v|)‖gv‖w.
Taking the limit as |v|→∞, we obtain ∥Mψ∥≥τψ. Therefore, ∥Mψ∥≥max{τψ,σψ}.


## 2.2. Isometries

In this section, we show there are no isometric multiplication operatorsMψ from the spaces ℒw or ℒw,0 to the spaces ℒ or ℒ0, respectively.AssumeMψ:ℒw→ℒ is an isometry. Then ∥ψ∥ℒ=∥Mψ1∥ℒ=1. On the other hand, |ψ(o)|=(1/2)∥Mψχo∥ℒ=(1/2)∥χo∥w=1. Thus supv∈T*Dψ(v)=∥ψ∥ℒ-|ψ(o)|=0, which implies that ψ is a constant of modulus 1. Yet, for v∈T*, letting fv=(1/(|v|+1))χv, we see that
(2.17)1=‖fv‖w=‖Mψfv‖L=1|v|+1,
which yields a contradiction. Therefore, we obtain the following result.Theorem 2.2.
There are no isometriesMψ from  ℒw to ℒ or  ℒw,0 to ℒ0, 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 compact operators fromℒw or ℒw,0 to ℒ.Lemma 2.3.
A bounded multiplication operatorMψ from  ℒw to ℒ is compact if and only if for every bounded sequence {fn} in ℒw converging to 0 pointwise, the sequence {∥ψfn∥ℒ}→0 as n→∞.Proof.
AssumeMψ is compact, and let {fn} be a bounded sequence in ℒw converging to 0 pointwise. Without loss of generality, we may assume ∥fn∥w≤1 for all n∈ℕ. Then the sequence {Mψfn}={ψfn} has a subsequence {ψfnk} which converges in the ℒ-norm to some function f∈ℒ. Clearly ψ(o)fnk(o)→ψ(o)f(o), and by part (a) of Lemma 1.1, for v∈T*, we have
(2.18)|ψ(v)fnk(v)-f(v)|≤|ψ(o)fnk(o)-f(o)|+|v|‖D(ψfnk-f)‖∞≤(1+|v|)‖ψfnk-f‖L.
Thus, ψfnk→f pointwise on T. Since fn→0 pointwise, it follows that f must be identically 0, which implies that ∥ψfnk∥ℒ→0. With 0 being the only limit point of {ψfn} in ℒ, it follows that ∥ψfn∥ℒ→0 as n→∞.
Conversely, assume every bounded sequence{fn} in ℒw converging to 0 pointwise has the property that ∥ψfn∥ℒ→0 as n→∞. Let {gn} be a sequence in ℒw with ∥gn∥w≤1 for all n∈ℕ. Then |gn(o)|≤1 for all n∈ℕ, and by part (a) of Lemma 1.2, for v∈T*, we obtain
(2.19)|gn(v)|≤(1+log⁡|v|)‖gn‖w≤1+log⁡|v|.
Thus, {gn} is uniformly bounded on finite subsets of T. So some subsequence {gnk} converges pointwise to some function g. Fix v∈T* and ɛ>0. Then for k sufficiently large, we have
(2.20)|g(v)-gnk(v)|<ɛ2|v|,|gnk(v-)-g(v-)|<ɛ2|v|.
We deduce
(2.21)|v|Dg(v)≤|v||g(v)-gnk(v)+gnk(v-)-g(v-)|+|v|Dgnk(v)≤|v||g(v)-gnk(v)|+|v||gnk(v-)-g(v-)|+|v|Dgnk(v)<ɛ+|v|Dgnk(v)≤ɛ+1,
for all k sufficiently large. So g∈ℒw. The sequence defined by fk=gnk-g is bounded in ℒw and converges to 0 pointwise. Thus by hypothesis, we obtain ∥ψfk∥ℒ→0 as k→∞. It follows that Mψgnk=ψgnk→ψg in the ℒ-norm, thus proving the compactness of Mψ.By an analogous argument, we obtain the corresponding compactness criterion forMψ from  ℒw,0 to ℒ0.Lemma 2.4.
A bounded multiplication operatorMψ from  ℒw,0 to ℒ0 is compact if and only if for every bounded sequence {fn} in  ℒw,0 converging to 0 pointwise, the sequence {∥ψfn∥ℒ}→0 as n→∞.The following result is a variant of Lemma1.3(a), which will be needed to prove a characterization of the compact multiplication operators from ℒw to ℒ and from ℒw,0 to ℒ0 (Theorem 2.6).Lemma 2.5.
Forf∈ℒw andv∈T(2.22)|f(v)|≤|f(o)|+2log⁡(1+|v|)sw(f),
where sw(f)=supw∈T*|w|Df(w).Proof.
Fixv∈T and argue by induction on n=|v|. For n=0, inequality (2.22) is obvious. So assume |v|=n>0 and |f(u)|≤|f(o)|+2log(1+|u|)sw(f) for all vertices u such that |u|<n. Then
(2.23)|f(v)|≤|f(v)-f(v-)|+|f(v-)|≤1|v|sw(f)+|f(o)|+2log⁡|v|sw(f)=|f(o)|+(1|v|+2log⁡|v|)sw(f).
Next, observe that 1/(|v|+1)≤log((|v|+1)/|v|), so
(2.24)1|v|≤2|v|+1≤2log⁡(|v|+1|v|).
Hence
(2.25)1|v|+2log⁡|v|≤2log⁡(|v|+1).
Inequality (2.22) now follows immediately from (2.23) and (2.25).Theorem 2.6.
LetMψ be a bounded multiplication operator from  ℒw to ℒ (or equivalently from  ℒw,0 to ℒ0). Then the following statements are equivalent:
(a)Mψ:ℒw→ℒ is compact.
(b)Mψ:ℒw,0→ℒ0 is compact.
(c)lim|v|→∞|ψ(v)|/(|v|+1)=0 and lim|v|→∞Dψ(v)log|v|=0.Proof.
We first prove (a)⇒(c). Assume Mψ:ℒw→ℒ is compact. It suffices to show that, for any sequence {vn} in T such that 2≤|vn|→∞, we have limn→∞|ψ(vn)|/(|vn|+1)=0 and limn→∞Dψ(vn)log|vn|=0. Let {vn} be such a sequence, and for each n∈ℕ, define fn=(1/(|vn|+1))χvn. Then fn(o)=0, fn→0 pointwise as n→∞, and ∥fn∥w=1. By Lemma 2.3, it follows that ∥ψfn∥ℒ→0 as n→∞. Furthermore
(2.26)‖ψfn‖L=sup⁡v∈T*|ψ(v)fn(v)-ψ(v-)fn(v-)|=|ψ(vn)fn(vn)|=|ψ(vn)||vn|+1.
Thus limn→∞|ψ(vn)|/(|vn|+1)=0.
Next, for eachn∈ℕ and v∈T, define
(2.27)gn(v)={0if|v|<|vn|,2log⁡|v|-log⁡|vn|if|vn|≤|v|<|vn|-1,log⁡|vn|if|v|≥|vn|-1.
Then Dgn(v)=0 if |v|≤|vn| or |v|>|vn|-1. In addition, if |vn|<|v|≤|vn|-1, then |v|Dgn(v)<4. Indeed, there are two possibilities. Either |vn|≤|v|-1, in which case
(2.28)|v|Dgn(v)=2|v|(log⁡|v|-log⁡(|v|-1))≤2|v||v|-1≤3,
or |v|-1<|vn|<|v|, in which case
(2.29)|v|Dgn(v)=|v|(2log⁡|v|-log⁡|vn|)≤(|vn|+1)log⁡(|vn|+1)2|vn|≤2(|vn|+1)|vn|≤2(1+12)<4.
Thus {∥gn∥w} is bounded, and {gn} converges to 0 pointwise. By Lemma 2.3, it follows that ∥ψgn∥ℒ→0 as n→∞. Moreover
(2.30)‖ψgn‖L≥|ψ(vn)gn(vn)-ψ(vn-)gn(vn-)|=Dψ(vn)log⁡|vn|.
Therefore limn→∞Dψ(vn)log|vn|=0.
To prove the implication (c)⇒(a), suppose lim|v|→∞Dψ(v)log|v|=0 and lim|v|→∞|ψ(v)|/(|v|+1)=0. Clearly, if ψ is identically 0, then Mψ is compact. So assume Mψ:ℒw→ℒ is bounded with ψ not identically 0. By Lemma 2.3, it suffices to show that if {fn} is bounded in ℒw converging to 0 pointwise, then ∥ψfn∥ℒ→0 as n→∞. Let {fn} be such a sequence, s=supn∈ℕ∥fn∥w, and fix ɛ>0. Note that
(2.31)lim⁡|v|→∞Dψ(v)log⁡(1+|v|)=lim⁡|v|→∞Dψ(v)log⁡|v|log⁡(1+|v|)log⁡|v|=0.
Thus there exists an M∈ℕ such that
(2.32)|fn(o)|<ɛ3s‖ψ‖L,Dψ(v)log⁡(1+|v|)<ɛ6s,|ψ(v)||v|+1<ɛ3s,
for |v|≥M. Using Lemma 2.5, for |v|>M, we have
(2.33)D(ψfn)(v)≤Dψ(v)|fn(v)|+Dfn(v-)|ψ(v-)|≤Dψ(v)(|fn(o)|+2log⁡(|v|+1))‖fn‖w+‖fn‖w|ψ(v-)||v|≤(‖ψ‖L|fn(o)|+2Dψ(v)log⁡(|v|+1)+|ψ(v-)||v|)‖fn‖w<ɛ.
On the other hand, on the setBM={v∈T:|v|≤M}, {fn} converges to 0 uniformly, and thus Dfn does as well. Moreover
(2.34)D(ψfn)(v)≤Dψ(v)|fn(v)|+|ψ(v-)|Dfn(v)≤‖ψ‖L|fn(v)|+max⁡|w|≤M|ψ(w)|Dfn(v)⟶0,
uniformly on BM. Therefore D(ψfn)→0 uniformly on T. Furthermore, the sequence {(ψfn)(o)} converges to 0 as n→∞. Hence ∥ψfn∥ℒ→0 as n→∞, proving that Mψ is compact.
Finally, note that the functionsfn and gn defined in the proof of (a)⇒(c) are in ℒw,0. So the equivalence of (b) and (c) is proved analogously.Recall theessential norm of a bounded operator S between Banach spaces 𝒳 and 𝒴 is defined as
(2.35)‖S‖e=inf⁡{‖S-K‖:KiscompactfromXtoY}.
For ψ a function on T, define the quantities
(2.36)A(ψ)=lim⁡n→∞sup⁡|v|≥n|ψ(v)||v|+1,B(ψ)=lim⁡n→∞sup⁡|v|≥nDψ(v)log⁡|v|.Theorem 2.7.
LetMψ be a bounded multiplication operator from  ℒw to ℒ. Then
(2.37)‖Mψ‖e≥max⁡{A(ψ),B(ψ)}.Proof.
For eachn∈ℕ, define fn=(1/(n+1))χn, where χn denotes the characteristic function of the set {v∈T:|v|=n}. Then fn∈ℒw,0, ∥fn∥w=1, and fn→0 pointwise. Thus, by Lemma 1.4, {fn} converges to 0 weakly in ℒw,0. Let 𝒦 be the set of compact operators from ℒw,0 to ℒ0, and let K∈𝒦. Then K is completely continuous [8], and so ∥Kfn∥ℒ→0 as n→∞. Thus
(2.38)∥Mψ-K∥≥limsup⁡n→∞‖(Mψ-K)fn‖L≥limsup⁡n→∞‖Mψfn‖L.
Now note that
(2.39)‖Mψfn‖L=sup⁡|v|=n|ψ(v)|n+1.
Hence
(2.40)‖Mψ‖e≥inf⁡{‖Mψ-K‖:K∈K}≥limsup⁡n→∞‖Mψfn‖L=lim⁡n→∞sup⁡|v|≥n|ψ(v)||v|+1=A(ψ).
We will now show that ∥Mψ∥e≥B(ψ). This estimate is clearly true if B(ψ)=0. So assume {vn} is a sequence in T such that 2≤|vn|→∞ as n→∞ and
(2.41)lim⁡n→∞Dψ(vn)log⁡|vn|=B(ψ).
For n∈ℕ and v∈T, define
(2.42)hn(v)={[log⁡(|v|+1)]2log⁡|vn|if0≤|v|<|vn|,log⁡|vn|if|v|≥|vn|.
Then hn(o)=0, hn(vn)=hn(vn-)=log|vn|, and
(2.43)|v|Dhn(v)={|v|log⁡|vn|log⁡(|v|+1|v|)log⁡[|v|(|v|+1)]if1≤|v|<|vn|,0if|v|≥|vn|.
The supremum of|v|Dhn(v) is attained at the vertices of length |vn|-1 and is given by
(2.44)sn=sup⁡v∈T*|v|Dhn(v)=(|vn|-1)log⁡(|vn||vn|-1)log⁡[(|vn|-1)|vn|]log⁡|vn|.
Since (|vn|-1)log(|vn|/(|vn|-1))≤1, we have
(2.45)(log⁡2)2log⁡|vn|≤‖hn‖w=sn≤log⁡[(|vn|-1)|vn|]log⁡|vn|<2.
By lettinggn=hn/∥hn∥w, we have gn∈ℒw,0, ∥gn∥w=1, and gn→0 pointwise. By Lemma 1.4, the sequence {gn} converges to 0 weakly in ℒw,0. Thus ∥Kgn∥ℒ→0 as n→∞. Therefore,
(2.46)‖Mψ-K‖≥limsup⁡n→∞‖(Mψ-K)gn‖L≥limsup⁡n→∞‖ψgn‖L.
For eachn∈ℕ, we have gn(vn)=gn(vn-)=log|vn|/sn. So
(2.47)D(ψgn)(vn)=1snDψ(vn)log⁡|vn|.
Since limn→∞sn=1, we have
(2.48)‖Mψ‖e≥inf⁡{‖Mψ-K‖:K∈K}≥limsup⁡n→∞sup⁡v∈T*D(ψgn)(v)≥lim⁡n→∞1snDψ(vn)log⁡|vn|=B(ψ).
Therefore, ∥Mψ∥e≥max{A(ψ),B(ψ)}.We now derive an upper estimate on the essential norm.Theorem 2.8.
LetMψ be a bounded multiplication operator from ℒw to ℒ. Then
(2.49)‖Mψ‖e≤A(ψ)+B(ψ).Proof.
Forn∈ℕ, define the operator Kn on ℒw by
(2.50)(Knf)(v)={f(v)if|v|≤n,f(vn)if|v|>n,
where f∈ℒw and vn is the ancestor of v of length n. For f∈ℒw, (Knf)(o)=f(o), and Knf∈ℒw,0. Let Bn={v∈T:|v|≤n}, and note that Knf attains finitely many values, whose number does not exceed the cardinality of Bn. Let {gk} be a sequence in ℒw such that ∥gk∥w≤1 for each k∈ℕ. Then a=supk∈ℕ|gk(o)|≤1, and |Kngk(o)|≤a. Furthermore, by part (a) of Lemma 1.3, for each v∈T* and for each k∈ℕ, we have |Kngk(v)|≤1+logn. Thus, some subsequence of {Kngk}k∈ℕ must converge to a function g on T attaining constant values on the sectors determined by the vertices of length n. It follows that this subsequence converges to g in ℒw as well, proving that Kn is a compact operator on ℒw. Since Mψ is bounded as an operator from ℒw to ℒ, it follows that MψKn:ℒw→ℒ is compact for all n∈ℕ.
Define the operatorJn=I-Kn, where I denotes the identity operator on ℒw. Then Jnf(o)=0, and for v∈T*, we have
(2.51)|v|D(Jnf)(v)=|v||(Jnf)(v)-(Jnf)(v-)|≤|v|Df(v)≤‖f‖w.
By part (a) of Lemma 1.3, we see that
(2.52)|(Jnf)(v)|≤(1+log⁡|v|)‖f‖w.
Using (2.51) and (2.52), we obtain
(2.53)‖(Mψ-MψKn)f‖L=‖(ψJnf)‖L=sup⁡|v|>n|ψ(v)(Jnf)(v)-ψ(v-)(Jnf)(v-)|≤sup⁡|v|>n[|(Jnf)(v)|Dψ(v)+|ψ(v-)|D(Jnf)(v)]=sup⁡|v|>n[|(Jnf)(v)|Dψ(v)+|ψ(v-)||v||v|D(Jnf)(v)]≤sup⁡|v|≥n[(1+log⁡|v|)Dψ(v)+|ψ(v)||v|+1]‖f‖w≤sup⁡|v|≥n[log⁡|v|Dψ(v)1+log⁡|v|log⁡|v|+|ψ(v)||v|+1]‖f‖w≤[sup⁡|v|≥nlog⁡|v|Dψ(v)1+log⁡nlog⁡n+sup⁡|v|≥n|ψ(v)||v|+1]‖f‖w.
Since
(2.54)‖Mψ‖e≤limsup⁡n→∞‖Mψ-MψKn‖=limsup⁡n→∞sup⁡∥f∥w=1‖(Mψ-MψKn)f‖L,
taking the limit as n→∞, we obtain
(2.55)‖Mψ‖e≤B(ψ)+A(ψ),
as desired.


