1Introduction FlaviaColonnaandSongxiaoLi WEIGHTEDCOMPOSITIONOPERATORSFROMTHEBESOVSPACESINTOTHEBLOCHSPACES

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WEIGHTED COMPOSITION OPERATORS FROM THE BESOV SPACES INTO THE BLOCH SPACES

Flavia Colonna and Songxiao Li

Abstract

Letϕbe an analytic self-map of the open unit diskDin the complex planeCand letube a fixed analytic function onD. The weighted composition operator is defined on the spaceH(D) of analytic functions onD by

uCϕf=u·(f◦ϕ), f∈H(D).

In this work, we characterize the bounded and the compact weighted composition operators from the Besov spaces Bp (1 < p < ∞) into the Bloch space as well as into the little Bloch space.

1 Introduction

LetDbe the open unit disk in the complex planeC.Denote byH(D) the class of all complex-valued functions analytic on D. An analytic self-map ϕ of D induces the composition operatorCϕ onH(D), defined byCϕ(f) =f ◦ϕforf analytic onD.

Let u be a fixed analytic function onD. The functions ϕand uinduce a linear operatoruCϕon the space H(D) as follows:

uCϕf =u·(f◦ϕ), f∈H(D),

where the dot denotes pointwise multiplication. An operator of the formuCϕ

is called aweighted composition operator. The functionsuandϕare called the symbolsof the operatoruC_ϕ. We may regard this operator as a generalization of a multiplication operator and a composition operator.

An interesting problem in operator theory is to provide a function theoretic characterization of the symbols of the bounded or the compact weighted composition operators on various spaces. For an in-depth study of the composition operators, see [5] and [17].

2000Mathematics Subject Classification. Primary 47B33, Secondary 30H30.

Key words and phrases. Weighted composition operator, Besov space, Bloch space The second author is supported by the NNSF of China(No. 11001107), NSF of Guangdong Province, China(No.10451401501004305).

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An analytic functionf inDis said to belong to theBloch spaceBif B(f) = sup

z∈D

(1− |z|²)|f⁰(z)|<∞.

The expression B(f) defines a seminorm while the natural norm is given by kfk_B =|f(0)|+B(f). Under this norm B is a Banach space. Let B0 denote the subspace ofB, called thelittle Bloch space, consisting of thosef ∈ Bwhich satisfy the condition

lim

|z|→1(1− |z|²)|f⁰(z)|= 0.

For p ∈ (1,∞), the analytic Besov space, denoted by B_p, is the set of all f ∈H(D) for which

L^p_p(f) = Z

D

|f⁰(z)|^p(1− |z|²)^p−2dA(z)<∞, wheredAis the normalized area measure onD. The correspondence

f 7→ kfkB_p =|f(0)|+Lp(f)

defines a norm which yields a Banach space structure onB_p. In particular,B₂ is the classical Dirichlet space with an equivalent norm.

The composition operators on the Bloch spaces B and B0 were studied by Madigan and Matheson in [13]. In [16], Ohno and Zhao extended their results by characterizing the bounded and the compact weighted composition operators on these spaces. In [14], Ohno characterized the bounded and the compact weighted composition operators betweenH^∞and the Bloch spaceB. The weighted composition operator from Bergman-type spaces and the Zygmund spaces into the Bloch spaces were investigated by the second author and Stevi´c in [10, 12], respectively.

The issues of boundedness and compactness of the composition operators and of the weighted composition operators between different analytic function spaces on the unit disc D, the unit ball, as well as on bounded homogeneous domains inCⁿ were studied by several authors, for example, in [1–4, 6–16, 18–

20, 24, 25, 27–30] (see also related references therein).

In this paper we study the weighted composition operators from the Besov spaceB_p into the Bloch spaceBand the little Bloch spaceB0.

Specifically, in Section 2, we characterize the bounded weighted composition operators from the Besov space into the Bloch space and, in Section 3, we give compactness criteria for such operators.

In Section 4, we characterize the bounded and the compact weighted composition operators fromBp into the little Bloch space.

