MULTIPLICATION OPERATORS ON WEIGHTED SPACES IN THE NON-LOCALLY CONVEX FRAMEWORK

Let X be a completely regular Hausdorff space, E a topological vector space, V a Nachbin family of weights on X, and CVo(X, E) the weighted space of continuous E-valued functions on X. Let 0 X C be a mapping, f E CVo(X, E) and define Me(f) Of (pointwise). In case E is a topological algebra, p X E is a mapping then define M,(f) pf (pointwise). The main purpose of this paper is to give necessary and sufficient conditions for Me and M, to be the multiplication operators on CV0 (X, E) where E is a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption that E is locally convex.


INTRODUCTION
The fundamental work on weighted spaces of continuous scalar-valued functions has been done mainly by Nachbin [9,10] in the 1960's. Since then it has been studied extensively for a variety of problems such as weighted approximation, characterization of the dual space, approximation property, description of inductive limit and of tensor-product, etc for both scalar-and vector-valued functions (for instance see [1][2][3][4][5][8][9][10][11][12][13][14]). Recently Singh and Summers [13] have studied the notion of composition operators on CVo(X, C). Later, Singh and Manhas [12] made an analogous study of multiplication operators on CVo(X, E), assuming E to be a locally convex space or a locally m-convex algebra. The purpose of this paper is to generalize the results of Singh and Manhas [12] to the case when E is a general topological vector space which is not necessarily locally convex. Section 3 contains our main results while section 2 is devoted to some technical preliminaries required for the development of our results 2. PRELIMINARIES Throughout this paper we shall assume, unless stated otherwise, that X is a completely regular Hausdorff space and E is a non-trivial Hausdorff topological vector space Let S + (X) denote the set of L.A. KHAN AND A. B THAHEEM all non-negative upper-semicontinuous functions on X, and let S(X) (respectively S-(X)), be the subset of S + (X) consisting of those functions vanishing at infinity (respectively having compact support) A Nachblnfamdy on X is a subset V of S + (X) such that, given u, v E V, there exist w E V and > 0 so that z, v < tw (pointwise); the elements of V are called weights. Let C(X,E) (Cb(X,E)) be the vector space of all continuous (and bounded) E-valued functions on X, and let CV(X, E) (CVo(X, E)) denote the subspace of C(X, E) consisting of those f such that vf is bounded (vanishes at infinity) for each v V When E C' (or R), these spaces are denoted by C(X), C(X), CV(X), and CVo(X) If q5 C(X) and a E, then q(R)a is a function in C(X,E) defined by (c/)(R)a)(z) (z)a(z X) If U and V are two Nachbin families on X and, for each u E U, there is a v V such that u < v, then we write U < V. If, for each z X, there is a v V with v(z) :/: O, we write V > 0. For any function 0" X C, we let ViOl {vlOlv w}. Given any Nachbin family V on X, the weighted topology w on CV(X, E) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form v(,a) {y e cv(x,) (y)(x) c_ where v V and G is a neighborhood of 0 in E; CV(X,E) endowed with w is called a wetghted space We mention that if V S (X), then CV(X, E) CVo(X, E) C(X, E) and w , the stricl topology and write as (C(X, E), ); if V S + (X), then CV(X, E) CVo(X, E) C(X, E) and w k, the compact-open topology and we write as (C(X, E), k). For more information on weighted spaces, we refer to 1-2, [9][10][11][12][13][14] when E is a scalar field or a locally convex space and to [1,[3][4][5]8] in the general setting.
Let 0" X--6' and q" X E be two mappings, and let L(X, E) be the vector space of all functions from X into E. The scalar multiplication on E and, in case E is an algebra, multiplication on E give rise to two linear mappings Mo and M e from CV(E,X) into L(X, E) defined by Mo(f) Of and Me(f)= f, where the product of functions is defined pointwise. If Mo and Mq, map CV(X,E)(CVo(X,E)) into itself and are continuous, they are called multiplication operators on CV (X, E) (CV0 (X, E)) induced by 0 and q, respectively.
A neighborhood G of 0 in E is called shrinkable if rG C_ int 17 for 0 _< r < 1. By ( [6], Theorems 4 and 5), every Hausdorff topological vector space has a base of shrinkable neighborhoods of 0 and also the Minkowski functional Pa of any such neighborhood G is continuous. Now let E be a topological algebra with jointly continuous multiplication and having W, a base of neighborhoods of 0. Then, given any G E W, there exists an H W such that HC_ G. (Here H {ab a, b H}.) A subset 6/ W is called idempotent (or multiplicative) if G c_ G Following Zelazko ([ 16], p. 31), E is said to be a locally idempotent algebra if it has a base of neighborhoods of 0 consisting of idempotent sets. It is easily seen that if G W is idempotent, then pa is submultiplicative pa(ab) <_ pa(a)pe(b) for all a, b E E; further, if E has an identity e, pc(e) _> 1. The notion of locally idempotent algebras is a strict generalization of the notion of locally m-convex algebras introduced by Michael [7] (see also [15, p. 3481).