## 3. Multiplication Operators fromℒ to ℒw

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


### 3.1. Boundedness and Operator Norm Estimates

Letψ be a function on the tree T. Define the quantities
(3.1)θψ=sup⁡v∈T*|v|2Dψ(v),ωψ=sup⁡v∈T(|v|+1)|ψ(v)|.Theorem 3.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:ℒ→ℒw is bounded.
(b)Mψ:ℒ0→ℒw,0 is bounded.
(c)θψ and ωψ are finite.
Furthermore, under the above conditions, one has(3.2)max⁡{θψ,ωψ}≤‖Mψ‖≤θψ+ωψ.Proof.
(a)⇒(c) Assume Mψ is bounded from ℒ to ℒw. The function fo=(1/2)χo∈ℒ and ∥fo∥ℒ=1. Thus
(3.3)|ψ(o)|=∥ψfo∥w≤∥Mψ∥.
Next, fixv∈T*. Then χv∈ℒ and ∥χv∥ℒ=1; so
(3.4)(|v|+1)|ψ(v)|=‖ψχv‖w≤‖Mψ‖.
Taking the supremum over all v∈T, from (3.3) and (3.4) we see that ωψ is finite and
(3.5)ωψ≤‖Mψ‖.
Withv∈T*, we now define
(3.6)fv(w)={|w|if|w|<|v|,|v|if|w|≥|v|.
Then fv∈ℒ, fv(o)=0 and ∥fv∥ℒ=1. By the boundedness of Mψ we obtain
(3.7)‖Mψ‖≥‖Mψfv‖w≥sup⁡1≤|w|≤|v||w||ψ(w)|w|-ψ(w-)(|w|-1)|≥sup⁡1≤|w|≤|v||w|2Dψ(w)-sup⁡1≤|w|≤|v||w||ψ(w-)|.
Therefore,
(3.8)|v|2Dψ(v)≤sup⁡1≤|w|≤|v||w|2Dψ(w)≤‖Mψ‖+ωψ.
Taking the supremum over all v∈T*, we obtain θψ<∞. From this and (3.5), we deduce the lower estimate
(3.9)‖Mψ‖≥max⁡{θψ,ωψ}.
(c)⇒(a) Assume θψ and ωψ are finite. Then, ψ∈ℒw, and by Lemma 1.1, for f∈ℒ with ∥f∥ℒ=1 and v∈T*, we have
(3.10)|v|D(ψf)(v)≤|v|Dψ(v)|f(v)|+|v||ψ(v-)|Df(v)≤|v|Dψ(v)|f(o)|+|v|2Dψ(v)‖Df‖∞+ωψ‖Df‖∞≤|v|Dψ(v)|f(o)|+(θψ+ωψ)‖Df‖∞.
Thus, ψf∈ℒw. Note that |f(o)|+∥Df∥∞=1 and
(3.11)‖ψ‖w=|ψ(o)|+sup⁡v∈T*|v|Dψ(v)≤ωψ+sup⁡v∈T*|v|2Dψ(v)=ωψ+θψ.
From this, we have
(3.12)‖ψf‖w≤‖ψ‖w|f(o)|+(θψ+ωψ)‖Df‖∞≤θψ+ωψ,
proving the boundedness of Mψ:ℒ→ℒw and the upper estimate
(3.13)‖Mψ‖≤θψ+ωψ.
(b)⇒(c) The proof is the same as for (a)⇒(c); since for v∈T*, the functions χv and fv used there belong to ℒ0.
(c)⇒(b) Assume θψ and ωψ are finite, and let f∈ℒ0. Then, by Lemma 1.1, for v∈T*, we have
(3.14)|v|D(ψf)(v)≤|v|Dψ(v)|f(v)|+|v||ψ(v-)|Df(v)≤|v|2Dψ(v)|f(v)||v|+|v||ψ(v-)|Df(v)≤θψ|f(v)||v|+ωψDf(v)⟶0
as |v|→∞. Thus, ψf∈ℒw,0. The proof of the boundedness of Mψ is similar to that in (c)⇒(a).


### 3.2. Isometries

In this section, we show there are no isometric multiplication operatorsMψ from the space ℒ to ℒw or from ℒ0 to ℒw,0.SupposeMψ:ℒ→ℒw is an isometry. Then ∥ψ∥w=∥Mψ1∥w=1. On the other hand,
(3.15)|ψ(o)|=12‖ψχo‖w=12‖χo‖L=1.
Thus supv∈T*|v|Dψ(v)=∥ψ∥w-|ψ(o)|=0, which implies that ψ is a constant of modulus 1. Now observe that, for v∈T*, we have
(3.16)1=‖χv‖L=‖Mψχ‖w=(|v|+1)|ψ(v)|=|v|+1,
which is a contradiction. Since χv∈ℒ0 for all v∈T, if Mψ:ℒ0→ℒw,0 is an isometry, then the above argument yields again a contradiction. Thus, we proved the following result.Theorem 3.2.
There are no isometriesMψ from ℒ to  ℒw or from ℒ0 to  ℒw,0.


