OPERATOR ON WEIGHTED SPACES OF CONTINUOUS FUNCTIONS
HAMID VAEZI
LetV be a system of weights on a completely regular Hausdorff spaceXandEbe a Hausdorff locally convex space. ThenCVo(X, E) is a weighted space of vector- valued continuous functions onXwith the topology derived from seminorms which are weighted analogues of the supremum norm. In this article we characterize a nearly open weighted composition operator πCφ onCVo(X, E) induced by π∈ C(X) and continuous self mapφonX.
AMS 2010 Subject Classification: 47B38, 47B33,46E40.
Key words: system of weights, locally convex spaces, weighted spaces of vector- valued continuous functions.
1. INTRODUCTION
LetXbe a completely regular Hausdorff space,V be a system of weights on X and let E be a Hausdorff locally convex space. Then CVo(X, E) and CVb(X, E) are Hausdorff locally convex spaces of E-valued continuous func- tions on X with the topology given by the seminorms which are weighted analogues of the supremum norm. If π is a function onX and φis a self map on X such that π.f ◦φ belongs to CVo(X, E) (or CVb(X, E)) whenever f ∈ CVo(X, E) (or CVb(X, E)), then the map takingf toπ.f◦φis a linear trans- formation on CVo(X, E) (or CVb(X, E)), where (π.f ◦φ)(x) = π(x).f(φ(x)) for every x ∈ X. If this linear transformation is also continuous, we call it the weighted composition operator on CVo(X, E) (or CVb(X, E)) induced by the pair (π, φ) and denote it by the symbol πCφ. For examples and details on weighted spaces of continuous functions and operators on them, we refer to [1], [2], [3], [4], [5], [10] and [11]. The class of weighted composition opera- tors has been the subject matter of several papers in recent years, see for example, [7], [8], [10], [12] and [13]. In this article we characterize a nearly open weighted composition operator πCφ on weighted spaces of vector valued continuous functions on X induced by the pair(π, φ).
MATH. REPORTS14(64),1 (2012), 107–114
2. PRELIMINARIES AND NOTATION
Let R+ be the set of all positive real numbers with the usual relative topology. Then a function v :X →R+ is called a weight on X if it is upper semicontinuous. A familyV of weights onX is called a Nachbin family [11], if for everyu, v∈V andα >0 there existsw∈V such thatαu≤wandαv ≤w (pointwise onX). If additionally,V satisfies the condition that for eachx∈X there existsvx∈V such thatvx(x)6= 0, it is called a system of weights onX. If U andV are two systems of weights onX, then we say thatU ≤V if for every u∈U there existsv∈V such thatu(x)≤v(x) for eachx∈X. IfU ≤V and V ≤U, then U and V are equivalent systems of weights on X and we denote this by U ∼V. Let C(X, E) denote the collection of all continuous functions fromX intoE andcs(E) denote the collection of all continuous seminorms on E. For a system V of weights on X, we now define the following spaces ofE- valued continuous functions on X :CVo(X, E) ={f ∈C(X, E) :vf vanishes at infinity on X for all v ∈ V} and CVb(X, E) = {f ∈ C(X, E) : vf(X) is bounded in E for all v∈V}.
Clearly CVo(X, E) andCVb(X, E) are vector spaces over K with point- wise linear operations while the upper-semicontinuity of the weights implies thatCVo(X, E)⊂CVb(X, E). For (v, p)∈V×cs(E) andf ∈C(X, E), we put kfkv,p= Sup{v(x)p(f(x)) :x∈X}.Thenk · kv,pis a seminorm onCVo(X, E) and on CVb(X, E), and the family{k · kv,p: (v, p)∈V ×cs(E)}of seminorms defines a locally convex Hausdorff topology on each of these spaces. These spaces with the corresponding topology are known as the weighted spaces of vector-valued continuous functions on X. In case E = K, we omit E from our notation and write, for example, CVo(X) in place ofCVo(X, E). We shall denote the closed unite ball inCVo(X, E) corresponding to the seminormk·kv,p by Bv,p. The spacesCVo(X) andCVb(X) were first introduced by Nachbin [4]
and the corresponding vector-valued analogues were subsequently studied in detail by Bierstedt ([1], [2]) and Prolla [5].
Ifπ is a function onXandφis a self map onXsuch thatπ.f◦φbelongs to CVo(X, E) (or CVb(X, E)) wheneverf ∈CVo(X, E) (or CVb(X, E)), then the map taking f to π.f ◦ φ is a linear transformation on CVo(X, E) (or CVb(X, E)), where (π.f◦φ)(x) =π(x).f(φ(x)) for everyx∈X. If this linear transformation is also continuous, we call it the weighted composition operator on CVo(X, E) (orCVb(X, E)) induced by the pair (π, φ) and denote it by the symbolπCφ. In case π= 1, the constant one function on X, we writeπCφ as Cφand call it the composition operator onCVo(X, E) (orCVb(X, E)) induced by φ. In case φ(x) =x for every x∈X, we write πCφ as Mπ and call it the multiplication operator on CVo(X, E) (orCVb(X, E)) induced by π.