Finally, in Section 5, we highlight the results for the component operators Cϕ and the multiplication operatorMu defined asMu(f) =u·f. In particular, we point out that, for 1 < p < ∞, a composition operator from Bp to B is compact if and only if it is compact as an operator acting on several other analytic function spaces (and likewise mapping into the Bloch space).

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Throughout this paper, we adopt the convention of denoting byCa positive constant which may differ from one occurrence to the next. The notationab means that there is a positive constantC such thata≤Cb. If bothaband bahold, we use the notationab.

2 Boundedness into B

In this section we characterize the bounded weighted composition operatoruCϕ: Bp → B. In order to prove the main results of this paper, we shall need the following lemmas.

Lemma 1. [26] Suppose thatz∈D,t >−1. Then It(z) =

Z

D

(1− |w|²)^t

|1−zw|¯^2+tdA(w)ln 2

1− |z|², as |z| →1.

Lemma 2. [4] For an analytic self map ϕ on D and a function u ∈ B, the following statements are equivalent:

(a) The sequence{kuϕⁿk_B}_n∈_N is bounded.

(b) sup

z∈D

(1− |z|²)|u(z)ϕ⁰(z)|

1− |ϕ(z)|² <∞.

(c) sup

w∈D

kuCϕL_ϕ(w)k_B<∞, where fora∈D,La(z) = a−z

1−az, z∈D. Forw∈D, define the function

fw(z) = ln 2

1− |w|² −1/p

ln 2 1−wz

, z∈D.

We shall now use the family{fϕ(λ):λ∈D} and the sequence (p_n)_n≥0 defined byp_n(z) =zⁿto characterize the bounded weighted composition operators from B_ptoB. Note that, however,p_nis unbounded inB_pand thus the characterizing condition of boundedness we obtain in terms ofuC_ϕp_n =uϕⁿ does not follow immediately from the boundedness ofuC_ϕ.

Theorem 1. Suppose that1< p <∞,u∈H(D)andϕis an analytic self-map ofD. Then the following statements are equivalent.

(a) uC_ϕ:B_p→ B is bounded.

(b) sup

λ∈D

kuCϕf_ϕ(λ)kB<∞ and L:= sup

n≥0

kuϕⁿkB<∞.

(c)

sup

z∈D

(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

<∞ (1)

and

sup

z∈D

(1− |z|²)|ϕ⁰(z)||u(z)|

1− |ϕ(z)|² <∞. (2)

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Proof. (a) ⇒(b) Suppose thatuCϕ:Bp→ B is bounded. Then applying this operator to the functionsf(z) =z andf(z) = 1, we obtain that the quantities

sup

z∈D

(1− |z|²)|u(z)ϕ⁰(z) +u⁰(z)ϕ(z)| and sup

z∈D

(1− |z|²)|u⁰(z)|

are finite. These facts and the boundedness of the functionϕ(z) indicate that M := sup

z∈D

(1− |z|²)|u(z)ϕ⁰(z)|<∞. (3) Further, forλ∈D, consider

g_λ(z) =L_ϕ(λ)(z) = ϕ(λ)−z

1−ϕ(λ)z, z∈D.

By the M¨obius invariance of the Besov space seminorm Lp, given f ∈ Bp, we have Lp(f ◦gλ) = Lp(f) for all λ ∈ D. In particular, a straightforward calculation shows that forf(z) =z,

L^p_p(gλ) =L^p_p(f) = Z

D

(1− |z|²)^p−2dA(z) = 1

p−1. (4)

Hence

sup

λ∈D

kgλkBp= sup

λ∈D

(|ϕ(λ)|+Lp(gλ))≤1 + 1 (p−1)^1/p. Moreover gλ(ϕ(λ)) = 0 and

g⁰_λ(ϕ(λ)) = 1 1− |ϕ(λ)|². Thus,

1 + 1

(p−1)^1/p

kuC_ϕk ≥ kuC_ϕg_λk_B≥ (1− |λ|²)|u(λ)ϕ⁰(λ)|

1− |ϕ(λ)|² ,

which yields (2). Sinceu=uCϕ1 ∈ B, we may apply Lemma 2 and conclude thatL <∞.