### 3.3. Compactness and Essential Norm

We now characterize the compact multiplication operators, but first we give a useful compactness criterion for multiplication operators fromℒ to ℒw or from ℒ0 to ℒw,0.Lemma 3.3.
A bounded multiplication operatorMψ from ℒ to  ℒw (ℒ0 to  ℒw,0) is compact if and only if for every bounded sequence {fn} in ℒ (ℒ0) converging to 0 pointwise, the sequence  ∥ψfn∥w converges to 0 as n→∞.Proof.
SupposeMψ is compact from ℒ to ℒw and {fn} is a bounded sequence in ℒ converging to 0 pointwise. Without loss of generality, we may assume ∥fn∥ℒ≤1 for all n∈ℕ. Since Mψ is compact, the sequence {ψfn} has a subsequence {ψfnk} that converges in the ℒw-norm to some function f∈ℒw.
By Lemma1.3, for v∈T* we have
(3.17)|ψ(v)fnk(v)-f(v)|≤(1+log⁡|v|)‖ψfnk-f‖w.
Thus, ψfnk→f pointwise on T*. Furthermore, since |ψ(o)fnk(o)-f(o)|≤∥ψfnk-f∥w, ψ(0)fnk(0)→f(0) as k→∞. Thus ψfnk→f pointwise on T. Since by assumption, fn→0 pointwise, it follows that f is identically 0, and thus ∥ψfnk∥w→0. Since 0 is the only limit point in ℒw of the sequence {ψfn}, we deduce that ∥ψfn∥w→0 as n→∞.
Conversely, suppose that every bounded sequence{fn} in ℒ that converges to 0 pointwise has the property that ∥ψfn∥w→0 as n→∞. Let {gn} be a sequence in ℒ such that ∥gn∥ℒ≤1 for all n∈ℕ. Then |gn(o)|≤1, and by part (a) of Lemma 1.1, for v∈T* we have |gn(v)|≤|v|. So {gn} is uniformly bounded on finite subsets of T. Thus there is a subsequence {gnk}, which converges pointwise to some function g.
Fixɛ>0 and v∈T*. Then |gnk(v)-g(v)|<ɛ/2 as well as |gnk(v-)-g(v-)|<ɛ/2 for k sufficiently large. Therefore, for all k sufficiently large, we have
(3.18)Dg(v)≤|g(v)-gnk(v)|+|gnk(v-)-g(v-)|+Dgnk(v)<ɛ+Dgnk(v).
Thus g∈ℒ. The sequence fnk=gnk-g is bounded in ℒ and converges to 0 pointwise. So ∥ψfnk∥w→0 as k→∞. Thus ψgnk→ψg in the ℒw-norm. Therefore, Mψ is compact.
The proof for the case ofMψ:ℒ0→ℒw,0 is similar.Theorem 3.4.
LetMψ be a bounded multiplication operator from ℒ to  ℒw (or equivalently from ℒ0 to  ℒw,0). Then the following are equivalent:
(a)Mψ:ℒ→ℒw is compact.
(b)Mψ:ℒ0→ℒw,0 is compact.
(c)lim|v|→∞|v|2Dψ(v)=0 and lim|v|→∞(|v|+1)|ψ(v)|=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|2Dψ(vn)=0 and limn→∞(|vn|+1)|ψ(vn)|=0. Let {vn} be such a sequence, and for n∈ℕ define fn=((|vn|+1)/|vn|)χvn. Clearly fn→0 pointwise, and ∥fn∥ℒ≤3/2. Using Lemma 3.3, we see that
(3.19)‖ψfn‖w⟶0asn⟶∞.
On the other hand, sincefn(o)=0 for all n∈ℕ, we have
(3.20)‖ψfn‖w=sup⁡v∈T*|v|D(ψfn)(v)≥|vn|(|vn|+1|vn|)|ψ(vn)|=(|vn|+1)|ψ(vn)|.
Hence limn→∞(|vn|+1)|ψ(vn)|=0.
Next, forn∈ℕ, define
(3.21)gn(v)={0if|v|<⌊|vn|2⌋,2|v|-|vn|+2if⌊|vn|2⌋≤|v|<|vn|,|vn|if|v|≥|vn|,
where ⌊x⌋ denotes the largest integer less than or equal to x. Then gn→0 pointwise, and ∥gn∥ℒ=2. Since gn(vn)=gn(vn-)=|vn|, we have
(3.22)‖ψgn‖w≥|vn||ψ(vn)gn(vn)-ψ(vn-)gn(vn-)|=|vn|2Dψ(vn).
By Lemma 3.3 we obtain limn→∞|vn|2Dψ(vn)≤limn→∞∥ψgn∥w=0.
(c)⇒(a) Suppose lim|v|→∞|v|2Dψ(v)=0 and lim|v|→∞(|v|+1)|ψ(v)|=0. Assume ψ is not identically zero, otherwise Mψ is trivially compact. By Lemma 3.3, to prove that Mψ is compact, it suffices to show that if {fn} is a bounded sequence in ℒ converging to 0 pointwise, then ∥ψfn∥w→0 as n→∞. Let {fn} be such a bounded sequence, s=supn∈ℕ∥fn∥ℒ, and fix ɛ>0. There exists M∈ℕ such that (|v|+1)|ψ(v)|<ɛ/2s and |v|2Dψ(v)<ɛ/2s for |v|≥M. For v∈T* and by Lemma 1.1, we have
(3.23)|v|D(ψfn)(v)≤|v||ψ(v)|Dfn(v)+|v|Dψ(v)|fn(v-)|≤|v||ψ(v)|Dfn(v)+|v|Dψ(v)(|fn(o)|+|v|‖Dfn‖∞)≤(|v|+1)|ψ(v)|Dfn(v)+|v|2Dψ(v)(|fn(o)|+‖Dfn‖∞)=(|v|+1)|ψ(v)|Dfn(v)+|v|2Dψ(v)‖fn‖L.
Since fn→0 uniformly on {v∈T:|v|≤M} as n→∞, so does Dfn. So, on the set {v∈T:|v|≤M}, |v|D(ψfn)(v)→0 as n→∞. On the other hand, on {v∈T:|v|≥M}, we have
(3.24)|v|D(ψfn)(v)≤(|v|+1)|ψ(v)|Dfn(v)+|v|2Dψ(v)‖fn‖L<ɛ.
So |v|D(ψfn)(v)→0 as n→∞. Since fn→0 pointwise, ψ(o)fn(o)→0 as n→∞. Thus ∥ψfn∥w→0 as n→∞. The compactness of Mψ follows at once from Lemma 3.3.
The proof of the equivalence of (b) and (c) is analogous.Forψ a function on T, define (3.25)A(ψ)=lim⁡n→∞sup⁡|v|≥n|v||ψ(v)|,B(ψ)=lim⁡n→∞sup⁡|v|≥n|v|2Dψ(v).Theorem 3.5.
LetMψ be a bounded multiplication operator from ℒ to  ℒw. Then
(3.26)‖Mψ‖e≥max⁡{A(ψ),12B(ψ)}.Proof.
Fixk∈ℕ, and for each n∈ℕ, consider the sets
(3.27)En,k={v∈T:n≤|v|≤kn,|v|even},On,k={v∈T:n≤|v|≤kn,|v|odd}.
Define the functions fn,k=χEn,k and gn,k=χOn,k. Then fn,k,gn,k∈ℒ0, ∥fn,k∥ℒ=∥gn,k∥ℒ=1, and fn,k and gn,k approach 0 pointwise as n→∞. By Lemma 1.2, the sequences {fn,k} and {gn,k} approach 0 weakly in ℒ0 as n→∞. Let 𝒦0 be the set of compact operators from ℒ0 to  ℒw,0, and note that every operator in 𝒦0 is completely continuous. Thus, if K∈𝒦0, then ∥Kfn,k∥w→0 and ∥Kgn,k∥w→0, as n→∞.
Therefore, ifK∈𝒦0, then
(3.28)‖Mψ-K‖≥limsup⁡n→∞‖(Mψ-K)fn,k‖w≥limsup⁡n→∞‖Mψfn,k‖w≥limsup⁡n→∞sup⁡v∈En,k(|v|+1)|ψ(v)|.
Similarly,(3.29)‖Mψ-K‖≥limsup⁡n→∞sup⁡v∈On,k(|v|+1)|ψ(v)|.
Therefore, combining (3.28) and (3.29), we obtain
(3.30)‖Mψ‖e=inf⁡{‖Mψ-K‖:K∈K0}≥limsup⁡n→∞sup⁡kn≥|v|≥n(|v|+1)|ψ(v)|≥limsup⁡n→∞sup⁡kn≥|v|≥n|v||ψ(v)|.
Letting k→∞, we obtain ∥Mψ∥e≥𝒜(ψ).
Next, we wish to show that∥Mψ∥e≥(1/2)ℬ(ψ). The result is clearly true if ℬ(ψ)=0. So assume there exists a sequence {vn} in T such that 2<|vn|→∞ as n→∞ and
(3.31)lim⁡n→∞|vn|2Dψ(vn)=B(ψ).
For n∈ℕ, define
(3.32)hn(v)={0ifv=o,(|v|+1)2|vn|if1≤|v|<|vn|,|vn|if|v|≥|vn|.
Clearly, hn(o)=0,hn(vn)=hn(vn-)=|vn|, and
(3.33)Dhn(v)={4|vn|if|v|=1,2|v|+1|vn|if1<|v|<|vn|,0if|v|≥|vn|.
The supremum of Dhn(v) is attained on the set {v∈T:|v|=|vn|-1}. Thus ∥hn∥ℒ=(2|vn|-1)/|vn|<2. Define gn=hn/∥hn∥ℒ, and observe that gn∈ℒ0, ∥gn∥ℒ=1, and gn→0 pointwise on T. By Lemma 1.2, gn→0 weakly in ℒ0. Thus ∥Kgn∥w→0 as n→∞ for any K∈𝒦0.
For eachn∈ℕ, gn(vn)=gn(vn-)=|vn|2/(2|vn|-1). Thus
(3.34)|vn|D(ψgn)(vn)=|vn||ψ(vn)gn(vn)-ψ(vn-)gn(vn-)|=|vn|2|vn|-1|vn|2Dψ(vn).
We deduce that(3.35)‖Mψ‖e=inf⁡{‖Mψ-K‖:K∈K0}≥limsup⁡n→∞‖(Mψ-K)gn‖w≥limsup⁡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|2Dψ(vn)≥12B(ψ).
Therefore,(3.36)‖Mψ‖e≥max⁡{A(ψ),12B(ψ)}.We next derive an upper estimate on the essential norm.Theorem 3.6.
LetMψ be a bounded multiplication operator from ℒ to ℒw. Then
(3.37)‖Mψ‖e≤A(ψ)+B(ψ).Proof.
For eachn∈ℕ, consider the operator Kn defined by
(3.38)(Knf)(v)={f(v)if|v|≤n,f(vn)if|v|>n,
for f∈ℒ, where vn is the ancestor of v of length n. Then (Knf)(o)=f(o), and Knf∈ℒw,0. Arguing as in the proof of Theorem 2.8, by the boundedness of Mψ, it follows that MψKn is a compact operator from ℒ to ℒw.
Define the operatorJn=I-Kn, where I is the identity operator I on ℒ. Then,
(3.39)D(Jnf)(v)≤Df(v)≤‖f‖L.
Since (Jnf)(v)=0 for |v|≤n, by Lemma 1.1, we obtain
(3.40)|(Jnf)(v)|≤|v|‖f‖L.
From these two estimates, we arrive at
(3.41)‖MψJnf‖w=sup⁡|v|>n|v||ψ(v)(Jnf)(v)-ψ(v-)(Jnf)(v-)|≤sup⁡|v|>n[|v|Dψ(v)|(Jnf)(v)|+|v||ψ(v-)|D(Jnf)(v)]≤sup⁡|v|>n|v|2Dψ(v)|(Jnf)(v)||v|+sup⁡|v|>n|v||ψ(v-)|D(Jnf)(v)≤sup⁡|v|>n|v|2Dψ(v)‖f‖L+sup⁡|v|>n|v||ψ(v-)|‖f‖L.
Since
(3.42)‖Mψ‖e≤limsup⁡n→∞‖Mψ-MψKn‖=limsup⁡n→∞sup⁡∥f∥L=1‖(Mψ-MψKn)f‖w=limsup⁡n→∞sup⁡∥f∥L=1‖MψJnf‖w,
from (3.41), taking the limit as n→∞, we obtain
(3.43)‖Mψ‖e≤B(ψ)+A(ψ),
as desired.


## 3.1. Boundedness and Operator Norm Estimates

Letψ be a function on the tree T. Define the quantities
(3.1)θψ=sup⁡v∈T*|v|2Dψ(v),ωψ=sup⁡v∈T(|v|+1)|ψ(v)|.Theorem 3.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:ℒ→ℒw is bounded.
(b)Mψ:ℒ0→ℒw,0 is bounded.
(c)θψ and ωψ are finite.
Furthermore, under the above conditions, one has(3.2)max⁡{θψ,ωψ}≤‖Mψ‖≤θψ+ωψ.Proof.
(a)⇒(c) Assume Mψ is bounded from ℒ to ℒw. The function fo=(1/2)χo∈ℒ and ∥fo∥ℒ=1. Thus
(3.3)|ψ(o)|=∥ψfo∥w≤∥Mψ∥.
Next, fixv∈T*. Then χv∈ℒ and ∥χv∥ℒ=1; so
(3.4)(|v|+1)|ψ(v)|=‖ψχv‖w≤‖Mψ‖.
Taking the supremum over all v∈T, from (3.3) and (3.4) we see that ωψ is finite and
(3.5)ωψ≤‖Mψ‖.
Withv∈T*, we now define
(3.6)fv(w)={|w|if|w|<|v|,|v|if|w|≥|v|.
Then fv∈ℒ, fv(o)=0 and ∥fv∥ℒ=1. By the boundedness of Mψ we obtain
(3.7)‖Mψ‖≥‖Mψfv‖w≥sup⁡1≤|w|≤|v||w||ψ(w)|w|-ψ(w-)(|w|-1)|≥sup⁡1≤|w|≤|v||w|2Dψ(w)-sup⁡1≤|w|≤|v||w||ψ(w-)|.
Therefore,
(3.8)|v|2Dψ(v)≤sup⁡1≤|w|≤|v||w|2Dψ(w)≤‖Mψ‖+ωψ.
Taking the supremum over all v∈T*, we obtain θψ<∞. From this and (3.5), we deduce the lower estimate
(3.9)‖Mψ‖≥max⁡{θψ,ωψ}.
(c)⇒(a) Assume θψ and ωψ are finite. Then, ψ∈ℒw, and by Lemma 1.1, for f∈ℒ with ∥f∥ℒ=1 and v∈T*, we have
(3.10)|v|D(ψf)(v)≤|v|Dψ(v)|f(v)|+|v||ψ(v-)|Df(v)≤|v|Dψ(v)|f(o)|+|v|2Dψ(v)‖Df‖∞+ωψ‖Df‖∞≤|v|Dψ(v)|f(o)|+(θψ+ωψ)‖Df‖∞.
Thus, ψf∈ℒw. Note that |f(o)|+∥Df∥∞=1 and
(3.11)‖ψ‖w=|ψ(o)|+sup⁡v∈T*|v|Dψ(v)≤ωψ+sup⁡v∈T*|v|2Dψ(v)=ωψ+θψ.
From this, we have
(3.12)‖ψf‖w≤‖ψ‖w|f(o)|+(θψ+ωψ)‖Df‖∞≤θψ+ωψ,
proving the boundedness of Mψ:ℒ→ℒw and the upper estimate
(3.13)‖Mψ‖≤θψ+ωψ.
(b)⇒(c) The proof is the same as for (a)⇒(c); since for v∈T*, the functions χv and fv used there belong to ℒ0.
(c)⇒(b) Assume θψ and ωψ are finite, and let f∈ℒ0. Then, by Lemma 1.1, for v∈T*, we have
(3.14)|v|D(ψf)(v)≤|v|Dψ(v)|f(v)|+|v||ψ(v-)|Df(v)≤|v|2Dψ(v)|f(v)||v|+|v||ψ(v-)|Df(v)≤θψ|f(v)||v|+ωψDf(v)⟶0
as |v|→∞. Thus, ψf∈ℒw,0. The proof of the boundedness of Mψ is similar to that in (c)⇒(a).


## 3.2. Isometries

In this section, we show there are no isometric multiplication operatorsMψ from the space ℒ to ℒw or from ℒ0 to ℒw,0.SupposeMψ:ℒ→ℒw is an isometry. Then ∥ψ∥w=∥Mψ1∥w=1. On the other hand,
(3.15)|ψ(o)|=12‖ψχo‖w=12‖χo‖L=1.
Thus supv∈T*|v|Dψ(v)=∥ψ∥w-|ψ(o)|=0, which implies that ψ is a constant of modulus 1. Now observe that, for v∈T*, we have
(3.16)1=‖χv‖L=‖Mψχ‖w=(|v|+1)|ψ(v)|=|v|+1,
which is a contradiction. Since χv∈ℒ0 for all v∈T, if Mψ:ℒ0→ℒw,0 is an isometry, then the above argument yields again a contradiction. Thus, we proved the following result.Theorem 3.2.
There are no isometriesMψ from ℒ to  ℒw or from ℒ0 to  ℒw,0.


