Bounded composition operators on T p

32 Chapter 4. Composition operators onTp

several open problems. Also, there is a number of articles dealing with composition operators on different transform spaces, see for example [1,2,22].

The study of composition operators on discrete function space was first initiated by Colonna et al. [7]. In that paper, Lipschitz space of a tree was investigated. Composition operators on weighted Banach spaces of an infinite tree were considered in [10].

In this chapter, we give equivalent conditions for the composition operator C_φ to be bounded on Tp and on Tp,0 spaces and compute their operator norms. We also characterize invertible composition operators and isometric composition operators on Tp and onTp,0 spaces. Also, we discuss the compactness ofC_φ onTp spaces and finally we prove that there are no compact composition operators on Tp,0 spaces.

This chapter is based on our articles [53,56].

4.2. Bounded composition operators onTp 33 for each n. Forw∈T, we define the weight functionW as follows:

W(w) :=







(q+ 1)q^|w|−1 ifw∈T \ {o},

1 ifw=o.

(4.2.2)

Note thatW(w) is nothing butc_|w|. For the boundedness ofC_φ, we will discuss case by case.

Theorem 4.2.1. Every self-map φ of T induces bounded composition operator C_φ on T∞ with kC_φk= 1.

Proof. For eachf ∈T∞ and every self-mapφof T, we have kC_φ(f)k_∞=kf◦φk_∞= sup

w∈φ(T)

|f(w)| ≤ kfk_∞.

Thus,C_φis bounded onT∞ with kC_φk ≤1.

For the converse, note that

kC_φ(χ_v)k_∞=kχ_v◦φk_∞= 1 =kχ_vk_∞

for each v ∈ φ(T), where χv denotes the characteristic function on {v}. It gives that kC_φk ≥1. Hence the result follows.

In order to study the boundedness of the composition operators onTp for 1≤p <∞, it is convenient to deal with the case q = 1 and q ≥ 2 independently. First, we begin with the caseq = 1.

Next, we consider composition operators on Tp for 1≤p <∞ over 2−homogeneous trees.

Theorem 4.2.2. Let T be a 2−homogeneous tree with root oand let Dn={a_n, bn} for each n∈N. Furthermore, let φ be a self-map ofT and C_φ be the composition operator onTp, 1≤p <∞. We have the following:

(1) If φ(o)6=o, then kC_φk^p = 2.

(2) If φ(o) =o, then any one of the following distinct cases must occur:

34 Chapter 4. Composition operators onTp

(a) Either φ≡o or for everyn∈N, if φmaps Dn bijectively ontoDm for some m∈N thenkC_φk^p= 1.

(b) If φ maps exactly one element of Dn to o for each n∈N thenkC_φk^p = ³₂. (c) Either there exists ann∈Nsuch that φ(a_n) =φ(b_n)6=o or if there exists an

n ∈N such that |φ(an)| and |φ(bn)| are not equal and different from 0 then kC_φk^p = 2.

Proof. From the growth estimate (see Lemma2.6.1) for 2−homogeneous trees, it follows that for eachn∈N0,

M_p^p(n, Cφf) = 1 cn

X

|v|=n

|f(φ(v))|^p≤2kfk^p for every f ∈Tp.

This yields that kC_φk^p ≤2. Thus every self-map φ of T induces a boundedC_φ on Tp

withkC_φk^p≤2.

Suppose that w=φ(o)6=o. For f = 2^1pχ_w, we havekfk= 1 andkC_φ(f)k^p= 2 and hence,kC_φk^p = 2.

Now suppose that φ(o) =o. Then we need to consider all the five possible cases.

Suppose that φ≡o. Then for eachn∈N0,

M_p^p(n, C_φf) =|f(o)|^p ≤ kfk^p for every f ∈Tp.

This yields that kC_φk^p ≤ 1. For f = χ_o, we obtain that kfk^p = kC_φ(f)k^p and thus, kC_φk^p = 1.

Suppose that for every n ∈ N, φ maps D_n bijectively onto D_m for some m ∈ N.

Then,M_p^p(n, Cφf) =M_p^p(m, f) for everyn∈Nand for some m∈N. Thus

kC_φfk^p ≤ kfk^p for every f ∈Tp,

which gives that kC_φk^p ≤ 1. As in the previous case, by considering f = χ_o, we get kC_φk^p = 1.

4.2. Bounded composition operators onTp 35 Suppose that φ maps exactly one element of Dn to ofor eachn∈N. Then, in view of growth estimate for 2−homogeneous trees along with this assumption, we see that

kC_φfk^p ≤ 3

2kfk^p for every f ∈Tp

which gives kC_φk^p ≤ 3/2. On the other hand, by assumption, either a1 or b1 maps to o. Without loss of generality, we assume that φ(a₁) = o. Take φ(b₁) = w and f =χo+ 2^1pχw. Then,kfk= 1 and

M_p^p(1, Cφf) = 3

2 =kC_φ(f)k^p. Thus,kC_φk^p = 3/2.