Furthermore, it follows easily from Lemma 1 that f_w ∈ B_p, and C :=

sup_w∈DkfwkBp is finite. Thus, by the boundedness of uC_ϕ : B_p → B, we have

kuC_ϕf_wk_B≤CkuC_ϕk.

Therefore,

sup

w∈D

kuCϕfwk_B<∞, which in particular yields (b).

(b)⇒(c) Assume (b) holds. Condition (2) follows immediately from Lemma 2, which can be applied sinceu∈ B(indeed, taking n= 0, we havekuk_B ≤L <

∞).

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Next observe that, forλ∈D, kuCϕf_ϕ(λ)kB ≥

(1− |λ|²)|u(λ)ϕ(λ)ϕ⁰(λ)|

1− |ϕ(λ)|²

ln 2 1− |ϕ(λ)|²

^−1/p

−(1− |λ|²)|u⁰(λ)|

ln 2

1− |ϕ(λ)|²

1−1/p .

Thus,

(1− |λ|²)|u⁰(λ)|

ln 2

1− |ϕ(λ)|² 1−1/p

≤ kuCϕf_ϕ(λ)kB+(1− |λ|²)|u(λ)ϕ(λ)ϕ⁰(λ)|

1− |ϕ(λ)|²

ln 2 1− |ϕ(λ)|²

−1/p

≤ kuCϕf_ϕ(λ)kB+(1− |λ|²)|u(λ)ϕ⁰(λ)|

1− |ϕ(λ)|²

ln 2

1− |ϕ(λ)|² −1/p

. (5)

Taking the supremum over allλ∈D, (5) yields (1).

(c) ⇒ (a) Let q denote the conjugate index of p, i.e. 1/p+ 1/q = 1. For f ∈B_p, using the H¨older inequality and Lemma 1, we obtain

|f(w)−f(0)|= Z

D

f⁰(z) w

1−w¯zdA(z)

≤ Z

D

|f⁰(z)|(1− |z|²)^1−2/p|w|(1− |z|²)^1−2/q

|1−w¯z| dA(z)

≤Z

D

|f⁰(z)|^p(1− |z|²)^p−2dA(z)^1/pZ

D

|w|^q(1− |z|²)^q−2

|1−w¯z|^q dA(z)^1/q

≤ kfkB_p

Z

D

(1− |z|²)^q−2

|1−w¯z|^q dA(z)^1/q

≤CkfkB_p

ln 2

1− |w|² 1−1/p

.

Since|f(0)| ≤ kfkBp, we deduce that

|f(w)| ≤CkfkBp

ln 2

1− |w|² 1−1/p

. (6)

The inequality

|f⁰(z)| ≤ C

1− |z|²kfkB_p, z∈D, (7) for some positive constantC, holds sinceBp is continuously embedded intoB.

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For an arbitraryz inDand forf ∈Bp, (6) and (7) show that (1− |z|²)|(uCϕf)⁰(z)|

≤ (1− |z|²)|u⁰(z)||f(ϕ(z))|+ (1− |z|²)|f⁰(ϕ(z))||u(z)ϕ⁰(z)|

≤ C(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

kfkB_p

+C(1− |z|²)|u(z)||ϕ⁰(z)|

1− |ϕ(z)|² kfkBp. (8)

Taking the supremum in (8) over allz∈Dand then utilizing conditions (1) and (2), we conclude that the operatoruCϕ:Bp→ B is bounded.

3 Compactness into B

We begin this section by stating three preliminary lemmas which will be used to characterize the compact weighted composition operators fromBp into the Bloch space.

Lemma 3. [4]For an analytic self mapϕonDand a functionu∈ Bsatisfying the condition

lim

|ϕ(z)|→1(1− |z|²)|u⁰(z)|= 0, the following statements are equivalent:

(a) lim

n→∞kuϕⁿkB= 0.

(b) lim

|ϕ(z)|→1

(1− |z|²)|u(z)ϕ⁰(z)|

1− |ϕ(z)|² = 0.

