(1)Banach Journal of MathematicalAnalysis ISSN electronic) www.emis.de/journals/BJMA/ WEIGHTED COMPOSITION OPERATORS BETWEEN VECTOR-VALUED LIPSCHITZ FUNCTION SPACES K

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www.emis.de/journals/BJMA/

WEIGHTED COMPOSITION OPERATORS BETWEEN VECTOR-VALUED LIPSCHITZ FUNCTION SPACES

K. ESMAEILI AND H. MAHYAR^∗ Communicated by K. Jarosz

Abstract. We give necessary and sufficient conditions for the boundedness and compactness of weighted composition operators between spaces of vector- valued Lipschitz functions. We then show that a bounded separating linear operator between these spaces is indeed a weighted composition operator.

1. Introduction and preliminaries

Let (X, d) be a compact metric space, (E,k · k) be a Banach space andα∈(0,1].

The space of all functions f :X →E for which p_α(f) = sup

kf(x)−f(y)k

d^α(x, y) :x, y ∈X, x6=y

<∞, is denoted by Lip_α(X, E). The subspace of those functions f with

d(x,y)→0lim

kf(x)−f(y)k d^α(x, y) = 0,

is denoted by lip_α(X, E). The spaces Lip_α(X, E) and lip_α(X, E) are Banach spaces when equipped with the norm kfk_α = kfk_X +p_α(f), where kfk_X = sup{kf(x)k : x ∈ X}. These are called vector-valued Lipschitz function spaces.

In the case where E is the scalar field of the complex numbersC, to simplify the notation, we write Lip_α(X) and lip_α(X) instead of Lip_α(X,C) and lip_α(X,C),

Date: Received: 4 May 2012; Revised: 8 June 2012; Accepted: 15 June 2012.

∗ Corresponding author.

2010Mathematics Subject Classification. Primary 47B33; Secondary 46E40, 47B38.

Key words and phrases. Vector-valued Lipschitz function space, weighted composition oper- ator, compact linear operator, separating map, disjointness preserving operator.

59

respectively. In this case, Lip_α(X) and lip_α(X) are Banach algebras which are also called Lipschitz algebras. The scalar-valued Lipschitz functions were first studied by de Leeuw [16] and Sherbert [17, 18]. The interested reader is also referred to [3, 6, 19] for further details on the subject. The spaces Lip_α(X, E) and lip_α(X, E) were first considered by Johnson [13]. Since then, there has been an extensive study on this subject.

In this paper we shall study a class of operators between vector-valued Lips- chitz function spaces known as weighted composition operators. Given X and Y two compact Hausdorff spaces and given E and F two Banach spaces, let S(X, E) be any subspace of C(X, E), the space of all continuous E-valued func- tions on X. A weighted composition operator between vector-valued function spaces is defined to be a linear operator T : S(X, E) → S(Y, F) of the form T f(y) = W_y(f(ϕ(y))) for every f ∈ S(X, E) and y∈Y whereW_y is a linear op- erator fromE intoF andϕ:Y →X is a function. In the scalar case, a weighted composition operator is a composition operator followed by a multiplier. The compactness of the weighted composition operators onC(X,C) has been charac- terized by Kamowitz [14]. Kamowitz and Scheinberg also determined necessary and sufficient conditions for composition operators on Lip_α(X) and lip_α(X) to be compact [15]. Jamison and Rajagopalan provided a necessary and sufficient con- dition for weighted composition operators on C(X, E) to be compact [7]. Chan has improved their results and characterized the compact weighted composition operators on the space of continuous functions defined on a locally compact Haus- dorff space vanishing at infinity [4]. In section 2, we give necessary and sufficient conditions for weighted composition operators between spaces of vector-valued Lipschitz functions to be compact.

It is also interesting to determine which types of operators have the form of weighted composition operators. We say that a linear operator T : S(X, E) → S(Y, F) has thedisjoint support property or isseparating, ifkT f(y)kkT g(y)k= 0 for all y ∈ Y whenever f, g ∈ S(X, E) satisfy kf(x)kkg(x)k = 0 for all x ∈ X.

We say that T is biseparating if it is bijective and both T and T⁻¹ are separat- ing. Note that a typical example of separating operators between function spaces are weighted composition operators and the standard problem is to determine whether these are the canonical examples. The notion of separating operator has been studied extensively. Separating linear maps between spaces of contin- uous scalar-valued functions were studied by Beckenstein, et al. [2], Font and Hern´andez [5], Jaroz [8] and Jeang and Wong [9]. Bounded separating linear op- erators on the space C(X, E) were studied by Jamison and Rajagopalan [7] and Chan [4]. The study of separating linear maps between spaces of scalar-valued Lipschitz functions was initiated by Wu in [20]. Jim´enez-Vargas in [10] obtained a representation of separating linear maps between scalar-valued Lipschitz algebras lip_α(X) and lip_α(Y) whenXand Y are compact andα∈(0,1). Biseparating lin- ear operators between spaces of vector-valued Lipschitz functions were studied by Araujo and Dubarbie in [1] and Jimen´ez-Vargas, et al. in [11] and [12]. In section 3, we shall show that bounded separating linear operators between vector-valued Lipschitz function spaces are weighted composition operators.

Given Banach spaces E and F, we denote by B(E, F) and K(E, F) the space of all bounded linear operators and compact linear operators from E into F, respectively. The dual space of a Banach spaceE is denoted byE^∗. In this note, the linear operators which we have considered are nonzero.

2. Compact weighted composition operators

In this section, we provide necessary and sufficient conditions for weighted composition operators between vector-valued Lipschitz spaces to be compact.