3. NEARLY OPEN WEIGHTED COMPOSITION OPERATORS
In this section we first exhibit some examples of weighted composi- tion operators πCφ on the weighted spaces induced by the pair (π, φ). Then we give a characterization of a nearly open weighted composition operator on CVo(X, E).
We note that ifπinduces a multiplication operator andφinduces a com- position operator, the pair (π, φ) will induce a weighted composition operator.
But it is interesting to observe that when one ofπorφdoes not induce the cor- responding operator, the pair (π, φ) may still induce a weighted composition operator (see Example 3.7).
Proposition 3.1. If Mπ is the multiplication operator on CVi(X, E) induced by π and Cφ is the composition operator on CVi(X, E) induced byφ, thenπCφis a weighted composition operator onCVi(X, E)induced by the pair (π, φ), where i∈ {0, b} .
Proposition 3.2 [9]. Let X be a completely regular Hausdorff space, E be a “lmc algebra” and V = {αχK : α ≥ 0, K ⊂ X, K is compact}. Let π ∈C(X) (or C(X, E)) andφ:X →X is continuous. Then (π, φ) induces a weighted composition operator πCφ on CVb(X, E) .
Corollary 3.3. Let X have the discrete topology, E be a Banach alge- bra and take V ={αχK :α ≥0, K ⊂X, K is finite}. Suppose π :X → E and φ : X → X are any functions. Then πCφ is the weighted composition operator on CVb(X, E) induced by (π, φ).
We will work under the following requirements:
(3.a)X is a completely regular Hausdorff space.
(3.b)E is a locally convex Hausdorff topological vector space such that there exists a vector s∈E for which p(s)6= 0 for every p∈cs(E).
(3.c)V is a system of weights onX.
(3.d) Corresponding to each x ∈ X, there exists an fx ∈ CVo(X) such that fx(x)6= 0.
For a function π ∈ C(X), the set V.|π|= {v.|π|: v ∈ V} is a Nachbin family on X. In case π is non-zero at each point of X, V.|π| is a system of weights on X. Again If φ is a continuous self map on X, then the set V ◦φ={v◦φ:v ∈V} is also a system of weights on X. If v is a weight on X and >0, we putN(v.|π|, ) ={x∈X :v(x)|π(x)| ≥}.
The following theorems are proved in [9]:
Theorem 3.4. Let π ∈ C(X) and φ : X → X is continuous. If πCφ : CVo(X, E)→CVb(X, E) is continuous, then V.|π| ≤V ◦φ.
Theorem 3.5. Letπ ∈C(X)andφ:X →Xis continuous. ThenπCφ: CVb(X, E)→ CVb(X, E) is a weighted composition operator on CVb(X, E) if and only if V.|π| ≤V ◦φ.
Example3.6. LetX=N, with discrete topology,Ebe a Banach algebra, and V ={αν :α≥0}, where ν(n) =nfor eachn∈X. Define π :X →R as π(n) = 1n for each n∈X and φ:X→X as
φ(x) =
√
n ifn is a perfect square, n otherwise.
Then it can be check that V.|π| ≤ V ◦φ and hence, from Theorem 3.5, we conclude that πCφ is a weighted composition operator onCVb(X, E).
Example 3.7. Let X = R+\0, with the usual relative topology, E be a Banach algebra and V = {αν : α ≥ 0}, where ν(x) = 1x for each x ∈ X.
Define π : X → R as π(x) = x2 for each x ∈ X and φ :X → X as φ(x) =
1
x for all x ∈ X. Then π does not induce a multiplication operator Mπ on CVb(X, E) (sinceV.|π| 6≤V), andφ does not induce a composition operator CφonCVb(X, E) (sinceV 6≤V◦φ). ButV.|π| ≤V◦φ. So Theorem 3.5 implies that (π, φ) induces a weighted composition operator πCφ on CVb(X, E).
We state the following Theorem (Theorem 3.6 in [9]) which characterizes the weighted composition operators on CVo(X, E) induced by a mapping π∈ C(X) and a continuous self mapφon X.
Theorem 3.8. Let π ∈C(X) and φ:X → X is continuous. Then the following statements are equivalent:
1. (π, φ) induces a weighted composition operator πCφ onCVo(X, E).
2. (π, φ)induces a weighted composition operatorπCφonCVb(X, E)and CVo(X, E) is invariant under πCφ.
