Weighted Composition Operator and Dynamical System on Weighted Function Spaces P.Chandra kala ∗ and D.Senthilkumar
†
Abstract Let X be a Hausdorff topological space. Let V be a system of weights on X. Let B(E) be the Banach algebra of all bounded linear operators on a Banach space E. Let CV0 (X, E), CVb (X, E) and LV0 (X),LVb (X) be the weighted locally convex spaces of continuous functions and cross-sections with a topology generated by semi-norms which are weighted analogues of the supremum norm. In this paper,we obtain dynamical system induced by weighted composition operators on weighted spaces of continuous functions and cross-sections.
Keywords : System of weights, weighted composition operators, dynamical systems, seminorm, operator valued mapping. 2000 AMS Subject Classification: 47 B 37,47 B 38, 47 B 07,47 D 03, 30 H 05.
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Introduction
Weighted composition operators have been appearing in a natural way on different spaces of continuous functions ,analytic functions and cross-sections. For example R.K.Singh and Manhas [1,2,4,5] have shown that weighted composition operators on spaces of continuous and cross-sections. This paper is a continuation of the papers [2,3,6]. For more details about weighted spaces of cross-sections see [6,7]. We have organized this paper into three sections. In section 2, we obtained dynamical system induced by weighted composition operators on weighted spaces continuous functions with preliminaries.In the last section we obtained dynamical system induced by weighted composition operators on weighted spaces of cross-sections with preliminaries. ∗
∗
† Department of Mathematics,Government Arts College, Coimbatore-641 018,Tamil Nadu,India. corresponding author e-mail:
[email protected]
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2
Dynamical system induced by weighted composition operators on weighted spaces of continuous functions
Preliminaries Let X be a Hausdorff topological space and C(X, E) be the collection of continuous functions from X into E. Let V be a set of non-negative upper-semi continuous functions on X. If V is a set of weights on X such that given any x ∈ X, there is some v ∈ V for which v(x) > 0. We write V > 0. A set V of weights on X is said to be directed upward provided for every pair u1 , u2 in V and α > 0 there exists v ∈ V such that αui ≤ v (pointwise on X) for i = 1, 2. By a system of weights,we mean a set V of weights on X with additionally satisfies V > 0. Let cs(E) be the set of all continuous functions from X into E. If V is a system of weights on X, then the pair (X, V ) is called the weighted topological system. Associated with each weighted topological system (X, V ), we have the weighted spaces of continuous E-valued functions defined as: CV0 (X, E) = {f ∈ C(X, E) : vf vanishes at infinity on X for each v ∈ V } CVb (X, E) = {f ∈ C(X, E) : vf (x) is bounded in E for all v ∈ V } . Let v ∈ V, q ∈ cs(E) and f ∈ C(X, E). If we define kf kv,q = sup{v(x)q(f (x)) : x ∈ X}, then k.kv can be regarded as a seminorm on either CV0 (X, E), CVb (X, E) and the family {k.kv,q : v ∈ V, q ∈ cs(E)} of seminorms defines a Hausdorff locally convex topology on each of these spaces. This topology will be denoted by ωv and the vector spaces CV0 (X, E) and CVb (X, E) endowed with ωv are called the weighted locally convex space of vector-valued continuous functions. It has a basis of closed absolutely convex neighborhoods of the origin of the form, Bv,q = {f ∈ CVb (X, E) : kf kv,q ≤ 1} . Also, CV0 (X, E) is a closed subspace of CVb (X, E). Theorem 2.1 [2] Let E be a locally convex Hausdorff space such that each convergent net in E is bounded. Let ψ ∈ C(X, B(E)) and T ∈ C(X, X). Then Wψ,T is a weighted composition operator on CVb (X, E) iff for every v ∈ V and p ∈ cs(E),there exists u ∈ V and q ∈ cs(E) such that v(x)p(ψx (y)) ≤ u(T (x))q(y) for every x ∈ X and y ∈ E . Remark: Let B(E) be the Banach algebra of all bounded linear operators on E. Then an operatorvalued map ψt : X → B(E) defined by ψt (x) = etg(x) for all t ∈ R and x ∈ X, where g ∈ Cb (X, B(E)) and kgk∞ = sup{kg(x)k : x ∈ X}. Also T : X → X is the self-map. Then 2
the weighted composition operator induced by ψt and T on the spaces of CV0 (X, E) and CVb (X, E) [2]. Theorem 2.2: Let V be an arbitrary system of weights on X. Let 5 : R × CVb (X, E) → C(X, E) be the function defined by 5(t, f ) = Wψt ,T f for all t ∈ R and f ∈ CVb (X, E). Then 5 is a linear dynamical system if for every v ∈ V and p ∈ cs(E), exists u ∈ V and q ∈ cs(E) such that v(x)p(ψx (y)) ≤ u(T (x))q(y) for every x ∈ X and y ∈ E. Proof: For every t ∈ R, Wψt ,T is a weighted composition operator on CVb (X, E). Thus it follows that, 5(t, f ) ∈ CVb (X, E) for all t ∈ R and f ∈ CVb (X, E). Clearly, 5 is linear and
5(0, f )(x) = Wψ0 ,T f (x) for all x ∈ X = ψ0 (x)f (T (x)) = f (x) for all x ∈ X. Therefore 5(0, f ) = f. Also 5(t + s, f )(x) = Wψt+s ,T f (x) for all x ∈ X = ψt+s (x)f (T (x)) = ψt (x)ψs (x)f (T (x)) = ψt (x)Wψs ,T f (x) = 5(t, 5(s, f ))(x). Therefore 5(t + s, f ) = 5(t, 5(s, f )) . Next,to show that 5 is linear dynamical system,it sufficient to show that 5 is separately continuous map. Let us first prove the continuity of 5 in the first argument.Let tn → t ∈ R. Then |tn − t| → 0 as n → ∞. We shall show that 5(tn , f ) → 5(t, f ) ∈ CVb (X, E).
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Let v ∈ V. Then, k 5 (tn , f ) − 5(t, f )kv = kWψtn ,T f − Wψt ,T f kv = sup{v(x)kψtn (x).f (T (x)) − ψt (x).f (T (x))k : x ∈ X} = sup{v(x)ketn g(x) f (T (x)) − etg(x) f (T (x))k : x ∈ X} ≤ sup{v(x)ketn g(x) f (T (x)).etg(x) e−tg(x) − etg(x) f (T (x))k : x ∈ X} ≤ sup{v(x)k(e(tn −t)g(x) − I)etg(x) f (T (x))k : x ∈ X} ≤ sup{v(x)k(e(tn −t)g(x) − I)kketg(x) f (x)k : x ∈ X} ≤ sup{v(x)k(e(tn −t)g(x) − I)k : x ∈ X} sup{v(x)k(etg(x) .f (x)k : x ∈ X} ≤ (e|tn −t|M ) − I)kf kv → 0 as |tn − t| → 0 This proves the continuity of 5 in the first argument. Now, we shall prove the continuity of 5 in the second argument. Let fα be a net in CVb (X, E) such that fα → f in CVb (X, E). Then kfα − f kv → 0 for all v ∈ V . We shall show that, 5(t, fα ) → 5(t, f ) ∈ CVb (X, E) . For this, let v ∈ V . Then, k 5 (t, fα ) − 5(t, f )kv = kWψt ,T fα − Wψt ,T f kv = sup{v(x)kψt (x).fα (T (x)) − ψt (x).f (T (x))k : x ∈ X} ≤ sup{v(x)kψt (x)(fα (T (x)) − f (T (x))k : x ∈ X} ≤ sup{v(x)kψt (x)kk((fα (T (x)) − f (T (x)))k : x ∈ X} ≤ sup{u(x)k((fα (T (x)) − f (T (x)))k : x ∈ X} = kfα − f ku → 0. This proves the continuity of 5 in the second argument and hence 5 is a (linear) dynamical system on the weighted space CVb (X, E).