## 3.3. Compactness and Essential Norm

We now characterize the compact multiplication operators, but first we give a useful compactness criterion for multiplication operators fromℒ to ℒw or from ℒ0 to ℒw,0.Lemma 3.3.
A bounded multiplication operatorMψ from ℒ to  ℒw (ℒ0 to  ℒw,0) is compact if and only if for every bounded sequence {fn} in ℒ (ℒ0) converging to 0 pointwise, the sequence  ∥ψfn∥w converges to 0 as n→∞.Proof.
SupposeMψ is compact from ℒ to ℒw and {fn} is a bounded sequence in ℒ converging to 0 pointwise. Without loss of generality, we may assume ∥fn∥ℒ≤1 for all n∈ℕ. Since Mψ is compact, the sequence {ψfn} has a subsequence {ψfnk} that converges in the ℒw-norm to some function f∈ℒw.
By Lemma1.3, for v∈T* we have
(3.17)|ψ(v)fnk(v)-f(v)|≤(1+log⁡|v|)‖ψfnk-f‖w.
Thus, ψfnk→f pointwise on T*. Furthermore, since |ψ(o)fnk(o)-f(o)|≤∥ψfnk-f∥w, ψ(0)fnk(0)→f(0) as k→∞. Thus ψfnk→f pointwise on T. Since by assumption, fn→0 pointwise, it follows that f is identically 0, and thus ∥ψfnk∥w→0. Since 0 is the only limit point in ℒw of the sequence {ψfn}, we deduce that ∥ψfn∥w→0 as n→∞.
Conversely, suppose that every bounded sequence{fn} in ℒ that converges to 0 pointwise has the property that ∥ψfn∥w→0 as n→∞. Let {gn} be a sequence in ℒ such that ∥gn∥ℒ≤1 for all n∈ℕ. Then |gn(o)|≤1, and by part (a) of Lemma 1.1, for v∈T* we have |gn(v)|≤|v|. So {gn} is uniformly bounded on finite subsets of T. Thus there is a subsequence {gnk}, which converges pointwise to some function g.
Fixɛ>0 and v∈T*. Then |gnk(v)-g(v)|<ɛ/2 as well as |gnk(v-)-g(v-)|<ɛ/2 for k sufficiently large. Therefore, for all k sufficiently large, we have
(3.18)Dg(v)≤|g(v)-gnk(v)|+|gnk(v-)-g(v-)|+Dgnk(v)<ɛ+Dgnk(v).
Thus g∈ℒ. The sequence fnk=gnk-g is bounded in ℒ and converges to 0 pointwise. So ∥ψfnk∥w→0 as k→∞. Thus ψgnk→ψg in the ℒw-norm. Therefore, Mψ is compact.
The proof for the case ofMψ:ℒ0→ℒw,0 is similar.Theorem 3.4.
LetMψ be a bounded multiplication operator from ℒ to  ℒw (or equivalently from ℒ0 to  ℒw,0). Then the following are equivalent:
(a)Mψ:ℒ→ℒw is compact.
(b)Mψ:ℒ0→ℒw,0 is compact.
(c)lim|v|→∞|v|2Dψ(v)=0 and lim|v|→∞(|v|+1)|ψ(v)|=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|2Dψ(vn)=0 and limn→∞(|vn|+1)|ψ(vn)|=0. Let {vn} be such a sequence, and for n∈ℕ define fn=((|vn|+1)/|vn|)χvn. Clearly fn→0 pointwise, and ∥fn∥ℒ≤3/2. Using Lemma 3.3, we see that
(3.19)‖ψfn‖w⟶0asn⟶∞.
On the other hand, sincefn(o)=0 for all n∈ℕ, we have
(3.20)‖ψfn‖w=sup⁡v∈T*|v|D(ψfn)(v)≥|vn|(|vn|+1|vn|)|ψ(vn)|=(|vn|+1)|ψ(vn)|.
Hence limn→∞(|vn|+1)|ψ(vn)|=0.
Next, forn∈ℕ, define
(3.21)gn(v)={0if|v|<⌊|vn|2⌋,2|v|-|vn|+2if⌊|vn|2⌋≤|v|<|vn|,|vn|if|v|≥|vn|,
where ⌊x⌋ denotes the largest integer less than or equal to x. Then gn→0 pointwise, and ∥gn∥ℒ=2. Since gn(vn)=gn(vn-)=|vn|, we have
(3.22)‖ψgn‖w≥|vn||ψ(vn)gn(vn)-ψ(vn-)gn(vn-)|=|vn|2Dψ(vn).
By Lemma 3.3 we obtain limn→∞|vn|2Dψ(vn)≤limn→∞∥ψgn∥w=0.
(c)⇒(a) Suppose lim|v|→∞|v|2Dψ(v)=0 and lim|v|→∞(|v|+1)|ψ(v)|=0. Assume ψ is not identically zero, otherwise Mψ is trivially compact. By Lemma 3.3, to prove that Mψ is compact, it suffices to show that if {fn} is a bounded sequence in ℒ converging to 0 pointwise, then ∥ψfn∥w→0 as n→∞. Let {fn} be such a bounded sequence, s=supn∈ℕ∥fn∥ℒ, and fix ɛ>0. There exists M∈ℕ such that (|v|+1)|ψ(v)|<ɛ/2s and |v|2Dψ(v)<ɛ/2s for |v|≥M. For v∈T* and by Lemma 1.1, we have
(3.23)|v|D(ψfn)(v)≤|v||ψ(v)|Dfn(v)+|v|Dψ(v)|fn(v-)|≤|v||ψ(v)|Dfn(v)+|v|Dψ(v)(|fn(o)|+|v|‖Dfn‖∞)≤(|v|+1)|ψ(v)|Dfn(v)+|v|2Dψ(v)(|fn(o)|+‖Dfn‖∞)=(|v|+1)|ψ(v)|Dfn(v)+|v|2Dψ(v)‖fn‖L.
Since fn→0 uniformly on {v∈T:|v|≤M} as n→∞, so does Dfn. So, on the set {v∈T:|v|≤M}, |v|D(ψfn)(v)→0 as n→∞. On the other hand, on {v∈T:|v|≥M}, we have
(3.24)|v|D(ψfn)(v)≤(|v|+1)|ψ(v)|Dfn(v)+|v|2Dψ(v)‖fn‖L<ɛ.
So |v|D(ψfn)(v)→0 as n→∞. Since fn→0 pointwise, ψ(o)fn(o)→0 as n→∞. Thus ∥ψfn∥w→0 as n→∞. The compactness of Mψ follows at once from Lemma 3.3.
The proof of the equivalence of (b) and (c) is analogous.Forψ a function on T, define (3.25)A(ψ)=lim⁡n→∞sup⁡|v|≥n|v||ψ(v)|,B(ψ)=lim⁡n→∞sup⁡|v|≥n|v|2Dψ(v).Theorem 3.5.
LetMψ be a bounded multiplication operator from ℒ to  ℒw. Then
(3.26)‖Mψ‖e≥max⁡{A(ψ),12B(ψ)}.Proof.
Fixk∈ℕ, and for each n∈ℕ, consider the sets
(3.27)En,k={v∈T:n≤|v|≤kn,|v|even},On,k={v∈T:n≤|v|≤kn,|v|odd}.
Define the functions fn,k=χEn,k and gn,k=χOn,k. Then fn,k,gn,k∈ℒ0, ∥fn,k∥ℒ=∥gn,k∥ℒ=1, and fn,k and gn,k approach 0 pointwise as n→∞. By Lemma 1.2, the sequences {fn,k} and {gn,k} approach 0 weakly in ℒ0 as n→∞. Let 𝒦0 be the set of compact operators from ℒ0 to  ℒw,0, and note that every operator in 𝒦0 is completely continuous. Thus, if K∈𝒦0, then ∥Kfn,k∥w→0 and ∥Kgn,k∥w→0, as n→∞.
Therefore, ifK∈𝒦0, then
(3.28)‖Mψ-K‖≥limsup⁡n→∞‖(Mψ-K)fn,k‖w≥limsup⁡n→∞‖Mψfn,k‖w≥limsup⁡n→∞sup⁡v∈En,k(|v|+1)|ψ(v)|.
Similarly,(3.29)‖Mψ-K‖≥limsup⁡n→∞sup⁡v∈On,k(|v|+1)|ψ(v)|.
Therefore, combining (3.28) and (3.29), we obtain
(3.30)‖Mψ‖e=inf⁡{‖Mψ-K‖:K∈K0}≥limsup⁡n→∞sup⁡kn≥|v|≥n(|v|+1)|ψ(v)|≥limsup⁡n→∞sup⁡kn≥|v|≥n|v||ψ(v)|.
Letting k→∞, we obtain ∥Mψ∥e≥𝒜(ψ).
Next, we wish to show that∥Mψ∥e≥(1/2)ℬ(ψ). The result is clearly true if ℬ(ψ)=0. So assume there exists a sequence {vn} in T such that 2<|vn|→∞ as n→∞ and
(3.31)lim⁡n→∞|vn|2Dψ(vn)=B(ψ).
For n∈ℕ, define
(3.32)hn(v)={0ifv=o,(|v|+1)2|vn|if1≤|v|<|vn|,|vn|if|v|≥|vn|.
Clearly, hn(o)=0,hn(vn)=hn(vn-)=|vn|, and
(3.33)Dhn(v)={4|vn|if|v|=1,2|v|+1|vn|if1<|v|<|vn|,0if|v|≥|vn|.
The supremum of Dhn(v) is attained on the set {v∈T:|v|=|vn|-1}. Thus ∥hn∥ℒ=(2|vn|-1)/|vn|<2. Define gn=hn/∥hn∥ℒ, and observe that gn∈ℒ0, ∥gn∥ℒ=1, and gn→0 pointwise on T. By Lemma 1.2, gn→0 weakly in ℒ0. Thus ∥Kgn∥w→0 as n→∞ for any K∈𝒦0.
For eachn∈ℕ, gn(vn)=gn(vn-)=|vn|2/(2|vn|-1). Thus
(3.34)|vn|D(ψgn)(vn)=|vn||ψ(vn)gn(vn)-ψ(vn-)gn(vn-)|=|vn|2|vn|-1|vn|2Dψ(vn).
We deduce that(3.35)‖Mψ‖e=inf⁡{‖Mψ-K‖:K∈K0}≥limsup⁡n→∞‖(Mψ-K)gn‖w≥limsup⁡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|2Dψ(vn)≥12B(ψ).
Therefore,(3.36)‖Mψ‖e≥max⁡{A(ψ),12B(ψ)}.We next derive an upper estimate on the essential norm.Theorem 3.6.
LetMψ be a bounded multiplication operator from ℒ to ℒw. Then
(3.37)‖Mψ‖e≤A(ψ)+B(ψ).Proof.
For eachn∈ℕ, consider the operator Kn defined by
(3.38)(Knf)(v)={f(v)if|v|≤n,f(vn)if|v|>n,
for f∈ℒ, where vn is the ancestor of v of length n. Then (Knf)(o)=f(o), and Knf∈ℒw,0. Arguing as in the proof of Theorem 2.8, by the boundedness of Mψ, it follows that MψKn is a compact operator from ℒ to ℒw.
Define the operatorJn=I-Kn, where I is the identity operator I on ℒ. Then,
(3.39)D(Jnf)(v)≤Df(v)≤‖f‖L.
Since (Jnf)(v)=0 for |v|≤n, by Lemma 1.1, we obtain
(3.40)|(Jnf)(v)|≤|v|‖f‖L.
From these two estimates, we arrive at
(3.41)‖MψJnf‖w=sup⁡|v|>n|v||ψ(v)(Jnf)(v)-ψ(v-)(Jnf)(v-)|≤sup⁡|v|>n[|v|Dψ(v)|(Jnf)(v)|+|v||ψ(v-)|D(Jnf)(v)]≤sup⁡|v|>n|v|2Dψ(v)|(Jnf)(v)||v|+sup⁡|v|>n|v||ψ(v-)|D(Jnf)(v)≤sup⁡|v|>n|v|2Dψ(v)‖f‖L+sup⁡|v|>n|v||ψ(v-)|‖f‖L.
Since
(3.42)‖Mψ‖e≤limsup⁡n→∞‖Mψ-MψKn‖=limsup⁡n→∞sup⁡∥f∥L=1‖(Mψ-MψKn)f‖w=limsup⁡n→∞sup⁡∥f∥L=1‖MψJnf‖w,
from (3.41), taking the limit as n→∞, we obtain
(3.43)‖Mψ‖e≤B(ψ)+A(ψ),
as desired.


## 4. Multiplication Operators fromℒw or ℒw,0 to L∞

In this section, we study the multiplication operatorsMψ from the weighted Lipschitz space or the little weighted Lipschitz space into L∞. We begin by characterizing the bounded operators and determining their operator norm. In addition, we characterize the bounded operators that are bounded from below and show that there are no isometries among them. Finally, we characterize the compact multiplication operators and determine the essential norm.