In what follows, we will assume thatX andY are compact metric spaces,Eand F are Banach spaces and α ∈ (0,1]. It is interesting to note that Lip₁(X, E)⊆ lip_α(X, E) ⊆ Lip_α(X, E) for α ∈ (0,1). For each scalar-valued function h on X andE-valued function f onX, one can define theE-valued function hf onX by (hf)(x) = h(x)f(x) for x ∈ X. If h ∈ Lip₁(X) and f ∈ Lip_α(X, E), then hf ∈ Lip_α(X, E). Similarly, it is true for lip_α(X, E). That is, both spaces Lip_α(X, E) and lip_α(X, E) are Lip₁(X)-modules. Moreover, Lip_α(X, E) is a Lip_α(X)-module and lip_α(X, E) is a lip_α(X)-module.

Identifying each e ∈ E with the constant function 1_e(x) = e for x ∈ X, the Banach space E can be considered as a subspace of Lip₁(X, E). Hence, given e ∈ E and f ∈ Lip_α(X), the function f_e defined by f_e(x) = f(x)e for x ∈ X belongs to Lip_α(X, E). Moreover, kf_ek_X = kfk_Xkek, p_α(f_e) = p_α(f)kek and hence kf_ek_α =kekkfk_α.

Recall that a weighted composition operator T : S(X, E)→ S(Y, F) is of the formT f(y) = W_y(f(ϕ(y))) whereW_y :E →F is a linear operator for eachy∈Y and ϕ:Y →X is a function. We denote by N the set of all yin Y for whichW_y is the zero operator. A map ϕ : Y → X is said to be a Lipschitz function on a subsetK ofY, if there exists a constantc > 0 such thatd(ϕ(y), ϕ(y⁰))≤cd(y, y⁰) for all y, y⁰ in K.

We first characterize a bounded weighted composition operator between vector- valued Lipschitz function spaces.

Proposition 2.1. LetS(X, E)⊆C(X, E)andS(Y, F)⊆C(Y, F)be two Banach spaces such that the topology of pointwise convergence is weaker than their norm topology. Suppose that T :S(X, E)→ S(Y, F)is a weighted composition operator of the form T f(y) = Wy(f(ϕ(y))). If Wy ∈ B(E, F) for each y ∈ Y, then T is bounded.

Proof. Let (f_n) be a sequence in S(X, E) that converges to zero and (T f_n) con- verges tog inS(Y, F). Then by the hypothesisf_n(ϕ(y))→0 and T f_n(y)→g(y) for every y ∈ Y. The boundedness of each Wy implies that Wy(fn(ϕ(y))) → 0.

Thereforeg = 0 and by the closed graph theorem, the map T is bounded.

Remark 2.2. The definition of Lipschitz normk · k_α asserts thatk · k_α-convergence implies pointwise convergence. Therefore one can conclude Proposition 2.1 for Lip_α(X, E) and lip_α(X, E).

Theorem 2.3. Let T : Lip_α(X, E)→Lip_α(Y, F) be a nonzero bounded weighted composition operator of the form T f(y) = W_y(f(ϕ(y))). Then W ∈

Lip_α(Y,B(E, F)) and ϕ is continuous on Y \N and Lipschitz on every compact subset of Y \N.

Proof. Let y∈Y. Then

kW_y(e)k=kT1_e(y)k ≤ kT1_ek_Y ≤ kT1_ek_α ≤ kTkkek,

for all e ∈ E. Hence Wy ∈ B(E, F) and kWyk ≤ kTk. We now show that W ∈Lip_α(Y,B(E, F)). To do this, let y₁, y₂ ∈Y with y₁ 6=y₂. Then

kWy1(e)−Wy2(e)k

d^α(y₁, y₂) = kT1e(y1)−T1e(y2)k

d^α(y₁, y₂) ≤p_α(T1_e)≤ kT1_ek_α ≤ kTkkek, for all e∈E. It follows that

kW_y1 −W_y2k

d^α(y₁, y₂) = sup

kek≤1

kW_y1(e)−W_y2(e)k

d^α(y₁, y₂) ≤ kTk, and W ∈Lip_α(Y,B(E, F)) with p_α(W)≤ kTk.

To show the continuity of ϕ on Y \N, we first prove for given f ∈ Lip_α(X), the scalar-valued map Ψ_f(y, e, u^∗) =< T f_e(y), u^∗ >is continuous onY ×E×F^∗. Fix a nonzero function f ∈Lip_α(X) and (y₀, e₀, u^∗₀)∈Y ×E×F^∗. Given ε >0, by the continuity of T f_e0 at y₀, there exists a neighborhood U₁ of y₀ in Y such that kT f_e0(y)−T f_e0(y₀)k < _3(1+ku^ε ∗

0k) for every y ∈ U₁. If δ < _3(1+ku^ε∗

0k)kTk, and U₂ :={e∈E :ke−e₀k< _kf^δ_k

α}, then

kT f_e−T f_e0k_α ≤ kTkkf_e−f_e0k_α ≤ kTkkfk_αke−e₀k< ε 3(1 +ku^∗₀k), for everye∈U₂. If U =U₁×U₂×U₃ where U₃ ={u^∗ ∈F^∗ :ku^∗−u^∗₀k< r}and r= ^ε₃(_3(1+ku^ε ∗

0k) +kT f_e0k_α)⁻¹, thenU is a neighborhood of (y₀, e₀, u^∗₀) and

|Ψ_f(y, e, u^∗)−Ψ_f(y₀, e₀, u^∗₀)|=|< T f_e(y), u^∗ >−< T f_e0(y₀), u^∗₀ >|

≤kT f_e(y)kku^∗−u^∗₀k+ku^∗₀kkT f_e(y)−T f_e0(y)k +ku^∗₀kkT f_e0(y)−T f_e0(y₀)k

<ku^∗−u^∗₀kkT f_ek_α+ku^∗₀kkT f_e−T f_e0k_α+ε 3

<( ε

3(1 +ku^∗₀k)+kT f_e0k_α)ku^∗−u^∗₀k+2ε 3 < ε, for all (y, e, u^∗)∈U.