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Dynamical system induced by weighted composition operators on weighted spaces of cross-sections
Preliminaries Let X be a Hausdorff topological space.A vector-fibration over X is a pair (X, (Fx ))x∈X , where each Fx is a vector space over the field K (where K=R (or)C). A cross-section over X is then any element of the Cartesian product πx∈X Fx . The Cartesian product πx∈X Fx is a made a vector space in the usual way and a vector space of cross-section over X is 4
by definition any vector subspace of πx∈X Fx .By a weight on X,we mean a function v on X such that v(x) is a semi-norm on Fx for each x ∈ X. By v ≤ u,we mean that vx ≤ ux for each x ∈ X. A set V of weights on X is said to be directed if, ∀ pair u, v ∈ V and λ > 0, ∃ ω ∈ V such that λu ≤ ω and λv ≤ ω. Here after we assume that each set of weights is directed. We write V > 0, if given x ∈ X and y ∈ Fx , there is some v ∈ V for which vx (y) > 0. A set V of weights on X which additionally satisfies V > 0 will be referred to as a system of weights on X. If f is a cross-section over X and v is a weight on X, then we will denote by v[f ] the positive-valued function on X which takes x into vx [f (y)]. We denote by L(X) a vector space of cross-sections over X. Now the weighted spaces of cross-sections over X with respect to the system of weights V on X are introduced as follows: LV0 (X) = {f ∈ L(Y ) : v[f ] is upper semi continuous and vanishes at infinity on X for each v ∈ V } and LVb (X) = {f ∈ L(Y ) : v[f ] is a bounded function on X for each v ∈ V }. Then LV0 (X) and LVb (X) are vector spaces and LV0 (X) ⊆ LVb (X). Now for v ∈ V and f ∈ L(X), if we put kf kv = sup{vy [f (x)] : x ∈ X}, then k.kv can be regarded as a seminorm on either LV0 (X) or LVb (X), and the family of seminorms {k.kv : v ∈ V } defines a Hausdorff locally convex topology on each of these spaces. This topology is denoted by ιv and the vector space endowed with ιv are called weighted locally convex spaces of cross-sections. Since V is a directed set of weights, ιv has a basis of closed absolutely convex neighborhood of the form Bv = {f ∈ LVb (X) : kf kv ≤ 1}. Theorem 3.1[2] S Let T : X → X be a map such that for every x ∈ X. FT (x) ⊂ Fx and π : x → x∈X (L(Fx )), where L(Fx ) denotes the linear space of all linear transformation from Fx into itself. Let S π : X → x∈X (L(Fx )) be a mapping such that for every x ∈ X, Π(x) ∈ L(Fx ). Then 0 0 W π ≤ W ◦ T . (i.e) for every w ∈ W ∃ w ∈ W such that wx (πx (y) ≤ wT (x)(y), for every x ∈ X and y ∈ FT (x). Remark: S For g ∈ Fb (X) and for each s ∈ R, define the map πs : X → x∈X (L(Fx )) as πs (x) = esAy (x) for every s ∈ R and the self map T : X → X. Theorem 3.2 Let g ∈ Fb (X) and let ∇g : R × LVb (X) → L(X) be the function defined by ∇g (s, f ) = Wπs ,T f for s ∈ R and f ∈ LVb (X). Then ∇g is a dynamical system on LVb (X). Proof: Since for each s ∈ R, Wπs ,T is a weighted composition operator on LVb (X). We can conclude 5
that, ∇g (s, f ) ∈ LVb (X), whenever s ∈ R and f ∈ LVb (X). Thus ∇g is a function from R × LVb (X) → L(X). Also, ∇g (0, f )(x) = Wπ0 ,T f (x) ∀ x ∈ X = π0 (x)f (T (x)) = e0Ay (x) f (T (x)) = f (x). Therefore ∇g (0, f ) = f and ∇g (s + t, f ) = ∇g (s, ∇g (t, f )). Next to show that ∇g is a dynamical system, it sufficient to show that ∇g is jointly continuous.Let {sn , fn } be sequence in R × LVb (X) such that {sn , fn } converges to {s, f }. Let v ∈ V be a weight. Then, k∇g (sn , fn ) − ∇g (s, f )kv = kWπsn ,T fn − Wπs ,T f kv = sup{v(x)kπsn (x)fn (T (x)) − πs (x)fn (T (x)) +πs (x)fn (T (x)) − πs (x)f (T (x)) : x ∈ X} ≤ sup{v(x)kπsn (x)fn (T (x)) − πs (x)fn (T (x))k : x ∈ X} + sup{v(x)kπs (x)fn (T (x)) − πs (x)f (T (x))k : x ∈ X} ≤ sup{v(x)kπsn (x) − πsn (x)kkfn (T (x))k : x ∈ X} + sup{v(x)kπs (x)kkfn (T (x)) − f (T (x))k : x ∈ X} ≤ (e|sn |M − e|s|M )kf kv + e|s|M kfn − f kv ≤ (e|sn −s|M − I)kf kv + e|s|M
kfn − f kv → 0 as |sn − s| → 0 and kfn − f k → 0. Therefore ∇g is a dynamical system on LVb (X). Acknowledgement. This research is part of the PhD work ”Dynamical system and composition operators”. The work was supported by Rajiv Gandhi National Fellowship under the University Grants Commission Fellowship and a bursary awarded to P.Chandra kala.Also I would like to thank Prof.Dr.S.Panayappan for continuous help to my research work.
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