(c) lim

|λ|→1sup

z∈D

(1− |z|²)|u(z)(CϕLλ)⁰(z)|= 0.

The following criterion for compactness follows from Lemma 3.7 of [24].

Lemma 4. Suppose that 1< p <∞,u∈H(D) andϕis an analytic self-map of D. The operator uC_ϕ : B_p → B is compact if and only if for any bounded sequence (f_n)_n∈_N in B_p which converges to zero uniformly on compact subsets ofD, we havekuCϕfnk_B→0 asn→ ∞.

We are now ready to prove the main result of this section.

Theorem 2. Suppose that1< p <∞,u∈H(D),ϕis an analytic self-map of D, anduCϕ:Bp→ Bis bounded. Then the following statements are equivalent.

(a) uC_ϕ:B_p→ B is compact.

(b)

lim

|ϕ(z)|→1kuCϕf_ϕ(z)k_B= 0 and lim

n→∞kuϕⁿk_B= 0. (9)

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(c)

lim

|ϕ(z)|→1(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

= 0 (10)

and

lim

|ϕ(z)|→1

(1− |z|²)|u(z)||ϕ⁰(z)|

1− |ϕ(z)|² = 0. (11)

Proof. (a)⇒(b) SupposeuC_ϕ:B_p→ Bis compact. Let (z_n)_n∈_Nbe a sequence in D such that |ϕ(z_n)| → 1 as n → ∞. If such a sequence does not exist conditions (10) and (11) are automatically satisfied, so suppose such a sequence exists. Forn∈Nandz∈D, define

fn(z) =f_ϕ(z_n₎=

ln 2

1− |ϕ(zn)|² −1/p

ln 2

1−ϕ(z_n)z

.

Then, the sequence{fn} converges to 0 uniformly on compact subsets ofDas n→ ∞and, as was observed in Section 2, sup_n∈_NkfnkB_p is finite. SinceuCϕ

is compact, by Lemma 4 we have

kuCϕfϕ(z_n)kB→0 as n→ ∞.

Thus, the first condition in (9) is satisfied. By Lemma 3, to prove thatkuϕⁿkB→ 0 asn→ ∞, it suffices to show that

lim

|ϕ(z)|→1(1− |z|²)|u⁰(z)|= 0 (12) and that (11) holds. Condition (12) follows at once from the boundedness of uC_ϕand condition (1).

To prove (11), assume (z_n)_n∈_N is a sequence inDsuch that |ϕ(z_n)| →1 as n→ ∞. Forn∈Nandz∈D, define

gn(z) =1− |ϕ(zn)|² 1−ϕ(z_n)z

ϕ(zn)−z 1−ϕ(z_n)z

for a sequence (zn)n∈N in D such that |ϕ(zn)| → 1 as n → ∞. Then (gn)n∈N

converges to 0 uniformly on every compact subset ofD,gn(ϕ(zn)) = 0 and g_n⁰(ϕ(z_n)) = 1

1− |ϕ(zn)|².

We now prove that (gn)_n∈_N is a bounded sequence in Bp. A straightforward calculation shows that

gn(L_ϕ(z_n₎(z)) =z−ϕ(zn)z², z∈D.

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Thus, by the conformal invariance of the Besov seminorm, using (4), we have L^p_p(gn) = L^p_p(gn◦L_ϕ(z_n₎) =

Z

|1−2ϕ(zn)z|^p(1− |z|²)^p−2dA(z)

≤ 3^p Z

(1− |z|²)^p−2dA(z) = 3^p p−1. Therefore,

kgnkBp=|gn(0)|+L_p(g_n)≤(1− |ϕ(zn)|²)|ϕ(zn)|+ 3

(p−1)^1/p ≤1 + 3 (p−1)^1/p, proving the boundedness of (gn)_n∈N inBp. Then

(1− |zn|²)|u(zn)ϕ⁰(z_n)|

1− |ϕ(zn)|² ≤sup

z∈D

(1− |z|²)|(uCϕg_n)⁰(z)| ≤ kuCϕg_nkB→0 (13) asn→ ∞.