Using the above result for the constant function f = 1, we conclude that the map Ψ₁(y, e, u^∗) =< T1_e(y), u^∗ >is continuous on Y ×E×F^∗ and therefore,

coz(Ψ1) ={(y, e, u^∗)∈Y ×E×F^∗ : Ψ1(y, e, u^∗)6= 0}

={(y, e, u^∗)∈Y ×E×F^∗ :< T1_e(y), u^∗ >=< W_y(e), u^∗ >6= 0}, is an open set inY ×E×F^∗. One can write,

f(ϕ(y)) = < T f_e(y), u^∗ >

< W_y(e), u^∗ > = < T f_e(y), u^∗ >

< T1_e(y), u^∗ > = Ψ_f(y, e, u^∗) Ψ₁(y, e, u^∗),

for every (y, e, u^∗) ∈ coz(Ψ₁) and f ∈ Lip_α(X). Therefore the continuity of Ψ_f and Ψ₁ implies that the map (y, e, u^∗) 7→ f(ϕ(y)) is continuous on coz(Ψ₁) for given f ∈Lip_α(X).

Let y ∈ Y \N. Then < W_y(e), u^∗ >6= 0 for some e ∈ E and u^∗ ∈ F^∗. Hence (y, e, u^∗) ∈ coz(Ψ₁). Suppose that (y_n) is a sequence in Y \ N converging to y. Then, (y_n, e, u^∗) ∈ coz(Ψ₁) for large enough n. If ϕ(y_n) does not converge to ϕ(y), there exists an open neighborhood V of ϕ(y) in X and a subsequence (yn_k) of (yn) such that ϕ(yn_k) ∈/ V for each k. On the other hand, considering the function f(x) = dist(x, X \ V), we note that f ∈ Lip_α(X), then by the above discussion the map (y, e, u^∗) 7→ f(ϕ(y)) is continuous on coz(Ψ₁). Hence f(ϕ(y_nk)) →f(ϕ(y)) as k → ∞. However, by the definition of f, f(ϕ(y_nk)) = 0 for each k and f(ϕ(y)) 6= 0 which is a contradiction. Therefore ϕ is continuous onY \N.

To finish the proof, it remains to be shown that the functionϕis Lipschitz on every compact subset of Y \N. Let K be a compact subset of Y \N. For each y ∈ Y, define f_y(x) = d^α(x, ϕ(y)) for all x ∈ X. Let e ∈ E be with kek = 1.

Therefore, (f_y)_e ∈ Lip_α(X, E) and k(f_y)_ek_α ≤ 1 + (diam(X))^α for every y ∈ Y. Lety₁, y₂ ∈Y with y₁ 6=y₂. Then

d^α(ϕ(y₁), ϕ(y₂))

d^α(y₁, y₂) kW_y2(e)k= kT(f_y1)_e(y₁)−T(f_y1)_e(y₂)k d^α(y₁, y₂)

≤p_α(T(f_y1)_e)≤ kT(f_y1)_ek_α ≤ckTk, for every e∈E with kek= 1 where c= 1 + (diam(X))^α. Therefore,

d^α(ϕ(y₁), ϕ(y₂))

d^α(y1, y2) kW_y2k= d^α(ϕ(y₁), ϕ(y₂)) d^α(y1, y2) sup

kek=1

kW_y2(e)k ≤ckTk.

The continuity of y7→ kW_yk implies thatγ = inf{kW_yk:y∈K}>0. Then d(ϕ(y₁), ϕ(y₂))

d(y₁, y₂) ≤

ckTk

γ

1/α

,

for everyy1, y2 ∈K with y1 6=y2 which implies thatϕis Lipschitz onK and the

proof of theorem is complete.

Using the same argument as in the proof of Theorem 2.3, we can obtain the similar result for lip_α(X, E).

Theorem 2.4. Let α ∈ (0,1) and let T : lip_α(X, E) → lip_α(Y, F) be a nonzero bounded weighted composition operator of the form T f(y) = W_y(f(ϕ(y))). Then W ∈ Lip_α(Y,B(E, F)) and ϕ is continuous on Y \ N and Lipschitz on every compact subset of Y \N.

Proof. Exactly the same as the proof of Theorem 2.3, one can show that W ∈ Lip_α(Y,B(E, F)) and ϕ is continuous on Y \N. To show that ϕ is Lipschitz on every compact subsetK of Y \N, we employ the function

fy1,y2(x) = (d(x, ϕ(y2))+d(ϕ(y1), ϕ(y2)))^α−d^α(ϕ(y1), ϕ(y2)) (x∈X, y1, y2 ∈Y).

In general, ifa >0, the functiong(t) = (t+a)^α−a^α,t≥0 has bounded derivative.

Then for any b > 0, g ∈ Lip₁([0, b]) ⊆ lip_α([0, b]). Fix y₁, y₂ ∈ Y with y₁ 6= y₂

and set a = d(ϕ(y₁), ϕ(y₂)) and b = diam(X). Then f_y1,y2(x) = g(d(x, ϕ(y₂))) for x∈X. Therefore, f_y1,y2 ∈ Lip₁(X)⊆lip_α(X),kf_y1,y2k_α ≤1 + (2 diam(X))^α, f_y1,y2(ϕ(y₂)) = 0 and f_y1,y2(ϕ(y₁)) = (2^α−1)d^α(ϕ(y₁), ϕ(y₂)).