### 4.1. Boundedness and Operator Norm

For a functionψ on T, define
(4.1)γψ=max⁡{|ψ(o)|,sup⁡v∈T*(1+log⁡|v|)|ψ(v)|}.Theorem 4.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:ℒw→L∞ is bounded.
(b)Mψ:ℒw,0→L∞ is bounded.
(c)supv∈T*log|v∥ψ(v)| is finite.
Furthermore, under the above conditions, one has∥Mψ∥=γψ.Proof.
The implication (a)⇒(b) is obvious.
(b)⇒(a) We begin by showing that, for each f∈ℒw, the function ψf is bounded. Since Mψ is bounded on ℒw,0, ψ=Mψ1∈L∞. Thus, if f is constant, then ψf∈L∞. Fix f∈ℒw, f nonconstant, v∈T, and set n=|v|. For w∈T, define
(4.2)fn(w)={f(w)if|w|≤n,f(wn)if|w|>n,
where wn is the ancestor of w of length n. Then fn∈ℒw,0, and ∥fn∥w≤∥f∥w. Thus, ψfn∈L∞ and
(4.3)‖ψfn‖∞≤‖Mψ‖‖f‖w.
So |ψ(v)f(v)|=|ψ(v)fn(v)|≤∥Mψ∥∥f∥w. Therefore, ψf∈L∞ and
(4.4)‖ψf‖∞≤‖Mψ‖‖f‖w,
proving the boundedness of Mψ as an operator from ℒw to L∞.
(a)⇒(c) Assume Mψ:ℒw→L∞ is bounded. Then ψ=Mψ1∈L∞, and
(4.5)‖Mψ‖≥‖ψ‖∞≥|ψ(o)|.
For v∈T, define f(v)=log(1+|v|). Then f(o)=0, and since for x≥1 the function x↦xlog((x+1)/x) is increasing and has limit 1 as x→∞, f∈ℒw and ∥f∥w=1. Thus
(4.6)‖Mψ‖≥‖ψf‖∞=sup⁡v∈T*log⁡(1+|v|)|ψ(v)|,
proving (c). Furthermore, from (4.5) and (4.6), we obtain
(4.7)‖Mψ‖≥γψ.
(c)⇒(a) Assume supv∈T*log|v||ψ(v)|<∞. Let f∈ℒw such that ∥f∥w=1. Then |ψ(o)f(o)|≤|ψ(o)|, and by Lemma 1.3, for v∈T*, we have
(4.8)|ψ(v)f(v)|≤(1+log⁡|v|)|ψ(v)|≤γψ.
Thus, ψf∈L∞ and
(4.9)‖ψf‖∞≤γψ,
proving the boundedness of Mψ as an operator from ℒw to L∞. Taking the supremum over all functions f∈ℒw such that ∥f∥w=1, from (4.9) we obtain ∥Mψ∥≤γψ. Therefore, from (4.7) we conclude that ∥Mψ∥=γψ.


### 4.2. Boundedness from Below

Recall that an operatorS from a Banach space 𝒳 to a Banach space 𝒴 is bounded below if there exists a constant C>0 such that for all x∈X(4.10)‖Sx‖≥C‖x‖.Theorem 4.2.
A bounded multiplication operatorMψ from  ℒw or  ℒw,0 to L∞ is bounded below if and only if
(4.11)inf⁡v∈T|ψ(v)||v|+1>0.Proof.
AssumeMψ is bounded below, and, arguing by contradiction, assume there exists v∈T such that ψ(v)=0. Then Mψχv 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, then ψ is nonvanishing.
Next assumeψ is nonvanishing and infv∈T|ψ(v)|/(|v|+1)=0. Then, there exists a sequence {vn} in T with 1≤|vn|→∞, such that |ψ(vn)|/(|vn|+1)→0 as n→∞. For n∈ℕ, define fn=(1/(|vn|+1))χvn. Then ∥fn∥w=1, but
(4.12)∥ψfn∥∞=|ψ(vn)||vn|+1⟶0.
Thus, Mψ is not bounded below.
Conversely, assumeinfv∈T|ψ(v)|/(|v|+1)=c>0 and that Mψ is not bounded below. Then, for each n∈ℕ, there exists fn∈ℒw such that ∥fn∥w=1 and ∥ψfn∥∞<1/n. Then, for each v∈T, we have
(4.13)c(|v|+1)|fn(v)|≤|ψ(v)fn(v)|<1n,
so that the sequence {gn} defined by gn(v)=(|v|+1)fn(v) converges to 0 uniformly.
On the other hand, forv∈T*, we have
(4.14)|v|Dfn(v)=||v||v|+1gn(v)-gn(v-)|≤|gn(v)|+|gn(v-)|⟶0
uniformly as n→∞. Since |ψ(o)fn(o)|<1/n, yet ∥fn∥w=1, this yields a contradiction.


### 4.3. Isometries

In this section, we show there are no isometries among the multiplication operators from the spacesℒw or ℒw,0 into L∞.SupposeMψ is an isometry from ℒw or ℒw,0 to L∞. Then, for v∈T the function fv=(1/(|v|+1))χv is in ℒw,0, ∥fv∥w=1, and
(4.15)1|v|+1|ψ(v)|=‖Mψfv‖∞=‖fv‖w=1.
Thus, |ψ(v)|=|v|+1. On the other hand, since Mψ is bounded, by Theorem 4.1, we have supv∈T*log|v||ψ(v)|<∞; so ψ(v)→0 as |v|→∞, which yields a contradiction. Thus, we proved the following result.Theorem 4.3.
There are no isometric multiplication operatorsMψ from  ℒw or  ℒw,0 to L∞.


### 4.4. Compactness and Essential Norm

We begin by giving a useful compactness criterion for the bounded operators fromℒw or ℒw,0 into L∞.Lemma 4.4.
A bounded multiplication operatorMψ from  ℒw to L∞ is compact if and only if for every bounded sequence {fn} in  ℒw converging to 0 pointwise, the sequence ∥ψfn∥∞ approaches 0 as n→∞.Proof.
AssumeMψ is compact on ℒw, and let {fn} be a bounded sequence in ℒw converging to 0 pointwise. By rescaling the sequence, if necessary, we may assume ∥fn∥w≤1 for all n∈ℕ. By the compactness of Mψ, {fn} has a subsequence {fnk} such that {ψfnk} converges in the norm to some function f∈L∞. In particular, ψfnk→f pointwise. Since by assumption, fn→0 pointwise, it follows that f must be identically 0. Thus, the only limit point of sequence {ψfn} in L∞ is 0. Hence ∥ψfn∥∞→0.
Conversely, assume that for every bounded sequence{fn} in ℒw converging to 0 pointwise, the sequence ∥ψfn∥∞ approaches 0 as n→∞. Let {gn} be a sequence in ℒw with ∥gn∥w≤1. Fix w∈T, and, by replacing gn with gn-gn(w), assume gn(w)=0 for all n∈ℕ. Then, for each v∈T, |gn(v)|=|gn(v)-gn(w)|≤d(v,w). Therefore, gn is uniformly bounded on finite subsets of T, and so some subsequence {gnk}k∈ℕ converges pointwise to some function g on T. Fix ɛ>0 and v∈T*. Then, |g(o)-gnk(o)|<ɛ/2, |gnk(v)-g(v)|<ɛ/(2|v|), and |gnk(v-)-g(v-)|<ɛ/(2|v|) for all k sufficiently large. Thus,
(4.16)|v|Dg(v)≤|v||g(v)-g(v-)-(gnk(v)-gnk(v-))|+|v|Dgnk(v)<ɛ+|v|Dgnk(v),
for k sufficiently large. Consequently, g∈ℒw, we have
(4.17)‖g‖w=|g(o)|+sup⁡v∈T*|v|Dg(v)≤|g(o)-gnk(o)|+|gnk(o)|+ɛ+sup⁡v∈T*Dgnk(v)<2ɛ+‖gnk‖w≤2ɛ+1.
Since ɛ was arbitrary, it follows that ∥g∥w≤1. Therefore, the sequence {fk} defined by fk=gnk-g is bounded in ℒw and converges to 0 pointwise, hence, by the hypothesis, ∥ψfk∥∞→0 as n→∞. We conclude that ψgnk→ψg in L∞, proving the compactness of Mψ.By an analogous argument, we obtain the corresponding compactness criterion forMψ:ℒw,0→L∞.Lemma 4.5.
A bounded multiplication operatorMψ from  ℒw,0 to L∞ is compact if and only if for every bounded sequence {fn} in  ℒw,0 converging to 0 pointwise, the sequence ∥ψfn∥∞ approaches 0 as n→∞.Theorem 4.6.
For a bounded operatorMψ from ℒw to L∞ (or equivalently from ℒw,0 to L∞) the following statements are equivalent:
(a)Mψ:ℒw→L∞ is compact.
(b)Mψ:ℒw,0→L∞ is compact.
(c)lim|v|→∞log|v||ψ(v)|=0.Proof.
(a)⇒(b) is trivial.
(b)⇒(c): Let {vn} be a sequence of vertices such that 1≤|vn|→∞ as n→∞. We need to show that
(4.18)lim⁡n→∞log⁡|vn||ψ(vn)|=0.
For n∈ℕ define
(4.19)fn(v)={0ifv=o,(log⁡|v|)2log⁡|vn|if1≤|v|<|vn|,log⁡|vn|if|v|≥|vn|.
Then {fn} converges to 0 pointwise. Using the fact that (|v|-1)(log|v|-log(|v|-1))≤1 for any choice of v in T* with |v|>1, we have
(4.20)|v|Dfn(v)=|v||v|-1(|v|-1)[(log⁡|v|)2-(log⁡(|v|-1))2]log⁡|vn|≤2(log⁡|v|+log⁡(|v|-1))log⁡|vn|≤4,
for 2≤|v|≤|vn|. Moreover, |v|Dfn(v)=0 for |v|=1 and for |v|>|vn|. Thus, fn∈ℒw,0, and {∥fn∥w} is bounded. By the compactness of Mψ as an operator from ℒw,0 to L∞ and by Lemma 4.5, we deduce
(4.21)log⁡|vn||ψ(vn)|≤‖ψfn‖∞⟶0
as n→∞.
(c)⇒(a) Assume {fn} is a sequence in ℒw converging to 0 pointwise and such that a=supn∈ℕ∥fn∥w<∞. By Lemma 1.3, for all v∈T* and all n∈ℕ, we have
(4.22)|ψ(v)fn(v)|≤a(1+log⁡|v|)|ψ(v)|.
Fix ɛ>0. There exists N∈ℕ such that N≥3, and for |v|≥N, log|v||ψ(v)|<ɛ/2a. Thus, for |v|≥N and for all n∈ℕ, |ψ(v)fn(v)|≤2alog|v||ψ(v)|<ɛ. On the other hand, since fn→0 pointwise, for each vertex v such that |v|<N and ψ(v)≠0, we obtain |fn(v)|<ɛ/|ψ(v)| for all n sufficiently large. Hence |ψ(v)fn(v)|<ɛ for all v∈T and all n sufficiently large. Therefore, ∥Mψfn∥∞→0 as n→∞, which, by Lemma 4.4, proves the compactness of Mψ.Next, we determine the essential norm of the bounded multiplication operatorsMψ from ℒw or ℒw,0 to L∞.Theorem 4.7.
LetMψ be a bounded multiplication operator from ℒw or ℒw,0 to L∞. Then
(4.23)‖Mψ‖e=lim⁡n→∞sup⁡|v|≥nlog⁡|v||ψ(v)|.Proof.
DefineA(ψ)=limn→∞sup|v|≥nlog|v||ψ(v)|. If A(ψ)=0, then by Theorem 4.6, Mψ is compact, hence its essential norm is 0. So assume A(ψ)>0. We first show that ∥Mψ∥e≥A(ψ). Let {vn} be a sequence in T such that 1≤|vn|→∞ as n→∞ and
(4.24)A(ψ)=lim⁡n→∞log⁡|vn||ψ(vn)|.
Fix p∈(0,1), and for each n∈ℕ, define
(4.25)fn,p(v)={0ifv=o,(log⁡|v|)p+1(log⁡|vn|)pif1≤|v|<|vn|,log⁡|vn|if|v|≥|vn|.
Then {fn,p} converges to 0 pointwise, fn,p∈ℒw,0, fn,p(vn)=log|vn|, and
(4.26)‖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.
By Lemma 1.4, {fn,p} converges to 0 weakly in ℒw,0. Let K be a compact operator from ℒw,0 (or equivalently, from ℒw) to L∞. Since compact operators are completely continuous, it follows that ∥Kfn,p∥∞→0 as n→∞. Thus,
(4.27)‖Mψ-K‖≥limsup⁡n→∞‖(Mψ-K)fn,p‖∞∥fn,p∥w≥1p+1limsup⁡n→∞‖Mψfn,p‖∞≥1p+1limsup⁡n→∞log⁡|vn||ψ(vn)|.
Taking the infimum over all such compact operators K and passing to the limit as p approaches 0, we obtain
(4.28)‖Mψ‖e≥lim⁡n→∞log⁡|vn||ψ(vn)|=A(ψ).
To prove the estimate∥Mψ∥e≤A(ψ), for each n∈ℕ and for f∈ℒw, define
(4.29)Knf(v)={f(v)if|v|≤n,f(vn)if|v|>n,
where vn is the ancestor of v of length n. In the proof of Theorem 2.8, it is was shown that Kn is a compact operator on ℒw. Since Mψ:ℒw→L∞ is bounded, it follows that MψKn is also compact as an operator from ℒw to L∞.
Letv∈T, and let w be a vertex in the path from o to v of length k≥1. Label the vertices from w to v by vj, j=k,…,|v|. Then for f∈ℒw with ∥f∥w=1, we have
(4.30)|f(v)-f(w)|≤∑j=k+1|v||f(vj)-f(vj-1)|≤∑j=k+1|v|1j≤log⁡|v|.
Thus
(4.31)‖(Mψ-MψKn)f‖∞=sup⁡|v|>n|ψ(v)||f(v)-f(vn)|≤sup⁡|v|>nlog⁡|v||ψ(v)|.
We deduce
(4.32)‖Mψ‖e≤sup⁡∥f∥w=1‖(Mψ-MψKn)f‖∞≤sup⁡|v|>nlog⁡|v||ψ(v)|.
Taking the limit asn→∞, we obtain ∥Mψ∥e≤A(ψ).