From (13) it follows that condition (11) holds, as desired.

(b)⇒ (c) Assume (b) holds and that (zn)n∈Nis a sequence in Dsuch that

|ϕ(zn)| →1 asn→ ∞. As noted in the proof of Theorem 1, we have kuCϕf_ϕ(z_n₎k_B ≥ sup

z∈D

(1− |z|²)|(uCϕf_ϕ(z_n₎)⁰(z)|

≥

(1− |z_n|²)|u(z_n)ϕ(z_n)ϕ⁰(z_n)|

1− |ϕ(zn)|²

ln 2

1− |ϕ(zn)|² −1/p

−(1− |zn|²)|u⁰(zn)|

ln 2

1− |ϕ(zn)|²

1−1/p .

Thus, from (9), we obtain lim

|ϕ(zn)|→1

(1− |zn|²)|u(zn)ϕ(zn)ϕ⁰(zn)|

1− |ϕ(z_n)|²

ln 2

1− |ϕ(z_n)|² −1/p

= lim

|ϕ(zn)|→1(1− |zn|²)|u⁰(zn)|

ln 2

1− |ϕ(zn)|² 1−1/p

, (14)

provided that one of these two limits exists.

As argued in (a) ⇒ (b), due to the boundedness of uC_ϕ, condition (12) holds, so we may apply Lemma 3. It follows that (11) holds.

From (11) we have

n→∞lim

(1− |z_n|²)|u(z_n)ϕ(z_n)ϕ⁰(z_n)|

1− |ϕ(zn)|²

ln 2 1− |ϕ(zn)|²

^−1/p

= 0.

Therefore by (14), we get lim

|ϕ(z)|→1(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

= 0,

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which yields (10).

(c)⇒(a) First assume conditions (10) and (11) hold. In light of Lemma 4, the compactness ofuCϕ will be proved if it can be shown thatkuCϕfnkB→0 as n → ∞ for any sequence (fn)n∈N bounded in Bp converging to 0 uniformly on compact subsets of D. Let (fn)n∈N be such a sequence and set Q= sup_n∈NkfnkBp.

By (10) and (11), for anyε >0, there is a constantδ, 0< δ <1, such that δ <|ϕ(z)|<1 implies

(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

< ε/Q and

|u(z)|(1− |z|²)|ϕ⁰(z)|

1− |ϕ(z)|² < ε/Q.

LetK={w∈D:|w| ≤δ}and setE={z∈D: δ <|ϕ(z)|<1}=D\ϕ⁻¹(K).

By the boundedness ofuCϕ:Bp→ B, which in particular yieldsu=uCϕ1∈ B, we have

B(uCϕfn) = sup

z∈D

(1− |z|²)|(uCϕfn)⁰(z)|

≤ sup

z∈D

(1− |z|²)|u⁰(z)fn(ϕ(z))|+ sup

z∈D

(1− |z|²)|u(z)f_n⁰(ϕ(z))ϕ⁰(z)|

≤ sup

z∈ϕ⁻¹(K)

(1− |z|²)|u⁰(z)f_n(ϕ(z))|+ sup

z∈E

(1− |z|²)|u⁰(z)f_n(ϕ(z))|

+ sup

z∈ϕ⁻¹(K)

(1− |z|²)|u(z)ϕ⁰(z)||f_n⁰(ϕ(z))|

+ sup

z∈E

(1− |z|²)|u(z)ϕ⁰(z)||f_n⁰(ϕ(z))|

≤ kuk_B sup

w∈K

|fn(w)|+M sup

w∈K

|f_n⁰(w)|

+Csup

z∈E

(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

kfnkB_p

+Csup

z∈E

(1− |z|²)|u(z)ϕ⁰(z)|

1− |ϕ(z)|² kfnkB_p

≤ kuk_B sup

w∈K

|fn(w)|+M sup

w∈K

|f_n⁰(w)|+Cε, whereM = sup_z∈_D(1− |z|²)|u(z)ϕ⁰(z)|.Therefore,

kuCϕfnk_B=|u(0)fn(ϕ(0))|+B(uCϕfn)