Lete∈E with kek= 1. Therefore, we have (f_y1,y2)_e∈lip_α(X, E) and d^α(ϕ(y₁), ϕ(y₂))

d^α(y₁, y₂) kW_y1(e)k= 1 2^α−1

kf_y1,y2(ϕ(y₁))W_y1(e)k d^α(y₁, y₂)

= 1

2^α−1

kT(f_y1,y2)_e(y₁)−T(f_y1,y2)_e(y₂)k d^α(y₁, y₂)

≤ 1

2^α−1p_α(T(f_y1,y2)_e)≤ 1

2^α−1kT(f_y1,y2)_ek_α ≤ckTk, where c= ₂α¹−1(1 + (2 diam(X))^α). Therefore,

d^α(ϕ(y₁), ϕ(y₂))

d^α(y₁, y₂) kW_y1k= d^α(ϕ(y₁), ϕ(y₂)) d^α(y₁, y₂) sup

kek=1

kW_y1(e)k ≤ckTk.

The continuity of y7→ kW_yk implies thatγ = inf{kW_yk:y∈K}>0. Then d(ϕ(y₁), ϕ(y₂))

d(y₁, y₂) ≤

ckTk

γ

1/α

,

for every y₁, y₂ ∈K with y₁ 6=y₂ which implies that ϕis Lipschitz on K.

The following example shows that in Theorems2.3and2.4,ϕis not necessarily Lipschitz on Y \N.

Example 2.5. Suppose X = [0,√

2],Y = [−1,1] and E is an arbitrary Banach space. Define W_y(e) = (1 +y)e and ϕ(y) = √

1 +y for every y ∈ [−1,1] and e ∈ E. Then N = {−1}. Let T : Lip_α(X, E) → Lip_α(Y, E) be the weighted composition operator induced by W and ϕ. Clearly, ϕ is continuous on [−1,1]

and it is not Lipschitz on (−1,1]. However,ϕ is Lipschitz on [−1 +δ,1] for each δ∈(0,2). Moreover, we haveT f(y) = (1+y)f(√

1 +y) for everyf ∈Lip_α(X, E) and y∈[−1,1] which is bounded.

To investigate the compactness of weighted composition operators between spaces of vector-valued Lipschitz functions, we need the following results. The first one is the generalized Arzela-Ascoli theorem for vector-valued continuous functions.

Theorem 2.6. [4, Theorem A] A subset H of C(X, E) is relatively compact if and only if the following conditions are satisfied:

(i) H is equicontinuous, and

(ii) H(x) = {f(x) :f ∈H} is relatively compact for every x∈X.

The following definition and theorem are provided in [15].

Definition 2.7. A map ϕ:Y →X will be called a supercontraction on K ⊆Y if ^d(ϕ(y),ϕ(y_d(y,y0)⁰⁾⁾ →0 whenever y, y⁰ ∈K and d(y, y⁰)→0

Theorem 2.8. [15, Theorem 1] Let T : Lip_α(X) → Lip_α(Y) be a composition operator of the form T f(y) = f(ϕ(y)) for every f ∈ Lip_α(X) and y ∈ Y where ϕ:Y →X. Then T is compact if and only if ϕ is a supercontraction.

It was established in [3] that lip_α(X)^∗∗= Lip_α(X) for 0 < α <1. As mentioned in [15, page 260], using this fact the following corollary follows from the above theorem.

Corollary 2.9. Let α ∈ (0,1) and let T : lip_α(X) → lip_α(Y) be a composition operator of the form T f(y) = f(ϕ(y)) for every f ∈ lip_α(X) and y ∈ Y where ϕ:Y →X. Then T is compact if and only if ϕ is a supercontraction.

Theorem2.8gives motivation to the following theorem which is our main result of this section.

Theorem 2.10. Let T : Lip_α(X, E)→Lip_α(Y, F) be a nonzero weighted compo- sition operator of the form T f(y) = W_y(f(ϕ(y))).

(i) If T is compact, then W ∈ Lip_α(Y,K(E, F)), and ϕ is continuous on Y \N and a supercontraction on every compact subset of Y \N.

(ii) If W ∈lip_α(Y,K(E, F)) and ϕ is a supercontraction on Y \N, then T is compact.

Proof. (i) By Theorem2.3,W ∈Lip_α(Y,B(E, F)). Thus it is enough to show that W_y ∈ K(E, F) for eachy ∈Y. To do this, let (e_n) be a bounded sequence in E.

The sequence (1_en) is bounded in Lip_α(X, E). Since T is compact, there exists a subsequence (e_nk) such that (T1_enk) converges in Lip_α(Y, F). In particular, (W_y(e_nk)) converges in F that is W_y ∈ K(E, F) for each y ∈Y.