3. (i) V.|π| ≤V ◦φ, and (ii) for each v ∈V, >0 and compact subset K of X, the set φ−1(K)∩N(v.|π|, ) is compact in X.
4. (i)V.|π| ≤V ◦φ, and (ii) for each v ∈V, >0 and u∈V such that v.|π| ≤u◦φ, the set φ−1(K)∩N(v.|π|, ) is compact whenever K is compact subset of N(u, ).
In the following theorem, we shall give a characterization of a nearly open weighted composition operator onCVo(X, E). Before proving the theorem, let us recall Ptak’s open mapping theorem [6]. According to that theorem, if CVo(X, E), say, happens to be B-complete (or fully complete), and ifφ:X → X induces a nearly open composition operatorCφ:CVo(X, E)→CVo(X, E) such that Cφ(CVo(X, E)) is dense in CVo(X, E), then Cφ is necessarily an open surjection. Moreover, it readily follows that if V is a system of weights generated by a single continuous weight on X and E is a Banach space, then Cφ will be open and surjective as soon as Cφ :CVo(X, E) → CVo(X, E) is a continuous nearly open operator with dense range. Now, we shall characterize
those weighted composition maps πCφ : CVo(X, E) → CVo(X, E) which are nearly open in the sense that given any v ∈ V and p ∈ cs(E), there exists u∈V and q ∈cs(E) such thatBu,q⊂πCφ(Bv,p).
Theorem 3.9.Let π ∈ C(X) and φ:X →X is continuous and πCφ : CVo(X, E) → CVo(X, E) is a weighted composition operator. Then πCφ is nearly open if and only if
(i) πCφ(CVo(X, E)) is dense inCVo(X, E);
(ii)V ◦φ≤V.|π|.
Proof. Necessity. Suppose that πCφ is nearly open. Let v ∈ V and p ∈ cs(E). Choose u ∈ V and q ∈ cs(E) such that Bu,q ⊂ πCφ(Bv,p). Now fix f ∈CVo(X, E) and set α=kfku,q+ 1. From this, it follows thatα−1f ∈Bu,q
and therefore there exists ag∈Bv,psuch thatπ.g◦φ∈α−1(f+Bv,p). Further, it implies that πCφ(αg)∈f+Bv,p and thusπ.αg◦φ∈f +Bv,p. This shows that πCφ(CVo(X, E)) is dense inCVo(X, E).
Now, we shall show thatV ◦φ≤V.|π|. By assumption (3.b) there exists a vectors∈E such thatp(s)6= 0, ∀p∈cs(E). Letγ = q(s)p(s). Thenγ >0. We claim that
v(φ(x))≤2γ|π(x)|u(x) ∀x∈X.
Letxo∈X be fixed. Set=u(xo). In case >0 the setG={x:u(x)< 32} is an open neighbourhood of xo and therefore, (by Lemma 2, p. 69 of [4]), there exists g ∈ CVo(X) such that 0≤ g ≤ 1, g(x0) = 1 and g(X\G) = 0.
Now we define h(x) = g(x)s for every x ∈ X. Then clearly h ∈CVo(X, E).
Setting α = (32q(s))−1 and h∗ = αh, then h∗ ∈ Bu,q. For w ∈ V such that w(xo)≥1, we can therefore find f ∈Bv,p such that
kπ.f ◦φ−h∗kw,p ≤ α|π(xo)|q(s) 2(v(φ(xo)) + 1). That is,
w(xo)p(π(xo)f(φ(xo))−αh(xo))≤ α|π(xo)|q(s) 2(v(φ(xo)) + 1). Further, it implies that
(3.1) p(π(xo)f(φ(xo))−αs)≤ α|π(xo)|q(s) 2(v(φ(xo)) + 1). Consequently, we have
|π(xo)| ≥v(φ(xo))p(f(φ(xo)))|π(xo)|
=v(φ(xo))p{αs−(αs−π(xo)f(φ(xo)))}
≥v(φ(xo)){p(αs)−p(αs−π(xo)f(φ(xo)))}
=v(φ(xo))αp(s)−v(φ(xo))p(π(xo)f(φ(xo))−αs).
(3.2)
From (3.1), it follows that
(3.3) −v(φ(xo))p(π(xo)f(φ(xo))−αs)≥v(φ(xo))−α|π(xo)|q(s) 2(v(φ(xo)) + 1). Using (3.3) in (3.2), we get
|π(xo)| ≥v(φ(xo))αp(s)−v(φ(xo)) α|π(xo)|q(s) 2(v(φ(xo)) + 1)
≥α[v(φ(xo))p(s)−
2|π(xo)|q(s)].
From this, it follows that
v(φ(xo))p(s)≤2|π(x0)|q(s).
Thus
v(φ(xo))≤2γ|π(xo)|u(xo), and in this case our claim is established.