≤ kuk_B sup

w∈K

|fn(w)|+M sup

w∈K

|f_n⁰(w)|+ Cε+|u(0)fn(ϕ(0))|.(15) SinceKis compact andfn→0 pointwise, it follows that,

n→∞lim sup

w∈K

|fn(w)|= 0

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and lim_n→∞|u(0)fn(ϕ(0))| = 0. On the other hand, by Cauchy’s estimates, sincefnconverges to zero uniformly on compact subsets ofD, so doesf_n⁰. From (15), lettingn→ ∞, we obtain

lim sup

n→∞

kuCϕfnkB≤Cε.

Sinceεis an arbitrary positive number, it follows that limn→∞kuCϕfnkB= 0.

Therefore,uC_ϕ:B_p→ B is compact.

4 Boundedness and compactness into the little Bloch space

We begin the section with two preliminary lemmas.

Lemma 5. Suppose that 1< p <∞,u∈H(D) andϕis an analytic self-map ofD. Then,

lim

|z|→1(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

= 0 (16)

if and only ifu∈ B₀ and

lim

|ϕ(z)|→1(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

= 0. (17)

Proof. Suppose that (16) holds, then

(ln 2)^1−1/p(1− |z|²)|u⁰(z)| ≤(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

→0 as|z| →1. Thus,u∈ B₀. Moreover, if|ϕ(z)| →1, then|z| →1, so (17) holds.

Conversely, suppose u ∈ B0 and (17) holds. Then, for every ε > 0, there existsr∈(0,1) such that

(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

< ε

whenr <|ϕ(z)|<1. Sinceu∈ B₀, there exists aσ∈(0,1), (1− |z|²)|u⁰(z)| ≤ε.

whenσ <|z|<1.

Therefore, whenσ <|z|<1 andr <|ϕ(z)|<1, we have (1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² ^1−1/p

< ε. (18)

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On the other hand, if|ϕ(z)| ≤randσ <|z|<1,then (1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

< (1− |z|²)|u⁰(z)|

ln 2 1−r²

1−1/p

<

ln 2 1−r²

1−1/p

ε. (19) Combining (18) with (19), we obtain the desired result.

The following lemma can be verified by using the method adopted in the above proof.

Lemma 6. Supposeu∈H(D) andϕis an analytic self-map ofD. Then lim

|z|→1

(1− |z|²)|u(z)ϕ⁰(z)|

1− |ϕ(z)|² = 0 if and only if

lim

|ϕ(z)|→1

(1− |z|²)|u(z)ϕ⁰(z)|

1− |ϕ(z)|² = 0 and

lim

|z|→1(1− |z|²)|u(z)ϕ⁰(z)|= 0.

Lemma 7. [13] A closed set K in B0 is compact if and only if it is bounded and satisfies

lim

|z|→1sup

f∈K

(1− |z|²)|f⁰(z)|= 0.

Using Lemmas 5, 6 and 7, and arguing as in the proof of Theorems 4.4 and 4.5 of [10], we derive the following two results. We omit the details.

Theorem 3. Suppose that1< p <∞,u∈H(D)andϕis an analytic self-map ofD. ThenuC_ϕ:B_p→ B₀ is bounded if and only if uC_ϕ:B_p→ B is bounded, u∈ B₀ and

lim

|z|→1(1− |z|²)|u(z)ϕ⁰(z)|= 0.

Theorem 4. Suppose that1< p <∞,u∈H(D)andϕis an analytic self-map ofD. ThenuC_ϕ:B_p→ B0 is compact if and only if

lim

|z|→1(1− |z|²)|u⁰(z)|

ln 2

1− |ϕ(z)|² 1−1/p

= 0 and

lim

|z|→1

(1− |z|²)|u(z)ϕ⁰(z)|

1− |ϕ(z)|² = 0.

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5 The component operators

We end the paper by discussing the boundedness and the compactness of the component operators, the multiplication operatorM_u and the operatorC_ϕ.