By Theorem2.3,ϕis continuous onY\N. To show thatϕis a supercontraction on every compact subset of Y \N, we first fix a point y₀ ∈ Y \N and e ∈ E with kek = 1 such that W_y0(e) 6= 0. The continuity of the function y 7→ W_y(e) from Y into F implies that, there exists δ >0 such that kW_y(e)k >0 for every y ∈Y with d(y, y₀)≤ δ. Consider the compact subset ∆ = B(y₀, δ) of Y where B(y₀, δ) = {y ∈ Y : d(y, y₀) < δ}. Then γ = infy∈∆kW_y(e)k > 0. Define S : Lip_α(X)→Lip_α(∆) by Sf =f◦ϕon ∆. We show that S is compact. To do this, let (f_n) be a sequence in Lip_α(X) such that kf_nk_α = 1. Thus (g_n) = ((f_n)_e) is a bounded sequence in Lip_α(X, E). By the compactness of T, there exists a subsequence (g_nk) of (g_n) such that (T g_nk) converges and hence it is a Cauchy sequence in Lip_α(Y, F). By the definition of S, we have

|Sf_nk(y)−Sf_nl(y)|=|f_nk(ϕ(y))−f_nl(ϕ(y))|

= kf_nk(ϕ(y))W_y(e)−f_nl(ϕ(y))W_y(e)k kW_y(e)k

≤ kT g_nk(y)−T g_nl(y)k

γ ≤ kT g_nk−T g_nlk_Y

γ ,

for every y∈∆. Hence

kSf_nk −Sf_nlk_∆≤ 1

γkT g_nk−T g_nlk_Y. (2.1)

Also, using the inequality p_α(W)≤ kTk, we obtain

|S(f_nk−f_nl)(y)−S(f_nk −f_nl)(y⁰)|

d^α(y, y⁰)

= k(fn_k −fn_l)(ϕ(y))Wy(e)−(fn_k−fn_l)(ϕ(y⁰))Wy(e)k kW_y(e)kd^α(y, y⁰)

≤ kT(g_nk−g_nl)(y)−T(g_nk−g_nl)(y⁰)k

kWy(e)kd^α(y, y⁰) + k(W_y−W_y⁰)(e)k

kWy(e)kd^α(y, y⁰)|(f_nk−f_nl)(ϕ(y⁰))|

≤ 1

γp_α(T(g_nk −g_nl)) + 1

γp_α(W)kSf_nk −Sf_nlk_∆

≤ 1

γp_α(T(g_nk −g_nl)) + 1

γ²kTkkT g_nk −T g_nlk_Y, for every y, y⁰ ∈∆ with y6=y⁰. Therefore,

p_α(Sf_nk −Sf_nl)≤ 1

γp_α(T(g_nk −g_nl)) + 1

γ²kTkkT g_nk−T g_nlk_Y. (2.2) From (2.1) and (2.2), we conclude that (Sf_nk) is a Cauchy and then a convergent sequence in Lip_α(∆). It implies that S is compact. By Theorem 2.8, ϕ|∆ is a supercontraction.

We have shown that for every y ∈ Y \N there exists some δ > 0 such that ϕis a supercontraction on B(y, δ). We now assume that K is a compact subset of Y \N. Then there exist y₁, . . . , y_n in K and positive numbers δ₁, . . . , δ_n such that {B(y_i,^δ₂ⁱ)}ⁿ_i=1 covers K and ϕis a supercontraction on eachB(y_i, δ_i). Then for any ε > 0, one can choose a positive number δ < min{^δ₂¹, . . . ,^δ₂ⁿ} such that

d(ϕ(y),ϕ(y⁰))

d(y,y⁰) < εfor everyy, y⁰ ∈K with 0< d(y, y⁰)< δ. This completes the proof of (i).

(ii) Using Proposition2.1and Remark2.2, one can say thatT is bounded. For compactness ofT, we assume that (f_n) is a bounded sequence in Lip_α(X, E) with kf_nk_α ≤ 1. Then boundedness of T and the fact that T f_n ∈ Lip_α(Y, F) for all n∈Nimply that (T f_n) is an equicontinuous subset ofC(Y, F), and compactness of each W_y (y ∈ Y) implies that (T f_n(y)) is relatively compact for each y ∈ Y. Therefore, by Theorem 2.6, (T f_n) is relatively compact in C(Y, F). Hence there exists a subsequence (f_nk) of (f_n) such that (T f_nk) converges in C(Y, F) and then it is a Cauchy sequence in C(Y, F). We will show that (T fn_k) is a Cauchy sequence in Lip_α(Y, F). Given ε >0, there exists a δ >0 such that

d(ϕ(y), ϕ(y⁰))

d(y, y⁰) <( ε

3kWk_α)^1/α, (2.3)

for every y, y⁰ ∈Y \N with 0< d(y, y⁰)< δ, kW_y −W_y⁰k

d^α(y, y⁰) < ε

6, (2.4)

for every y, y⁰ ∈Y with 0< d(y, y⁰)< δ, and sup

y∈Y

kT f_nk(y)−T f_nl(y)k< 1

2δ^αε, (2.5)

for large enough k, l, since ϕ is a supercontraction onY \N, the weight W is in lip_α(Y,K(E, F)) and (T f_nk) is a Cauchy sequence in C(Y, F), respectively.

Lety, y⁰ ∈Y and k, l be large enough. We consider three cases.

Case 1. If d(y, y⁰)≥δ, using (2.5) we obtain kT f_nk(y)−T f_nl(y)−T f_nk(y⁰) +T f_nl(y⁰)k

d^α(y, y⁰) ≤ 2

δ^α sup

y∈Y

kT f_nk(y)−T f_nl(y)k< ε.

Case 2. If y, y⁰ ∈ Y \N with 0 < d(y, y⁰) < δ and ϕ(y) 6= ϕ(y⁰), then applying (2.3) and (2.4) we get

kT f_nk(y)−T f_nl(y)−T f_nk(y⁰) +T f_nl(y⁰)k d^α(y, y⁰)

=kWy(fn_k(ϕ(y)))−Wy(fn_l(ϕ(y)))−Wy⁰(fn_k(ϕ(y⁰))) +Wy⁰(fn_l(ϕ(y⁰)))k d^α(y, y⁰)

≤kWykkfn_k(ϕ(y))−fn_k(ϕ(y⁰))k

d^α(y, y⁰) + kWy −Wy⁰kkfn_k(ϕ(y⁰))−fn_l(ϕ(y⁰))k d^α(y, y⁰)

+kW_ykkf_nl(ϕ(y⁰))−f_nl(ϕ(y))k d^α(y, y⁰)

≤kWk_Yp_α(f_nk)d^α(ϕ(y), ϕ(y⁰)) d^α(y, y⁰) + ε

6(kf_nkk_X +kf_nlk_X) +kWk_Yp_α(f_nl)d^α(ϕ(y), ϕ(y⁰))

d^α(y, y⁰)

<2kWk_α· ε

3kWk_α + 2ε 6 =ε.