## 4.1. Boundedness and Operator Norm

For a functionψ on T, define
(4.1)γψ=max⁡{|ψ(o)|,sup⁡v∈T*(1+log⁡|v|)|ψ(v)|}.Theorem 4.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:ℒw→L∞ is bounded.
(b)Mψ:ℒw,0→L∞ is bounded.
(c)supv∈T*log|v∥ψ(v)| is finite.
Furthermore, under the above conditions, one has∥Mψ∥=γψ.Proof.
The implication (a)⇒(b) is obvious.
(b)⇒(a) We begin by showing that, for each f∈ℒw, the function ψf is bounded. Since Mψ is bounded on ℒw,0, ψ=Mψ1∈L∞. Thus, if f is constant, then ψf∈L∞. Fix f∈ℒw, f nonconstant, v∈T, and set n=|v|. For w∈T, define
(4.2)fn(w)={f(w)if|w|≤n,f(wn)if|w|>n,
where wn is the ancestor of w of length n. Then fn∈ℒw,0, and ∥fn∥w≤∥f∥w. Thus, ψfn∈L∞ and
(4.3)‖ψfn‖∞≤‖Mψ‖‖f‖w.
So |ψ(v)f(v)|=|ψ(v)fn(v)|≤∥Mψ∥∥f∥w. Therefore, ψf∈L∞ and
(4.4)‖ψf‖∞≤‖Mψ‖‖f‖w,
proving the boundedness of Mψ as an operator from ℒw to L∞.
(a)⇒(c) Assume Mψ:ℒw→L∞ is bounded. Then ψ=Mψ1∈L∞, and
(4.5)‖Mψ‖≥‖ψ‖∞≥|ψ(o)|.
For v∈T, define f(v)=log(1+|v|). Then f(o)=0, and since for x≥1 the function x↦xlog((x+1)/x) is increasing and has limit 1 as x→∞, f∈ℒw and ∥f∥w=1. Thus
(4.6)‖Mψ‖≥‖ψf‖∞=sup⁡v∈T*log⁡(1+|v|)|ψ(v)|,
proving (c). Furthermore, from (4.5) and (4.6), we obtain
(4.7)‖Mψ‖≥γψ.
(c)⇒(a) Assume supv∈T*log|v||ψ(v)|<∞. Let f∈ℒw such that ∥f∥w=1. Then |ψ(o)f(o)|≤|ψ(o)|, and by Lemma 1.3, for v∈T*, we have
(4.8)|ψ(v)f(v)|≤(1+log⁡|v|)|ψ(v)|≤γψ.
Thus, ψf∈L∞ and
(4.9)‖ψf‖∞≤γψ,
proving the boundedness of Mψ as an operator from ℒw to L∞. Taking the supremum over all functions f∈ℒw such that ∥f∥w=1, from (4.9) we obtain ∥Mψ∥≤γψ. Therefore, from (4.7) we conclude that ∥Mψ∥=γψ.


## 4.2. Boundedness from Below

Recall that an operatorS from a Banach space 𝒳 to a Banach space 𝒴 is bounded below if there exists a constant C>0 such that for all x∈X(4.10)‖Sx‖≥C‖x‖.Theorem 4.2.
A bounded multiplication operatorMψ from  ℒw or  ℒw,0 to L∞ is bounded below if and only if
(4.11)inf⁡v∈T|ψ(v)||v|+1>0.Proof.
AssumeMψ is bounded below, and, arguing by contradiction, assume there exists v∈T such that ψ(v)=0. Then Mψχv 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, then ψ is nonvanishing.
Next assumeψ is nonvanishing and infv∈T|ψ(v)|/(|v|+1)=0. Then, there exists a sequence {vn} in T with 1≤|vn|→∞, such that |ψ(vn)|/(|vn|+1)→0 as n→∞. For n∈ℕ, define fn=(1/(|vn|+1))χvn. Then ∥fn∥w=1, but
(4.12)∥ψfn∥∞=|ψ(vn)||vn|+1⟶0.
Thus, Mψ is not bounded below.
Conversely, assumeinfv∈T|ψ(v)|/(|v|+1)=c>0 and that Mψ is not bounded below. Then, for each n∈ℕ, there exists fn∈ℒw such that ∥fn∥w=1 and ∥ψfn∥∞<1/n. Then, for each v∈T, we have
(4.13)c(|v|+1)|fn(v)|≤|ψ(v)fn(v)|<1n,
so that the sequence {gn} defined by gn(v)=(|v|+1)fn(v) converges to 0 uniformly.
On the other hand, forv∈T*, we have
(4.14)|v|Dfn(v)=||v||v|+1gn(v)-gn(v-)|≤|gn(v)|+|gn(v-)|⟶0
uniformly as n→∞. Since |ψ(o)fn(o)|<1/n, yet ∥fn∥w=1, this yields a contradiction.


## 4.3. Isometries

In this section, we show there are no isometries among the multiplication operators from the spacesℒw or ℒw,0 into L∞.SupposeMψ is an isometry from ℒw or ℒw,0 to L∞. Then, for v∈T the function fv=(1/(|v|+1))χv is in ℒw,0, ∥fv∥w=1, and
(4.15)1|v|+1|ψ(v)|=‖Mψfv‖∞=‖fv‖w=1.
Thus, |ψ(v)|=|v|+1. On the other hand, since Mψ is bounded, by Theorem 4.1, we have supv∈T*log|v||ψ(v)|<∞; so ψ(v)→0 as |v|→∞, which yields a contradiction. Thus, we proved the following result.Theorem 4.3.
There are no isometric multiplication operatorsMψ from  ℒw or  ℒw,0 to L∞.


## 4.4. Compactness and Essential Norm

We begin by giving a useful compactness criterion for the bounded operators fromℒw or ℒw,0 into L∞.Lemma 4.4.
A bounded multiplication operatorMψ from  ℒw to L∞ is compact if and only if for every bounded sequence {fn} in  ℒw converging to 0 pointwise, the sequence ∥ψfn∥∞ approaches 0 as n→∞.Proof.
AssumeMψ is compact on ℒw, and let {fn} be a bounded sequence in ℒw converging to 0 pointwise. By rescaling the sequence, if necessary, we may assume ∥fn∥w≤1 for all n∈ℕ. By the compactness of Mψ, {fn} has a subsequence {fnk} such that {ψfnk} converges in the norm to some function f∈L∞. In particular, ψfnk→f pointwise. Since by assumption, fn→0 pointwise, it follows that f must be identically 0. Thus, the only limit point of sequence {ψfn} in L∞ is 0. Hence ∥ψfn∥∞→0.
Conversely, assume that for every bounded sequence{fn} in ℒw converging to 0 pointwise, the sequence ∥ψfn∥∞ approaches 0 as n→∞. Let {gn} be a sequence in ℒw with ∥gn∥w≤1. Fix w∈T, and, by replacing gn with gn-gn(w), assume gn(w)=0 for all n∈ℕ. Then, for each v∈T, |gn(v)|=|gn(v)-gn(w)|≤d(v,w). Therefore, gn is uniformly bounded on finite subsets of T, and so some subsequence {gnk}k∈ℕ converges pointwise to some function g on T. Fix ɛ>0 and v∈T*. Then, |g(o)-gnk(o)|<ɛ/2, |gnk(v)-g(v)|<ɛ/(2|v|), and |gnk(v-)-g(v-)|<ɛ/(2|v|) for all k sufficiently large. Thus,
(4.16)|v|Dg(v)≤|v||g(v)-g(v-)-(gnk(v)-gnk(v-))|+|v|Dgnk(v)<ɛ+|v|Dgnk(v),
for k sufficiently large. Consequently, g∈ℒw, we have
(4.17)‖g‖w=|g(o)|+sup⁡v∈T*|v|Dg(v)≤|g(o)-gnk(o)|+|gnk(o)|+ɛ+sup⁡v∈T*Dgnk(v)<2ɛ+‖gnk‖w≤2ɛ+1.
Since ɛ was arbitrary, it follows that ∥g∥w≤1. Therefore, the sequence {fk} defined by fk=gnk-g is bounded in ℒw and converges to 0 pointwise, hence, by the hypothesis, ∥ψfk∥∞→0 as n→∞. We conclude that ψgnk→ψg in L∞, proving the compactness of Mψ.By an analogous argument, we obtain the corresponding compactness criterion forMψ:ℒw,0→L∞.Lemma 4.5.
A bounded multiplication operatorMψ from  ℒw,0 to L∞ is compact if and only if for every bounded sequence {fn} in  ℒw,0 converging to 0 pointwise, the sequence ∥ψfn∥∞ approaches 0 as n→∞.Theorem 4.6.
For a bounded operatorMψ from ℒw to L∞ (or equivalently from ℒw,0 to L∞) the following statements are equivalent:
(a)Mψ:ℒw→L∞ is compact.
(b)Mψ:ℒw,0→L∞ is compact.
(c)lim|v|→∞log|v||ψ(v)|=0.Proof.
(a)⇒(b) is trivial.
(b)⇒(c): Let {vn} be a sequence of vertices such that 1≤|vn|→∞ as n→∞. We need to show that
(4.18)lim⁡n→∞log⁡|vn||ψ(vn)|=0.
For n∈ℕ define
(4.19)fn(v)={0ifv=o,(log⁡|v|)2log⁡|vn|if1≤|v|<|vn|,log⁡|vn|if|v|≥|vn|.
Then {fn} converges to 0 pointwise. Using the fact that (|v|-1)(log|v|-log(|v|-1))≤1 for any choice of v in T* with |v|>1, we have
(4.20)|v|Dfn(v)=|v||v|-1(|v|-1)[(log⁡|v|)2-(log⁡(|v|-1))2]log⁡|vn|≤2(log⁡|v|+log⁡(|v|-1))log⁡|vn|≤4,
for 2≤|v|≤|vn|. Moreover, |v|Dfn(v)=0 for |v|=1 and for |v|>|vn|. Thus, fn∈ℒw,0, and {∥fn∥w} is bounded. By the compactness of Mψ as an operator from ℒw,0 to L∞ and by Lemma 4.5, we deduce
(4.21)log⁡|vn||ψ(vn)|≤‖ψfn‖∞⟶0
as n→∞.
(c)⇒(a) Assume {fn} is a sequence in ℒw converging to 0 pointwise and such that a=supn∈ℕ∥fn∥w<∞. By Lemma 1.3, for all v∈T* and all n∈ℕ, we have
(4.22)|ψ(v)fn(v)|≤a(1+log⁡|v|)|ψ(v)|.
Fix ɛ>0. There exists N∈ℕ such that N≥3, and for |v|≥N, log|v||ψ(v)|<ɛ/2a. Thus, for |v|≥N and for all n∈ℕ, |ψ(v)fn(v)|≤2alog|v||ψ(v)|<ɛ. On the other hand, since fn→0 pointwise, for each vertex v such that |v|<N and ψ(v)≠0, we obtain |fn(v)|<ɛ/|ψ(v)| for all n sufficiently large. Hence |ψ(v)fn(v)|<ɛ for all v∈T and all n sufficiently large. Therefore, ∥Mψfn∥∞→0 as n→∞, which, by Lemma 4.4, proves the compactness of Mψ.Next, we determine the essential norm of the bounded multiplication operatorsMψ from ℒw or ℒw,0 to L∞.Theorem 4.7.
LetMψ be a bounded multiplication operator from ℒw or ℒw,0 to L∞. Then
(4.23)‖Mψ‖e=lim⁡n→∞sup⁡|v|≥nlog⁡|v||ψ(v)|.Proof.
DefineA(ψ)=limn→∞sup|v|≥nlog|v||ψ(v)|. If A(ψ)=0, then by Theorem 4.6, Mψ is compact, hence its essential norm is 0. So assume A(ψ)>0. We first show that ∥Mψ∥e≥A(ψ). Let {vn} be a sequence in T such that 1≤|vn|→∞ as n→∞ and
(4.24)A(ψ)=lim⁡n→∞log⁡|vn||ψ(vn)|.
Fix p∈(0,1), and for each n∈ℕ, define
(4.25)fn,p(v)={0ifv=o,(log⁡|v|)p+1(log⁡|vn|)pif1≤|v|<|vn|,log⁡|vn|if|v|≥|vn|.
Then {fn,p} converges to 0 pointwise, fn,p∈ℒw,0, fn,p(vn)=log|vn|, and
(4.26)‖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.
By Lemma 1.4, {fn,p} converges to 0 weakly in ℒw,0. Let K be a compact operator from ℒw,0 (or equivalently, from ℒw) to L∞. Since compact operators are completely continuous, it follows that ∥Kfn,p∥∞→0 as n→∞. Thus,
(4.27)‖Mψ-K‖≥limsup⁡n→∞‖(Mψ-K)fn,p‖∞∥fn,p∥w≥1p+1limsup⁡n→∞‖Mψfn,p‖∞≥1p+1limsup⁡n→∞log⁡|vn||ψ(vn)|.
Taking the infimum over all such compact operators K and passing to the limit as p approaches 0, we obtain
(4.28)‖Mψ‖e≥lim⁡n→∞log⁡|vn||ψ(vn)|=A(ψ).
To prove the estimate∥Mψ∥e≤A(ψ), for each n∈ℕ and for f∈ℒw, define
(4.29)Knf(v)={f(v)if|v|≤n,f(vn)if|v|>n,
where vn is the ancestor of v of length n. In the proof of Theorem 2.8, it is was shown that Kn is a compact operator on ℒw. Since Mψ:ℒw→L∞ is bounded, it follows that MψKn is also compact as an operator from ℒw to L∞.
Letv∈T, and let w be a vertex in the path from o to v of length k≥1. Label the vertices from w to v by vj, j=k,…,|v|. Then for f∈ℒw with ∥f∥w=1, we have
(4.30)|f(v)-f(w)|≤∑j=k+1|v||f(vj)-f(vj-1)|≤∑j=k+1|v|1j≤log⁡|v|.
Thus
(4.31)‖(Mψ-MψKn)f‖∞=sup⁡|v|>n|ψ(v)||f(v)-f(vn)|≤sup⁡|v|>nlog⁡|v||ψ(v)|.
We deduce
(4.32)‖Mψ‖e≤sup⁡∥f∥w=1‖(Mψ-MψKn)f‖∞≤sup⁡|v|>nlog⁡|v||ψ(v)|.
Taking the limit asn→∞, we obtain ∥Mψ∥e≤A(ψ).