5.1 The operator M

_u

In the special case whenϕ(z) =z, forz∈D, condition (1) implies thatu∈ B0. From Theorems 1 and 3 we deduce the following result.

Corollary 1. Letube analytic onDand1< p <∞. The following statements are equivalent.

(a) Mu:Bp→ Bis bounded.

(b)Mu:Bp→ B0 is bounded.

(c) u∈H^∞ andsup_z∈_D(1− |z|²)|u⁰(z)|

ln_1−|z|² 2

1−1/p

<∞.

From Theorems 2 and 4, we obtain the following corollary.

Corollary 2. Letube analytic onDand1< p <∞. The following statements are equivalent.

(a) Mu:Bp→ Bis compact.

(b)Mu:Bp→ B0 is compact.

(c) uis identically 0.

5.2 The operator C

_ϕ

By the Schwartz-Pick Lemma, we see that the operatorC_ϕ:B_p→ Bis bounded for any analytic self-mapϕofDand 1< p <∞.

From Theorem 2, we obtain the following result which was proved in [4] in the special case of the Dirichlet spaceD.

Corollary 3. Suppose that 1 < p < ∞ and ϕ is an analytic self-map of D. Then the following statements are equivalent:

(a) Cϕ:Bp→ Bis compact.

(b) lim

n→∞kϕⁿkB= 0.

(c) lim

|ϕ(z)|→1

(1− |z|²)|ϕ⁰(z)|

1− |ϕ(z)|² = 0.

Recall that the space BMOA is defined as the Banach space of functionsf in the Hardy spaceH² such that

kfk∗= sup

λ∈D

kf◦L_λ−f(λ)kH²<∞ with normkfkBM OA=|f(0)|+kfk∗,where

kgk²_H2= sup

0<r<1

1 2π

Z 2π

0

|g(re^iθ)|²dθ.

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In [23], Theorem 3.1, Tjani characterized the compact composition operators from the Besov spaces, the space BMOA, and the Bloch space intoB in terms of the family of automorphisms{Lλ: λ∈D}.

Theorem 5. [23]Let ϕbe an analytic self-map of Dand letX=Bp(1< p <

∞),BMOA, or B. ThenC_ϕ:X → Bis a compact operator if and only if lim

|λ|→1kCϕL_λkB= 0.

Furthermore, in [4], it was shown that a composition operatorC_ϕ:H^∞→ B is compact if and only if

n→∞lim kϕⁿk_B = 0.

From this, Theorem 5, and Corollary 3, we obtain the following result:

Corollary 4. Suppose that 1 < p < ∞ and ϕ is an analytic self-map of D. Then the following statements are equivalent:

(a) Cϕ:Bp→ Bis compact.

(b)Cϕ:BM OA→ B is compact.

(c) Cϕ:B → B is compact.

(d)Cϕ:H^∞→ Bis compact.

From Theorem 3 and using the remarks in [13] (see Section 1), we obtain the following result.

Corollary 5. Suppose that 1 < p < ∞ and ϕ is an analytic self-map of D. Then the following statements are equivalent.

(a) Cϕ:Bp→ B0 is bounded.

(b)Cϕ:B0→ B0 is bounded.

(c) ϕ∈ B0.

Finally, from Theorem 4 and Theorem 1 in [13], we can easily arrive at the following corollary.

Corollary 6. Suppose that 1 < p < ∞ and ϕ is an analytic self-map of D. Then the following statements are equivalent.

(a) Cϕ:Bp→ B0 is compact.

(b)Cϕ:B0→ B0 is compact.

(b) lim

|z|→1

(1− |z|²)|ϕ⁰(z)|

1− |ϕ(z)|² = 0.

6 Acknowledgment

We wish to thank the referee for his/her useful suggestions for the improvement of the manuscript.

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Flavia Colonna: Department of Mathematical Sciences George Mason University Fairfax, VA 22030

E-mail address:[email protected]

Songxiao Li: Department of Mathematics, JiaYing University, 514015, Meizhou, GuangDong, China

E-mail address: [email protected]; [email protected]