Case 3. If y∈Y \N and y⁰ ∈N with 0< d(y, y⁰)< δ, applying (2.4), we obtain kT f_nk(y)−T f_nl(y)−T f_nk(y⁰) +T f_nl(y⁰)k

d^α(y, y⁰) =kT f_nk(y)−T f_nl(y)k d^α(y, y⁰)

≤kW_ykkf_nk(ϕ(y))−f_nl(ϕ(y))k d^α(y, y⁰)

≤2kW_y −W_y⁰k d^α(y, y⁰) < ε

3 < ε.

Therefore, (T f_nk) is a Cauchy sequence in Lip_α(Y, F) from which we have T is

compact.

Using Corollary 2.9, similar to the proof of Theorem 2.10, one can conclude the following results for vector-valued little Lipschitz function spaces lip_α(X, E).

Theorem 2.11. Let α∈ (0,1) and let T : lip_α(X, E)→ lip_α(Y, F) be a nonzero weighted composition operator of the form T f(y) = W_y(f(ϕ(y))).

(i) If T is compact, then W ∈ Lip_α(Y,K(E, F)) and ϕ is a supercontraction on every compact subset of Y \N.

(ii) If W ∈lip_α(Y,K(E, F)) and ϕ is a supercontraction on Y \N, then T is compact.

3. The separating operators from lip_α(X, E) into lip_α(Y, F) In this section, we characterize the representation of bounded separating linear operators between spaces of vector-valued little Lipschitz functions on compact metric spaces. For this, we need the following result which is a modified version of Theorem 2.2 in [10]. In fact, by imposing an extra assumption on T to be bounded, we provide a simpler statement of the mentioned theorem that serves our purpose in the present paper.

In what follows, we will assume that X and Y are compact metric spaces, E and F are Banach spaces and α ∈ (0,1). For each y ∈ Y, let δ_y be the linear functional on lip_α(Y) defined by δ_y(f) = f(y).

Theorem 3.1. [10, Theorem 2.2]Letα∈(0,1)and let T be a bounded separating linear operator from lip_α(X) into lip_α(Y). If Y₀ = {y ∈ Y : δ_y ◦T = 0} and Y₀^c is the complement of Y₀, then there exist a continuous map ϕ : Y₀^c → X, and a non-vanishing function h ∈ lip_α(Y₀^c) such that T f(y) = h(y)f(ϕ(y)) for every y∈Y₀^c and f ∈lip_α(X), and T f(y) = 0 for every y∈Y₀ and f ∈lip_α(X).

First, using Theorem 3.1 and a similar argument as in [7, Theorem 1], we show that a bounded separating linear operator T : lip_α(X, E)→lip_α(Y, F) is a weighted composition operator on the span{f_e : f ∈ lip_α(X), e ∈ E}. Then we generalized this fact.

Theorem 3.2. Let α ∈ (0,1) and let T : lip_α(X, E) → lip_α(Y, F) be a nonzero bounded separating linear operator. Then there exist a functionW :Y → B(E, F) and a map ϕ:Y →X continuous on Y \N such that

T fe(y) = Wy(fe(ϕ(y)))

for every f ∈ lip_α(X), e ∈ E and y ∈ Y where N = {y ∈ Y : W_y = 0} is the kernel of W. Moreover, W ∈ Lip_α(Y,B(E, F)) and ϕ is Lipschitz on every compact subset of Y \N.

Proof. Given a fixed e ∈ E and u^∗ ∈ F^∗, consider the map T_e,u^∗ : lip_α(X) → lip_α(Y) by T_e,u^∗f(y) =< T f_e(y), u^∗ >. Then T_e,u^∗ is a bounded linear map and kT_e,u^∗k ≤ 2ku^∗kkTkkek. We show that T_e,u^∗ is a separating map. To do this, let f, g∈lip_α(X) withf g = 0 onX. Thenkf_e(x)kkg_e(x)k=|f(x)||g(x)|kek² = 0 for every x∈X. Hence, due to the separating property ofT, kT f_e(y)kkT g_e(y)k= 0 for every y∈Y. Therefore,

|T_e,u^∗f(y)T_e,u^∗g(y)|=|< T f_e(y), u^∗ >||< T g_e(y), u^∗ >|

≤ ku^∗k²kT f_e(y)kkT g_e(y)k= 0,

for every y ∈Y. That is, T_e,u^∗ is a bounded separating linear map from lip_α(X) into lip_α(Y). Thus, Theorem 3.1 ensures the existence of a map ϕ_e,u^∗ :Y₀^c →X and a non-vanishing functionh_e,u^∗ :Y₀^c →Cwith T_e,u^∗f(y) =h_e,u^∗(y)f(ϕ_e,u^∗(y))

for every y ∈ Y₀^c and f ∈ lip_α(X), and T_e,u^∗f = 0 on Y₀ for every f ∈ lip_α(X).