## 5. Multiplication Operators fromL∞ to ℒw or ℒw,0

In this last section, we study the multiplication operatorsMψ from L∞ into the weighted Lipschitz space or the little weighted Lipschitz space. We first characterize the bounded operators and determine the operator norm. We also show there are no isometries among such operators. Finally, we characterize the compact multiplication operators and determine the essential norm.


### 5.1. Boundedness and Operator Norm

For a functionψ on T, define
(5.1)ηψ=|ψ(o)|+sup⁡v∈T*|v|[|ψ(v)|+|ψ(v-)|].Theorem 5.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:L∞→ℒw is bounded.
(b)supv∈T|v||ψ(v)|<∞.
Furthermore, under these conditions, one has(5.2)‖Mψ‖=ηψ.Proof.
(a)⇒(b) Assume Mψ:L∞→ℒw is bounded. Fix v∈T*. Since χv∈L∞ and ∥χv∥∞=1, the function ψχv∈ℒw, so
(5.3)|v||ψ(v)|<(|v|+1)|ψ(v)|=sup⁡w∈T*|w|D(ψχv)(w)≤‖Mψ‖.
Thus, supv∈T|v∥ψ(v)| is finite.
(b)⇒(a) Suppose supv∈T|v||ψ(v)|<∞. Let f∈L∞ such that ∥f∥∞=1. Then
(5.4)‖Mψf‖w≤|ψ(o)|+sup⁡v∈T*|v|[|ψ(v)|+|ψ(v-)|]<∞.
Thus, Mψ is bounded and ∥Mψ∥≤ηψ.
We next show that∥Mψ∥≥ηψ. The inequality is obvious if ψ is identically 0. For ψ not identically 0 and for v∈T, define
(5.5)f(v)={0ifψ(v)=0,ψ(v)¯|ψ(v)|ifψ(v)≠0,|v|even,-ψ(v)¯|ψ(v)|ifψ(v)≠0,|v|odd.
Then ∥f∥∞=1, and for v∈T*, D(ψf)(v)=|ψ(v)|+|ψ(v-)|, so that
(5.6)‖Mψf‖w=|ψ(o)|+sup⁡v∈T*|v|[|ψ(v)|+|ψ(v-)|].
Thus, ∥Mψ∥≥ηψ, completing the proof.In the next result, we characterize the bounded multiplication operators fromL∞ to ℒw,0.Theorem 5.2.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:L∞→ℒw,0 is bounded.
(b)lim|v|→∞|v||ψ(v)|=0.
Furthermore, under these conditions, one has,(5.7)‖Mψ‖=ηψ.Proof.
(a)⇒(b) Assume Mψ:L∞→ℒw,0 is bounded. Applying Mψ to the constant function 1, we obtain ψ=Mψ1∈ℒw,0. On the other hand, if 𝒪={v∈T:|v|isodd}, then ψχ𝒪∈ℒw,0, so for v∈T*, we have
(5.8)|v||ψ(v)|=|v||ψ(v)|DχO(v)≤|v|D(ψχO)(v)+|v|Dψ(v)|χO(v-)|≤|v|D(ψχO)(v)+|v|Dψ(v)⟶0,
as |v|→∞, proving (b).
(b)⇒(a) Suppose |v||ψ(v)|→0 as |v|→∞. First observe that
(5.9)|v|Dψ(v)≤|v||ψ(v)|+|v||v|-1(|v|-1)|ψ(v-)|≤|v||ψ(v)|+2(|v|-1)|ψ(v-)|⟶0,
as |v|→∞. Then for f∈L∞ and v∈T*, we have
(5.10)|v|D(ψf)(v)≤|v||ψ(v)|Df(v)+|v|Dψ(v)|f(v-)|≤(2|v||ψ(v)|+|v|Dψ(v))‖f‖∞⟶0,
as |v|→∞. Thus, ψf∈ℒw,0. The proof of the boundedness of Mψ and of the formula ∥Mψ∥=ηψ is similar to the case when Mψ:L∞→ℒw.


### 5.2. Isometries

As for all other multiplication operators in this paper, there are no isometries among the multiplication operators fromL∞ into ℒw or ℒw,0.AssumeMψ is an isometry from L∞ to ℒw or ℒw,0. Then, for v∈T the function fv=(1/(|v|+1))χv is in ℒw,0 with ∥Mψχv∥w=∥χv∥∞=1. In particular, it follows that |ψ(o)|=1/2, and for v∈T*,
(5.11)(|v|+1)|ψ(v)|=1.
Thus, |ψ(v)|=1/(|v|+1). On the other hand, taking as a test function f the characteristic function of the set {v∈T:|v|≤1}, we obtain
(5.12)1=‖f‖∞=‖Mψf‖w=|ψ(o)|+max⁡{sup⁡|v|=1|ψ(v)-ψ(o)|,sup⁡|v|=12|ψ(v)|}≥32,
which yields a contradiction. Therefore, we obtain the following result.Theorem 5.3.
There are no isometric multiplication operatorsMψ from L∞ to  ℒw or  ℒw,0.


### 5.3. Compactness and Essential Norm

The following two results are compactness criteria for multiplication operators fromL∞ into ℒw or ℒw,0 similar to those given in the previous sections.Lemma 5.4.
A bounded multiplication operatorMψ from L∞ to  ℒw is compact if and only if for every bounded sequence {fn} in L∞ converging to 0 pointwise, the sequence ∥ψfn∥w approaches 0 as n→∞.Proof.
AssumeMψ is compact, and let {fn} be a bounded sequence in L∞ converging to 0 pointwise. By rescaling the sequence, if necessary, we may assume ∥fn∥∞≤1 for all n∈ℕ. By the compactness of Mψ, {fn} has a subsequence {fnk} such that {ψfnk} converges in the ℒw-norm to some function f∈ℒw. Since by Lemma 1.3, for v∈T*,
(5.13)|ψ(v)fnk(v)-f(v)|≤(1+log⁡|v|)‖ψfnk-f‖w
and |ψ(o)fnk(o)-f(o)|≤∥ψfnk-f∥w, it follows that ψfnk→f pointwise. Since by assumption, fn→0 pointwise, the function f must be identically 0. Thus, the only limit point of the sequence {ψfn} in ℒw is 0. Hence ∥ψfn∥w→0 as n→∞.
Conversely, suppose∥ψfn∥w approaches 0 as n→∞ for every bounded sequence {fn} in L∞ converging to 0 pointwise. Let {gn} be a sequence in L∞ with ∥gn∥∞≤1. Then some subsequence {gnk} converges to a bounded function g. Thus, the sequence fnk=gnk-g converges to 0 uniformly, and ∥fn∥∞ is bounded. By the hypothesis, it follows that ∥ψfnk∥w→0 as k→∞. Thus, ψgnk→ψg in ℒw. Therefore, Mψ is compact.By an analogous argument, we obtain the corresponding result forMψ:L∞→ℒw,0.Lemma 5.5.
A bounded multiplication operatorMψ from L∞ to  ℒw,0 is compact if and only if for every bounded sequence {fn} in L∞ converging to 0 pointwise, the sequence ∥ψfn∥w approaches 0 as n→∞.Theorem 5.6.
For a bounded operatorMψ from L∞ to  ℒw, the following statements are equivalent:
(a)Mψ is compact.
(b)lim|v|→∞|v||ψ(v)|=0.Proof.
(a)⇒(b) Assume Mψ is compact. Let {vn} be a sequence in T such that |vn|→∞ as n→∞. For n∈ℕ, let fn denote the characteristic function of the set {w∈T:|w|≥|vn|}. Then ∥fn∥∞=1 and fn→0 pointwise. By Lemma 5.4 and the compactness of Mψ, it follows that
(5.14)|vn||ψ(vn)|=|vn|D(ψfn)(vn)≤‖Mψfn‖w⟶0
as n→∞.
(b)⇒(a) Assume lim|v|→∞|v∥ψ(v)|=0 and that ψ is not identically 0. In particular, ψ is bounded. Let {fn} be a sequence in L∞ converging pointwise to 0 such that ∥fn∥∞ is bounded above by some positive constant C. Then corresponding to ɛ>0, there exists N∈ℕ such that |v||ψ(v)|<ɛ/4C for all vertices v such that |v|≥N. Therefore, for |v|>N and n∈ℕ, we have
(5.15)|v|D(ψfn)(v)≤|v||ψ(v)|Dfn(v)+|v|Dψ(v)|fn(v-)|<ɛ.
Furthermore, the sequence {fn} converges to 0 uniformly on the set {v∈T:|v|≤N} so that |fn(v)|<ɛ/4N∥ψ∥∞ for all n sufficiently large. Hence |v|D(ψfn)(v)<ɛ for all v∈T* and all n sufficiently large. Consequently, ∥ψfn∥w→0 as n→∞. Using Lemma 5.4, we deduce that Mψ is compact.Since the above proof is also valid whenMψ is a bounded operator from L∞ to ℒw,0, through the application of Lemma 5.5, from Theorems 5.2 and 5.6 we obtain the following result.Corollary 5.7.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:L∞→ℒw is compact.
(b)Mψ:L∞→ℒw,0 is bounded.
(c)Mψ:L∞→ℒw,0 is compact.
(d)lim|v|→∞|v||ψ(v)|=0.We now determine the essential norm of the bounded multiplication operators fromL∞ to ℒw.Theorem 5.8.
LetMψ:L∞→ℒw be bounded. Then
(5.16)‖Mψ‖e=lim⁡n→∞sup⁡|v|≥n|v|[|ψ(v)|+|ψ(v-)|].Proof.
SetB(ψ)=limn→∞sup|v|≥n|v|[|ψ(v)|+|ψ(v-)|]. In the case B(ψ)=0, then lim|v|→∞|v|ψ(v)=0, so by Theorem 5.6, Mψ is compact, and thus ∥Mψ∥e=0. So assume B(ψ)>0. Then there exists a sequence {vn} in T such that 1≤|vn|→∞ as n→∞ and
(5.17)B(ψ)=lim⁡n→∞|vn|[|ψ(vn)|+|ψ(vn-)|].
For each n∈ℕ let fn be the function on T defined by
(5.18)fn(v)={0if|v|<|vn|orψ(v)=0,ψ(v)¯|ψ(v)|if|v|≥|vn|,|v|iseven,ψ(v)≠0,-ψ(v)¯|ψ(v)|otherwise.
Then∥fn∥∞=1, and {fn} converges to 0 pointwise. Thus, for any compact operator K:L∞→ℒw, there exists a subsequence {fnk} such that ∥Kfnk∥w→0 as k→∞. Thus
(5.19)‖Mψ-K‖≥limsup⁡k→∞‖(Mψ-K)fnk‖w≥limsup⁡k→∞‖ψfnk‖w=limsup⁡k→∞sup⁡|v|≥|vnk||v|[|ψ(v)|+|ψ(v-)|]=B(ψ).
Therefore, ∥Mψ∥e≥B(ψ).
We now show that∥Mψ∥e≤B(ψ). For each n∈ℕ, define the operator Kn on L∞ by
(5.20)Knf(v)={f(v)if|v|≤n,0if|v|>n.
Then, for v∈T*, we have
(5.21)|v|D(Knf)(v)={|v|Df(v)for1≤|v|≤n,(n+1)|f(v-)|for|v|=n+1,0for|v|>n+1.
Thus, Knf∈ℒw with ∥Knf∥w≤|f(o)|+2n∥f∥∞.
Assume{fk} is a sequence in L∞ with ∥fk∥∞≤1. Then there exists a subsequence {fkj} converging pointwise to some function f∈L∞. Thus,
(5.22)‖Knfkj-Knf‖w=|fkj(o)-f(o)|+max⁡{sup⁡1≤|v|≤n|v|D(fkj-f)(v),sup⁡|v|=n+1|v||fkj(v-)-f(v-)|}≤|fkj(o)-f(o)|+2nmax⁡{sup⁡1≤|v|≤nD(fkj-f)(v),sup⁡|v|=n+1|fkj(v-)-f(v-)|}.
So ∥Knfkj-Knf∥w→0 as j→∞. Therefore, Kn is compact, and thus, since Mψ is bounded, MψKn is also compact.
Forf∈L∞, we have
(5.23)‖(Mψ-MψKn)f‖w=sup⁡|v|>n|v||ψ(v)f(v)-ψ(v-)f(v-)+ψ(v-)Knf(v-)|=max⁡{sup⁡|v|=n+1|v||ψ(v)||f(v)|,sup⁡|v|>n+1|v||ψ(v)f(v)-ψ(v-)f(v-)|}≤sup⁡|v|>n|v|[|ψ(v)|+|ψ(v-)|]‖f‖∞.
Therefore, we obtain
(5.24)‖Mψ‖e≤limsup⁡n→∞‖Mψ-MψKn‖=limsup⁡n→∞sup⁡∥f∥∞=1‖(Mψ-MψKn)f‖w≤B(ψ),
thus completing the proof.