Now assume that there exists an n∈Nsuch that w =φ(a_n) = φ(b_n) 6=o. We have already observed that kC_φk^p ≤2. Forf = 2^1pχw, we have

kfk= 1 and kC_φ(f)k^p = 2 and therefore,kC_φk^p = 2.

Finally, assume that there exists ann∈Nsuch that|φ(an)|and|φ(bn)|are not equal and are different from 0. Now, we take

f = 2^1p(χ_u+χ_v),

whereφ(a_n) =u and φ(b_n) =v. It follows that kfk= 1 and kC_φ(f)k^p = 2, which gives thatkC_φk^p= 2. The proof is complete.

Corollary 4.2.3. For every self-map φof 2−homogeneous treeT,C_φ is bounded onTp

with kC_φk^p ≤2, 1≤p <∞.

Theorem 4.2.4. If T is a (q+ 1)−homogeneous tree withq≥2 such that

sup

n∈N



 X

|v|=n

q^|φ(v)|−n



<∞, (4.2.3)

thenCφ is bounded on Tp,1≤p <∞.

36 Chapter 4. Composition operators onTp

Proof. For n∈N,w∈T andf ∈Tp, by Lemma2.6.1on growth estimate, we have M_p^p(n, C_φf) ≤ 1

c_n X

|v|=n

(q+ 1)q^|φ(v)|−1kfk^p = ^X

|v|=n

q^|φ(v)|−nkfk^p.

Moreover,

M_p^p(0, Cφf) =|f(φ(o))|^p≤(q+ 1)q^|φ(o)|−1kfk^p and thus,

kC_φfk^p ≤max







(q+ 1)q^|φ(o)|−1, sup

n∈N

(^X

|v|=n

q^|φ(v)|−n)





 kfk^p

showing that C_φis bounded onTp.

Theorem 4.2.5. LetT be a(q+1)−homogeneous tree and1≤p <∞. IfC_φis bounded onTp, then

w∈Tsup sup

n∈N0

W(w)

c_n N_φ(n, w)≤ kC_φk^p.

Proof. For each w ∈T, define fw = {W(w)χw}^1p,where W is defined in (4.2.2). It is easy to verify that for everyw∈T,M_p(n, f_w) = 1 when n=|w|and 0 otherwise. This observation gives that kf_wk= 1 for all w∈T. Now, for each fixedw∈ T, we have for n∈N0,

M_p^p(n, Cφfw) = 1 cn

X

|v|=n

W(w)χw(φ(v))

= 1

cn

X

|v|=n φ(v)=w

W(w) = W(w) cn

Nφ(n, w)

which yields that

kC_φf_wk^p = sup

n∈N0

W(w)

c_n N_φ(n, w). Consequently,

kC_φk^p = sup

kfk=1

kC_φ(f)k^p ≥sup

w∈T

kC_φ(f_w)k^p = sup

w∈T sup

n∈N0

W(w) cn

N_φ(n, w) and the desired conclusion follows.

4.2. Bounded composition operators onTp 37 Corollary 4.2.6. If Cφ is bounded on Tp, then

supⁿq^|w|−nN_φ(n, w) :w∈T \ {o}, n∈N o

is finite.

Proof. For w∈T\ {o} and n∈N, we note that W(w)

c_n =q^|w|−n. The desired result follows by Theorem4.2.5.

Corollary 4.2.7. If φ fixes the root, namely, φ(o) =o, then kC_φk ≥1.

Proof. Let f be the characteristic function on the root o. Clearly, kfk = 1 and M_p(0, C_φf) =|f(φ(o))|=|f(o)|= 1. We see that

kC_φk= sup

kgk=1

kC_φ(g)k ≥ kC_φ(f)k ≥M_p(0, C_φf) = 1 and the result follows.

Corollary 4.2.8. If φ does not fix the root, i.e. φ(o)6=o, then kC_φk^p ≥(q+ 1)q^|φ(o)|−1.

Proof. Let w=φ(o) and, as before, considerfw={W(w)χw}^1p. Now, we observe that kf_wk= 1 and M_p^p(0, Cφfw) =|f_w(φ(o))|^p= (q+ 1)q^|w|−1

which shows thatkC_φ(fw)k^p≥(q+ 1)q^|w|−1 and the desired conclusion follows.