Note thatY₀ depends oneandu^∗. In fact,Y₀ ={y∈Y :δ_y◦T_e,u^∗ = 0}. Extend- ing h_e,u^∗ to Y by defining zero on Y₀, one can sayT_e,u^∗f(y) = h_e,u^∗(y)f(ϕ_e,u^∗(y)) for every f ∈ lip_α(X), y ∈ Y and any extension of ϕ_e,u^∗ to Y. Take f = 1, the constant function in lip_α(X). Then h_e,u^∗(y) =T_e,u^∗1(y) =< T1_e(y), u^∗ >for every y ∈ Y. We now define W : Y → B(E, F), by W_y(e) = T1_e(y) for every y ∈ Y and e ∈ E. It is easy to see that W is well-defined, kWyk ≤ kTk for every y ∈ Y. Similar to the proof of Theorem 2.3, W ∈ Lip_α(Y,B(E, F)). By the definition of W, one can write h_e,u^∗(y) =< W_y(e), u^∗ > and then

< T f_e(y), u^∗ >=< W_y(e), u^∗ > f(ϕ_e,u^∗(y))

=< f(ϕ_e,u^∗(y))W_y(e), u^∗ >=< W_y(f_e(ϕ_e,u^∗(y))), u^∗ >, for every (e, u^∗, y)∈E×F^∗ ×Y and f ∈lip_α(X).

We now show that ϕ_e,u^∗ is independent of e and u^∗ onY \N where N ={y∈ Y :W_y = 0}. Given y₀ ∈Y \N, choose e∈E whit W_y0(e) = T1_e(y₀)6= 0. Then there exists u^∗ ∈F^∗ such that < T1_e(y₀), u^∗ >6= 0. We claim that for every t^∗ in F^∗with< T1_e(y₀), t^∗ >6= 0, we haveϕ_e,u^∗(y₀) = ϕ_e,t^∗(y₀). To prove this claim, let x₁ =ϕ_e,u^∗(y₀) andx₂ =ϕ_e,t^∗(y₀). Ifx₁ 6=x₂, then there exist open neighborhoods U1 of x1 and U2 of x2 such that U1∩U2 =∅. Define fi(x) = dist(x, X \Ui) and note thatfi ∈lip_α(X) and fi(xi)6= 0 for each i= 1,2. Moreover, f1(x)f2(x) = 0 for every x∈X. Hence,

k(f1)e(x)kk(f2)e(x)k=|f1(x)||f2(x)|kek² = 0, for all x∈X. Thus, by the separating property of T,

kT(f₁)_e(y)kkT(f₂)_e(y)k= 0, for all y∈Y. On the other hand,

< T(f₁)_e(y₀), u^∗ >=< T1_e(y₀), u^∗ > f₁(ϕ_e,u^∗(y₀)) =< T1_e(y₀), u^∗ > f₁(x₁)6= 0, and

< T(f₂)_e(y₀), t^∗ >=< T1_e(y₀), t^∗ > f₂(ϕ_e,t^∗(y₀)) =< T1_e(y₀), t^∗ > f₂(x₂)6= 0.

Therefore,

0<|< T(f₁)_e(y₀), u^∗ >||< T(f₂)_e(y₀), t^∗ >|

≤ ku^∗kkt^∗kkT(f₁)_e(y₀)kkT(f₂)_e(y₀)k= 0,

which is a contradiction. Hence ifWy(e) = (T1e)(y)6= 0, then< T1e(y), u^∗ >6= 0 for someu^∗ ∈F^∗ and ϕ_e,u^∗(y) = ϕ_e,t^∗(y) for everyt^∗ in{t^∗ ∈F^∗ :< T1_e(y), t^∗ >6=

0}. Thus, for a fixede ∈E, one can define ϕ_e on {y∈Y :W_y(e) =T1_e(y)6= 0}

byϕ_e(y) =ϕ_e,u^∗(y) where< T1_e(y), u^∗ >6= 0. Similarly, one can show thatϕ_e(y) does not depend onefor everyy∈Y\N, that is, ify ∈Y ande₁, e₂ ∈Esuch that W_y(e₁) 6= 0 and W_y(e₂) 6= 0, then ϕ_e1(y) = ϕ_e2(y). We are then able to define the functionϕ:Y \N →X byϕ(y) =ϕ_e(y) whereW_y(e) = T1_e(y)6= 0 for some e∈E. Hence, T f_e(y) =W_y(f_e(ϕ(y))) ify ∈Y \N and T f_e(y) = 0 if y∈ N, for every f ∈ lip_α(X) and e ∈ E. Therefore, one can write T f_e(y) =W_y(f_e(ϕ(y))) for every y∈Y, f ∈lip_α(X),e∈E and any extension of ϕto Y.

Similar to the proof of Theorem 2.3, one can show that ϕ is continuous on Y \N and Lipschitz on every compact subset of Y \N.

As an immediate application of Theorem 3.2, we provide the following result which characterizes the general form of a bounded separating linear operator T : lip_α(X, E)→lip_α(Y, F) under certain conditions on X and E.

Corollary 3.3. Let X be a compact metric space, E be a Banach space and α ∈ (0,1). Suppose that the linear span of {f_e : f ∈ lip_α(X), e ∈ E} is dense in lip_α(X, E). Then every bounded separating linear operator T : lip_α(X, E) → lip_α(Y, F) is a weighted composition operator T f(y) = W_y(f(ϕ(y))) for every f ∈ lip_α(X, E) and y ∈ Y, where ϕ and W are the same as found in Theorem 3.2.

Note that the density of the linear span of {f_e : f ∈ lip_α(X), e ∈ E} in lip_α(X, E) is not very restrictive and there are many cases with such property.

For instance, as follows from [13, page 167 and Cor. 5.17], if X is an infinite compact set inRⁿ and E is a dual space of a Banach space, then the linear span of {f_e :f ∈lip_α(X), e∈E} is dense in lip_α(X, E).