## 5.1. Boundedness and Operator Norm

For a functionψ on T, define
(5.1)ηψ=|ψ(o)|+sup⁡v∈T*|v|[|ψ(v)|+|ψ(v-)|].Theorem 5.1.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:L∞→ℒw is bounded.
(b)supv∈T|v||ψ(v)|<∞.
Furthermore, under these conditions, one has(5.2)‖Mψ‖=ηψ.Proof.
(a)⇒(b) Assume Mψ:L∞→ℒw is bounded. Fix v∈T*. Since χv∈L∞ and ∥χv∥∞=1, the function ψχv∈ℒw, so
(5.3)|v||ψ(v)|<(|v|+1)|ψ(v)|=sup⁡w∈T*|w|D(ψχv)(w)≤‖Mψ‖.
Thus, supv∈T|v∥ψ(v)| is finite.
(b)⇒(a) Suppose supv∈T|v||ψ(v)|<∞. Let f∈L∞ such that ∥f∥∞=1. Then
(5.4)‖Mψf‖w≤|ψ(o)|+sup⁡v∈T*|v|[|ψ(v)|+|ψ(v-)|]<∞.
Thus, Mψ is bounded and ∥Mψ∥≤ηψ.
We next show that∥Mψ∥≥ηψ. The inequality is obvious if ψ is identically 0. For ψ not identically 0 and for v∈T, define
(5.5)f(v)={0ifψ(v)=0,ψ(v)¯|ψ(v)|ifψ(v)≠0,|v|even,-ψ(v)¯|ψ(v)|ifψ(v)≠0,|v|odd.
Then ∥f∥∞=1, and for v∈T*, D(ψf)(v)=|ψ(v)|+|ψ(v-)|, so that
(5.6)‖Mψf‖w=|ψ(o)|+sup⁡v∈T*|v|[|ψ(v)|+|ψ(v-)|].
Thus, ∥Mψ∥≥ηψ, completing the proof.In the next result, we characterize the bounded multiplication operators fromL∞ to ℒw,0.Theorem 5.2.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:L∞→ℒw,0 is bounded.
(b)lim|v|→∞|v||ψ(v)|=0.
Furthermore, under these conditions, one has,(5.7)‖Mψ‖=ηψ.Proof.
(a)⇒(b) Assume Mψ:L∞→ℒw,0 is bounded. Applying Mψ to the constant function 1, we obtain ψ=Mψ1∈ℒw,0. On the other hand, if 𝒪={v∈T:|v|isodd}, then ψχ𝒪∈ℒw,0, so for v∈T*, we have
(5.8)|v||ψ(v)|=|v||ψ(v)|DχO(v)≤|v|D(ψχO)(v)+|v|Dψ(v)|χO(v-)|≤|v|D(ψχO)(v)+|v|Dψ(v)⟶0,
as |v|→∞, proving (b).
(b)⇒(a) Suppose |v||ψ(v)|→0 as |v|→∞. First observe that
(5.9)|v|Dψ(v)≤|v||ψ(v)|+|v||v|-1(|v|-1)|ψ(v-)|≤|v||ψ(v)|+2(|v|-1)|ψ(v-)|⟶0,
as |v|→∞. Then for f∈L∞ and v∈T*, we have
(5.10)|v|D(ψf)(v)≤|v||ψ(v)|Df(v)+|v|Dψ(v)|f(v-)|≤(2|v||ψ(v)|+|v|Dψ(v))‖f‖∞⟶0,
as |v|→∞. Thus, ψf∈ℒw,0. The proof of the boundedness of Mψ and of the formula ∥Mψ∥=ηψ is similar to the case when Mψ:L∞→ℒw.


## 5.2. Isometries

As for all other multiplication operators in this paper, there are no isometries among the multiplication operators fromL∞ into ℒw or ℒw,0.AssumeMψ is an isometry from L∞ to ℒw or ℒw,0. Then, for v∈T the function fv=(1/(|v|+1))χv is in ℒw,0 with ∥Mψχv∥w=∥χv∥∞=1. In particular, it follows that |ψ(o)|=1/2, and for v∈T*,
(5.11)(|v|+1)|ψ(v)|=1.
Thus, |ψ(v)|=1/(|v|+1). On the other hand, taking as a test function f the characteristic function of the set {v∈T:|v|≤1}, we obtain
(5.12)1=‖f‖∞=‖Mψf‖w=|ψ(o)|+max⁡{sup⁡|v|=1|ψ(v)-ψ(o)|,sup⁡|v|=12|ψ(v)|}≥32,
which yields a contradiction. Therefore, we obtain the following result.Theorem 5.3.
There are no isometric multiplication operatorsMψ from L∞ to  ℒw or  ℒw,0.


## 5.3. Compactness and Essential Norm

The following two results are compactness criteria for multiplication operators fromL∞ into ℒw or ℒw,0 similar to those given in the previous sections.Lemma 5.4.
A bounded multiplication operatorMψ from L∞ to  ℒw is compact if and only if for every bounded sequence {fn} in L∞ converging to 0 pointwise, the sequence ∥ψfn∥w approaches 0 as n→∞.Proof.
AssumeMψ is compact, and let {fn} be a bounded sequence in L∞ converging to 0 pointwise. By rescaling the sequence, if necessary, we may assume ∥fn∥∞≤1 for all n∈ℕ. By the compactness of Mψ, {fn} has a subsequence {fnk} such that {ψfnk} converges in the ℒw-norm to some function f∈ℒw. Since by Lemma 1.3, for v∈T*,
(5.13)|ψ(v)fnk(v)-f(v)|≤(1+log⁡|v|)‖ψfnk-f‖w
and |ψ(o)fnk(o)-f(o)|≤∥ψfnk-f∥w, it follows that ψfnk→f pointwise. Since by assumption, fn→0 pointwise, the function f must be identically 0. Thus, the only limit point of the sequence {ψfn} in ℒw is 0. Hence ∥ψfn∥w→0 as n→∞.
Conversely, suppose∥ψfn∥w approaches 0 as n→∞ for every bounded sequence {fn} in L∞ converging to 0 pointwise. Let {gn} be a sequence in L∞ with ∥gn∥∞≤1. Then some subsequence {gnk} converges to a bounded function g. Thus, the sequence fnk=gnk-g converges to 0 uniformly, and ∥fn∥∞ is bounded. By the hypothesis, it follows that ∥ψfnk∥w→0 as k→∞. Thus, ψgnk→ψg in ℒw. Therefore, Mψ is compact.By an analogous argument, we obtain the corresponding result forMψ:L∞→ℒw,0.Lemma 5.5.
A bounded multiplication operatorMψ from L∞ to  ℒw,0 is compact if and only if for every bounded sequence {fn} in L∞ converging to 0 pointwise, the sequence ∥ψfn∥w approaches 0 as n→∞.Theorem 5.6.
For a bounded operatorMψ from L∞ to  ℒw, the following statements are equivalent:
(a)Mψ is compact.
(b)lim|v|→∞|v||ψ(v)|=0.Proof.
(a)⇒(b) Assume Mψ is compact. Let {vn} be a sequence in T such that |vn|→∞ as n→∞. For n∈ℕ, let fn denote the characteristic function of the set {w∈T:|w|≥|vn|}. Then ∥fn∥∞=1 and fn→0 pointwise. By Lemma 5.4 and the compactness of Mψ, it follows that
(5.14)|vn||ψ(vn)|=|vn|D(ψfn)(vn)≤‖Mψfn‖w⟶0
as n→∞.
(b)⇒(a) Assume lim|v|→∞|v∥ψ(v)|=0 and that ψ is not identically 0. In particular, ψ is bounded. Let {fn} be a sequence in L∞ converging pointwise to 0 such that ∥fn∥∞ is bounded above by some positive constant C. Then corresponding to ɛ>0, there exists N∈ℕ such that |v||ψ(v)|<ɛ/4C for all vertices v such that |v|≥N. Therefore, for |v|>N and n∈ℕ, we have
(5.15)|v|D(ψfn)(v)≤|v||ψ(v)|Dfn(v)+|v|Dψ(v)|fn(v-)|<ɛ.
Furthermore, the sequence {fn} converges to 0 uniformly on the set {v∈T:|v|≤N} so that |fn(v)|<ɛ/4N∥ψ∥∞ for all n sufficiently large. Hence |v|D(ψfn)(v)<ɛ for all v∈T* and all n sufficiently large. Consequently, ∥ψfn∥w→0 as n→∞. Using Lemma 5.4, we deduce that Mψ is compact.Since the above proof is also valid whenMψ is a bounded operator from L∞ to ℒw,0, through the application of Lemma 5.5, from Theorems 5.2 and 5.6 we obtain the following result.Corollary 5.7.
For a functionψ on T, the following statements are equivalent:
(a)Mψ:L∞→ℒw is compact.
(b)Mψ:L∞→ℒw,0 is bounded.
(c)Mψ:L∞→ℒw,0 is compact.
(d)lim|v|→∞|v||ψ(v)|=0.We now determine the essential norm of the bounded multiplication operators fromL∞ to ℒw.Theorem 5.8.
LetMψ:L∞→ℒw be bounded. Then
(5.16)‖Mψ‖e=lim⁡n→∞sup⁡|v|≥n|v|[|ψ(v)|+|ψ(v-)|].Proof.
SetB(ψ)=limn→∞sup|v|≥n|v|[|ψ(v)|+|ψ(v-)|]. In the case B(ψ)=0, then lim|v|→∞|v|ψ(v)=0, so by Theorem 5.6, Mψ is compact, and thus ∥Mψ∥e=0. So assume B(ψ)>0. Then there exists a sequence {vn} in T such that 1≤|vn|→∞ as n→∞ and
(5.17)B(ψ)=lim⁡n→∞|vn|[|ψ(vn)|+|ψ(vn-)|].
For each n∈ℕ let fn be the function on T defined by
(5.18)fn(v)={0if|v|<|vn|orψ(v)=0,ψ(v)¯|ψ(v)|if|v|≥|vn|,|v|iseven,ψ(v)≠0,-ψ(v)¯|ψ(v)|otherwise.
Then∥fn∥∞=1, and {fn} converges to 0 pointwise. Thus, for any compact operator K:L∞→ℒw, there exists a subsequence {fnk} such that ∥Kfnk∥w→0 as k→∞. Thus
(5.19)‖Mψ-K‖≥limsup⁡k→∞‖(Mψ-K)fnk‖w≥limsup⁡k→∞‖ψfnk‖w=limsup⁡k→∞sup⁡|v|≥|vnk||v|[|ψ(v)|+|ψ(v-)|]=B(ψ).
Therefore, ∥Mψ∥e≥B(ψ).
We now show that∥Mψ∥e≤B(ψ). For each n∈ℕ, define the operator Kn on L∞ by
(5.20)Knf(v)={f(v)if|v|≤n,0if|v|>n.
Then, for v∈T*, we have
(5.21)|v|D(Knf)(v)={|v|Df(v)for1≤|v|≤n,(n+1)|f(v-)|for|v|=n+1,0for|v|>n+1.
Thus, Knf∈ℒw with ∥Knf∥w≤|f(o)|+2n∥f∥∞.
Assume{fk} is a sequence in L∞ with ∥fk∥∞≤1. Then there exists a subsequence {fkj} converging pointwise to some function f∈L∞. Thus,
(5.22)‖Knfkj-Knf‖w=|fkj(o)-f(o)|+max⁡{sup⁡1≤|v|≤n|v|D(fkj-f)(v),sup⁡|v|=n+1|v||fkj(v-)-f(v-)|}≤|fkj(o)-f(o)|+2nmax⁡{sup⁡1≤|v|≤nD(fkj-f)(v),sup⁡|v|=n+1|fkj(v-)-f(v-)|}.
So ∥Knfkj-Knf∥w→0 as j→∞. Therefore, Kn is compact, and thus, since Mψ is bounded, MψKn is also compact.
Forf∈L∞, we have
(5.23)‖(Mψ-MψKn)f‖w=sup⁡|v|>n|v||ψ(v)f(v)-ψ(v-)f(v-)+ψ(v-)Knf(v-)|=max⁡{sup⁡|v|=n+1|v||ψ(v)||f(v)|,sup⁡|v|>n+1|v||ψ(v)f(v)-ψ(v-)f(v-)|}≤sup⁡|v|>n|v|[|ψ(v)|+|ψ(v-)|]‖f‖∞.
Therefore, we obtain
(5.24)‖Mψ‖e≤limsup⁡n→∞‖Mψ-MψKn‖=limsup⁡n→∞sup⁡∥f∥∞=1‖(Mψ-MψKn)f‖w≤B(ψ),
thus completing the proof.



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*Source: 472495-2011-06-05.xml*