Next, we consider composition operators onTpfor 1≤p <∞over (q+1)−homogeneous tree with q≥2. A self-map φof T is calledbounded if{|φ(v)|:v∈T} is a bounded set inN0. From Theorem 4.2.4, it is easy to see that every bounded self-map of T induces

38 Chapter 4. Composition operators onTp

a bounded composition operator. Recall that Nm,n denotes the maximum ofNφ(n, w) over|w|=m.

Theorem 4.2.9. If T is a(q+ 1)−homogeneous tree with q ≥2 andφ is a self-map of T such thatsup

v∈T|φ(v)|=M, then kC_φk^p ≤cM. Moreover, kC_φk^p =cM if and only if

n∈supN0

N_M,n c_n = 1.

Proof. For n∈N0 and f ∈Tp, by Lemma 2.6.1on growth estimate, we have

M_p^p(n, C_φf)≤c_Mkfk^p. Thus,φinduces a bounded C_φwith kC_φk^p ≤c_M.

Let us now prove the equality case. Suppose that

n∈supN0

N_M,n c_n = 1.

Then there are two cases. First we consider the case N_M,k =c_k for some k∈N0. This means that φ : D_k → DM is a constant, say, φ(v) = w ∈ DM for all v ∈ D_k. For f = (c_M)^p1χ_w, we have

kfk= 1 and kC_φ(f)k^p=cM

which proves that kC_φk^p =cM.

Next, we suppose thatN_M,k6=c_kfor allk∈N0. Then, there is a sequence{n_k}such that

N_M,nk cnk

→1 ask→ ∞.

For each k∈N, choose w_k ∈D_M such that N_M,nk =N_φ(n_k, w_k). Take f_k= (c_M)^1pχ_wk so that kf_kk= 1 and

cM

NM,nk

cn_k

≤M_p^p(nk, Cφ(f))≤ kC_φ(f)k^p≤ kC_φk^p.

By allowing k→ ∞, we getcM ≤ kC_φk^p, and thus,kC_φk^p =cM in either case.

4.2. Bounded composition operators onTp 39 For the converse part, we assume that kC_φk^p=cM. Suppose on the contrary that

n∈supN0

NM,n

c_n ≤δ <1.

Then, N_M,n ≤ δc_n for every n. Note that, there are at least N_M,n vertices from D_n mapped intoDM and therefore, there are at mostcn−NM,n vertices ofDn mapped into {v: |v|< M} for each n. For f ∈Tp, we obtain

M_p^p(n, C_φf) = 1 c_n

X

|φ(v)|=M

|v|=n

|f(φ(v))|^p+ 1 c_n

X

|φ(v)|<M

|v|=n

|f(φ(v))|^p

≤ NM,n

c_n c_Mkfk^p+cn−NM,n

c_n cM−1kfk^p

≤ (1 + (q−1)δ)c_M−1kfk^p. Therefore,

kC_φk^p ≤(1 + (q−1)δ)cM−1 < cM, which is a contradiction. Hence,kC_φk^p =cM if and only if sup

n∈N0

NM,n

cn = 1.

Now, we consider general self-maps on (q+ 1)−homogeneous trees.

Theorem 4.2.10. Let T be a (q+ 1)−homogeneous tree and 1 ≤p <∞. Then C_φ is bounded on Tp if and only if

α= sup

n∈N0

(1 c_n

∞

X

m=0

Nm,ncm

)

<∞.

Moreover, kC_φk^p =α.

Proof. Assume that α <∞.First we show that C_φ is bounded on Tp. To do this, for n∈N0 and f ∈Tp, we find that

M_p^p(n, C_φf) = 1 c_n











∞

X

m=0

X

|φ(v)|=m

|v|=n

|f(φ(v))|^p











≤ ( 1

c_n

∞

X

m=0

N_m,nc_m )

kfk^p

≤ αkfk^p,

40 Chapter 4. Composition operators onTp

which yields thatC_φ is bounded onTp and

kC_φk^p≤α = sup

n∈N0

(1 cn

∞

X

m=0

Nm,ncm

)

. (4.2.4)

Conversely, suppose that Cφ is bounded onTp. In order to show thatα is finite, we fixn∈N0. For eachm∈N0, choose v_m ∈D_m such thatN_φ(n, v_m) =N_m,n. Take

f =

∞

X

m=0

(cm)^1pχvm,

so that kfk= 1 and

M_p^p(n, C_φf) = 1 cn

∞

X

m=0

N_m,nc_m, which gives that

α= sup

n∈N0

(1 c_n

∞

X

m=0

Nm,ncm

)

≤ kC_φk^p, (4.2.5)

and hence the desired result follows. Moreover, by (4.2.4) and (4.2.5), it follows that kC_φk^p =α. The proof is complete.