To characterize a bounded separating linear operatorT : lip_α(X, E)→lip_α(Y, F) in the general case, we need the following lemma.

Lemma 3.4. Let z ∈X and set

J_z ={f ∈lip_α(X, E) :z /∈coz(f) = supp(f)}, M_z ={f ∈lip_α(X, E) :f(z) = 0}.

Then J_z is a dense subspace of M_z.

Proof. Let f ∈ M_z and ε > 0. By the definition of lip_α(X, E), there exists δ ∈ (0,1) such thatkf(x₁)−f(x₂)k< εd^α(x₁, x₂) for everyx₁, x₂ ∈Xwithd(x₁, x₂)<

δ. Let U = B(z,^δ₄) and V = B(z,^δ₂). Define h(x) = min{⁴_δdist(x, U),1} and g = hf. Note that h ∈ Lip₁(X), 0 ≤ h ≤ 1, h = 0 on U, h = 1 on X \V and k1−hk_X = 1. Then g ∈ lip_α(X, E) since lip_α(X, E) is a Lip₁(X)-module, and g ∈J_z since g = 0 onU. Also f =g on X\V and

kf(x)k=kf(x)−f(z)k< εd^α(x, z)< ε(δ 2)^α, for every x∈V. Therefore,

kf−gkX = sup

x∈V

kf(x)−g(x)k= sup

x∈V

kf(x)k|1−h(x)|< ε(δ

2)^α < ε.

Next we show that p_α(f−g)≤5ε. Let x₁, x₂ ∈X with x₁ 6=x₂. Ifx1, x2 ∈V, then by [19, Proposition 1.5.5], we have

|h(x₁)−h(x₂)|

d^α(x₁, x₂) ≤p₁(h)(diam(V))^1−α ≤ 4

δδ^1−α = 4 δ^α,

which implies

k(f −g)(x₁)−(f−g)(x₂)k

d^α(x₁, x₂) ≤kf(x₁)k|h(x₁)−h(x₂)|

d^α(x₁, x₂)

+|1−h(x₂)|kf(x₁)−f(x₂)k d^α(x₁, x₂)

<ε(δ 2)^α· 4

δ^α +ε <5ε.

Ifx₁ ∈X\V and x₂ ∈V, then d(x₁, x₂)≥ ^δ₂ −d(x₂, z) and k(f−g)(x₁)−(f−g)(x₂)k

d^α(x₁, x₂) = kf(x₂)k(1−⁴_δdist(x₂, U)) d^α(x₁, x₂)

≤ε(δ

2)^α 2− ⁴_δd(x₂, z) (^δ₂ −d(x₂, z))^α

= 2ε(δ−2d(x₂, z)

δ )^1−α ≤2ε.

Therefore,pα(f−g)≤5ε. Thus kf −gkα =kf −gkX +pα(f−g)≤6ε.

Theorem 3.5. Let α ∈ (0,1) and let T : lip_α(X, E) → lip_α(Y, F) be a nonzero bounded separating linear operator. Then T is a weighted composition operator T f(y) = W_y(f(ϕ(y))) for every f ∈ lip_α(X, E) and y ∈Y \N, where ϕ and W are defined as in Theorem 3.2.

Proof. In the proof of Theorem 3.2, we have shown that T f_e(y) =W_y(f_e(ϕ(y))) for every f ∈ lip_α(X), e ∈ E and y ∈ Y. We now show that this still holds for every f ∈lip_α(X, E) and y∈Y \N.

Fix y ∈ Y \N. Let f ∈ J_ϕ(y). Then there exists δ > 0 with B(ϕ(y), δ)∩ coz(f) =∅. We choose e ∈E with kek= 1 such that W_y(e)6= 0. Define g(x) =

1

δdist(x, X\B(ϕ(y), δ)) forx∈X. Theng_e ∈lip_α(X, E) andkf(x)kkg_e(x)k= 0 for everyx∈X. The separating property ofT implies that,kT f(z)kkT g_e(z)k= 0 for every z ∈ Y. In particular, kT f(y)kkT g_e(y)k = 0. On the other hand, by what we have proved in Theorem 3.2, T g_e(y) = g(ϕ(y))W_y(e) = W_y(e) 6= 0.

Therefore,kT f(y)k= 0 and T f(y) = 0.

We now assume that f ∈ M_ϕ(y). By Lemma 3.4, there exists a sequence (f_n) inJ_ϕ(y) converging tof in lip_α(X, E). Then the sequence (T f_n) converges to T f in lip_α(Y, F). In particular, T f(y) = limT f_n(y). Therefore, T f(y) = 0.

Finally, let f ∈ lip_α(X, E) and e =f(ϕ(y)). Then g =f −1_e is in M_ϕ(y) and hence T f(y)−T1_e(y) =T g(y) = 0. Therefore,

T f(y) =T1_e(y) =W_y(e) =W_y(f(ϕ(y))).

This completes the proof of the theorem.

Remark 3.6. By imposing certain conditions on T, the kernel N of W will be empty. In this case, the operatorT in Theorem3.5, will be weighted composition of the formT f(y) = W_y(f(ϕ(y))) for everyy∈Y, andϕwill be continuous onY. For instance, N =∅, if either T1_e = 1_u for some nonzero elements e∈ E, u∈ F or if the family{T1_e:e∈E} vanishes at no point of Y.

Acknowledgement. The authors would like to thank the referee for many valuable and useful comments and suggestions which have improved this paper.

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Department of Mathematics, Kharazmi University, 50, Taleghani Ave., 15618, Tehran, iran.

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