Ifu(x0) = 0 and v(φ(xo))>0, we set = 4γ|π(xv(φ(xo))
o)| and G={x :u(x)<
}. ThenGis an open neighbourhood ofxo and therefore, (by Lemma 2, p. 69 of [4]), there existsg∈CVo(X) such that 0≤g≤1,g(x0) = 1 andg(X\G) = 0. Define h(x) = g(x)s for every x ∈ X. Then clearly h ∈ CVo(X, E). Set α = (q(s))−1 and h∗ =αh. Then h∗ ∈Bu,q. Forw∈V such thatw(xo)≥1, we can therefore find f ∈Bv,p such that
kπ.f ◦φ−h∗kw,p ≤ 1 2γ. That is,
w(xo)p(π(xo)f(φ(xo))−αh(xo))≤ 1 2γ. Further, it implies that
p(π(xo)f(φ(xo))−αs)≤ 1 2γ. Consequently, similar to the previous case we have
|π(xo)| ≥v(φ(xo))αp(s)−v(φ(xo))p(π(xo)f(φ(xo))−αs)
≥v(φ(xo))αp(s)−v(φ(xo)) 1 2γ
=v(φ(xo))1
γ −v(φ(xo)) 1
2γ = v(φ(xo)) 2γ . Thus
v(φ(xo))≤2|π(x0)|γ.
Hence
v(φ(xo))≤ v(φ(xo))
2 ,
which is a contradiction. With this the proof of the necessity part of the theorem is complete.
Sufficiency. Suppose that πCφ(CVo(X, E)) is dense in CVo(X, E) and V ◦φ ≤V.|π|. We will prove that for any v ∈ V and p ∈ cs(E), there exist u ∈ V and q ∈cs(E) such that Bu,q ⊂ πCφ(Bv,p). Take anyf ∈CVo(X, E) for whichπCφ(f)∈Bu,q, i.e.,kπ.f◦φku,q<1. SinceπCφ(CVo(X, E)) is dense in CVo(X, E), to show that Bu,q ⊂ πCφ(Bv,p), it is enough to verify that πCφ(f) =π.f ◦φ∈πCφ(Bv,p). Now, given w∈V andp0∈cs(E), we put
K1 ={x∈X:p0(π(x).f(φ(x)))w(x)≥1}
and
K2 ={x∈X:p(f(x))v(x)≥1}.
Then K1 and K2 both are compact sets. Since V ◦φ ≤ V.|π|, then we can assume that for any v ∈V and p ∈ cs(E), there exist u ∈V and q ∈ cs(E) such that v(φ(x))p(y) ≤ |π(x)|u(x)q(y), ∀x∈X,∀y ∈E. So for everyx ∈X we have
v(φ(x))p(f(φ(x)))≤ |π(x)|u(x)q(f(φ(x)))<1.
Thus φ(x) 6∈ K2. Since φ is necessarily continuous, φ(K1) is compact and φ(K1)∩K2 = ∅. Choose g ∈ Cb(X) such that 0 ≤ g ≤ 1, g(K2) = 1 and g(φ(K1)) = 0. We put h=f−gf. Thenh∈CVo(X, E). Now for everyx∈X we have
v(x)p(h(x)) =v(x)p(f(x)−g(x)f(x)) =|1−g(x)|v(x)p(f(x))<1.
Thus h∈Bv,pand therefore it follows that
p0(π(x).f(φ(x))−π(x).h(φ(x)))w(x) =|π(x)|p0((f −h)(φ(x)))w(x) =
=|π(x)|p0(g(φ(x))f(φ(x)))w(x) =|π(x)|g(φ(x))p0(f(φ(x)))w(x)<1, for every x ∈ X. It implies that for any p0 ∈ cs(E) and w ∈ V, we have kπ.f ◦φ−π.h◦φkw,p0 < 1. Thus π.f ◦φ ∈ πCφ(Bv,p). This completes the proof of the theorem.
The next result is an immediate consequence of Theorem 3.9 coupled with Theorem 3.8. Before stating this result we remember that ifV ◦φ≤V.|π|and V.|π| ≤V ◦φ, then they are equivalent and we denote this by V ◦φ∼V.|π|.
Corollary 3.10. Let π ∈ C(X) and φ: X → X is continuous. Then (π, φ)induce a nearly open weighted composition operator onCVo(X, E)if and only if
(i) πCφ(CVo(X, E)) is dense inCVo(X, E);
(ii)V ◦φ∼V.|π|;
(iii) for every v ∈ V, > 0 and compact set K ⊂ X, the set φ−1(K)∩ N(v.|π|, ) is compact in X.
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Received 23 May 2010 University of Tabriz
Faculty of Mathematical Sciences Tabriz, Iran
hvaezi@tabrizu.ac.ir