CHARACTERIZATION OF MULTIPLICATION OPERATORS
In this section, we give necessary and sufficient conditions for Mo and M to be the multiplication operators on the weighted space CVo(X, E). (These results hold also for the space CVb(X, E) with slight modification in the proofs and are therefore omitted.) To avoid trivial cases we assume that the Nachbin family V on X satisfies the following conditions (*) V > 0; (**)corresponding to each z X, there exists an h= CVo(X) such that h=(z) 0 (This holds in particular, when each v in V vanishes at infinity or X is locally compact.) TItEOREM 3.1. For a mapping 0 X --) (7, the following are equivalent: (a) O is continuous and El0[ _< v; (b) Ms is a multiplication operator on CV0 (X, E).
PROOF. Let W be a base of closed, balanced, and shrinkable neighborhoods of 0 in E.
(a) = (b). We first show that Mo maps CVo(X, E) into itself. Let f CVo(X, E), and let v V and G W Choose u V such that viol _< u There exists a compact set K c_ X such that for all zX\K Hence vMo(f) vanishes at infinity; further, since 0 is continuous, Ms(f) CVo(X,E). To prove the continuity of Mo, let {fo) be a net in CVo(X,E) with fo 0 Let v, G and u be chosen as above. Choose an index a0 such that fo N(u, G) for all o > a0 Then it follows that Ofo N(v, G) for all a > a0. Thus Mo(f,) O. So M0 is continuous at 0 and hence, by linearity, it is continuous on CVo(X, E). We next consider the case of the operator Me. PROOF. We may assume that W consists of closed, balanced, shrinkable, and idempotent sets L A KHANANDA B THAHEEM (a) = (b) We first show that Me maps CVo(X, E) into itself Let f e CVo(X, E), and let v e V and G E W. Choose u 6 V such that Vpc o p < u There exists a compact set K c_ X such that u(z)f(z) 6 G for all z 6 X\K. Since Pc is submultiplicative, for any z 6 X\K, we have pC(v(z)(z)f(z)) <_ v(z)pc((:r,))pc(f(x)) <_ u(z)pc(f(z)) <_ 1; hence M(f) 6 CVo(X,E). Using again the submultiplicativity of Pc, the continuity of M follows in the same way as in the proof of Theorem 1.
(b) = (a). Let {zo} be a net in X such that z, a: E X. Choose an h CVo(X) with h(:r.) 0 Since Me is a self-map on CVo(X,E), it follows that the function b(h (R)a) from X into E is continuous.
Finally, we apply the above results to the cases: V S+(X) and V S (X) and obtain the following.
(i) If0-X C is a continuous mapping, then M0 is a multiplication operator on (C(X, E), k).
(ii) If E is a Hausdorff locally idempotent algebra with identity e and %b" X E a continuous mapping, then Me is a multiplication operator on (C(X, E), k). PROOF. (i) In view of Theorem 1, we only need to verify that Vll _< v, where V S + (X). Let v 6 V. Choose a compact set K C_ X with v(x) 0 for all x 6 X\K. Let s sup{10(z)[ z e K} and t sup{v(x)'x 6 K}, and let u st XK. Then u 6 V and clearly v(z)lO(z)l <_ u(z) for all x6X.
(ii) Let W be a base of neighborhoods of 0 in E consisting of closed, balanced, shrinkable, and idempotent sets. In view of Theorem 2, we only need to verify that Vpc o <_ V for every G 6 W, where V S2 (X). Let v 6 V and G 6 W. Choose a compact set K g X with v(z) 0 for all z 6 X\K. Let s sup{pc(p(z))'z 6 K} and t sup{v(z)'z 6 K}, and let u stXK. Then u 6 V and clearly v(z)pG((z)) < u(z) for all z 6 X. This completes the proofofthe theorem RE1MARK. The above result need not hold for the space (Cb(X,E),). To see this, consider X R+, E C, and V S-(X) Let t9 p" X 7 be a mapping given by 19(z) z 2 (z 6 X), (z 6 X) Then v(z)lS(z)l z for all z E x. Since each u 6 V is a and let v 6 V be given by v(z)bounded function, vll u for every u 6 v. Hence VlSI _< v does not hold and so, by Theorem 1, M0 is not a multiplication operator on (C'b(X),/). The same is also true for the space (Cb(X), u), where u is the uniform topology, since/ < u. However, if 6 and q., are bounded continuous functions, then it is easily seen that Ms and M are always multiplication operators on CVo(X, E) for any Nachbin family V ACKNOWLEDGMENT. One of the authors (A. B. Thaheem) wishes to acknowledge the support provided by King Fahd University of Petroleum and Minerals during this research.

Call for Papers
Intermodal transport refers to the movement of goods in a single loading unit which uses successive various modes of transport (road, rail, water) without handling the goods during mode transfers. Intermodal transport has become an important policy issue, mainly because it is considered to be one of the means to lower the congestion caused by single-mode road transport and to be more environmentally friendly than the single-mode road transport. Both considerations have been followed by an increase in attention toward intermodal freight transportation research. Various intermodal freight transport decision problems are in demand of mathematical models of supporting them. As the intermodal transport system is more complex than a single-mode system, this fact offers interesting and challenging opportunities to modelers in applied mathematics. This special issue aims to fill in some gaps in the research agenda of decision-making in intermodal transport.
The mathematical models may be of the optimization type or of the evaluation type to gain an insight in intermodal operations. The mathematical models aim to support decisions on the strategic, tactical, and operational levels. The decision-makers belong to the various players in the intermodal transport world, namely, drayage operators, terminal operators, network operators, or intermodal operators.
Topics of relevance to this type of decision-making both in time horizon as in terms of operators are: • Intermodal terminal design • Infrastructure network configuration • Location of terminals • Cooperation between drayage companies • Allocation of shippers/receivers to a terminal • Pricing strategies • Capacity levels of equipment and labour • Operational routines and lay-out structure • Redistribution of load units, railcars, barges, and so forth • Scheduling of trips or jobs • Allocation of capacity to jobs • Loading orders • Selection of routing and service Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/jamds/guidelines.html. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/, according to the following timetable:

Manuscript Due
June 1, 2009 First Round of Reviews September 1, 2009