Multiplication operators on weighted spaces of continuous functions
Let \(X\) be a completely regular Hausdorff space, and let \(E\) be a locally convex Hausdorff space over the field \(K\) of real numbers \(\mathbb{R}\) or complex numbers \(\mathbb{C}\). The topology of \(E\) is derived from a family of semi-norms \(\mathbb{P}\). The space of \(E\)-valued continuous functions in \(X\) is denoted by \(C (X,E)\), the algebra of continuous operators on \(E\) is denoted by \(B(E)\), and a Nachbin space \(V\) on \(X\) is a class of upper semi-continuous functions on \(X\) such that, if \(v_1,v_2\) are in \(V\) and \(\lambda>0\), then there is a function \(v\) in \(V\) with max \((\lambda v_1,\lambda v_2)\leq v\), and for each \(x\in X\), there is a \(v\) in \(V\) such that \(v(x)\neq 0\), and related spaces \(CV(X,E)\), \(CV_0 (X,E)\) defined by \(CV(X,E)= \{f\in C(X,E): vP(f)\in B(X)\), for \(P\in P\) and \(v \in V\}\); \(CV_0(X,E)= \{f\in CV(X,E): vP(f)\in B_0(X)\) for \(P\in\mathbb{P}\), \(v\in V\}\), where \(B_0(X)= \{g\in B(X): g(x)\to 0\) at \(\infty\}\). If \(\psi:X\to B(E)\), then the operator \({\mathcal M}_\psi\) defined by \({\mathcal M}_\psi(f)(x) =\psi(x)f(x)\), is said to be a multiplication operator. The main results of this paper include the following Theorem: If \(F\) is an \(EV\)-solid subspace of \(CV(X,E)\), \(X \) is \(a\) \(\gamma_R\)-space for some property \(\gamma\), and if \(\psi:X\to B_\gamma (E)\) is continuous, then \({\mathcal M}_\psi\) is a multiplication operator on \(F\) if and only if for \(v\in V\) and \(P\in\mathbb{P}\), there are functions \(u\) in \(V\) and \(Q\) in \(\mathbb{P}\) such that \(v(x)P (\psi_x(a))\leq u(x)Q(a)\), for \(a\in E\) and \(x\in coz(F)\), where \(coz(F)= \{x\in X:f(x)\neq 0\) for some \(f\in F\}\), and \(F\) is \(EV\)-solid if for \(g\in F\), \(v\in V\), \(P\in\mathbb{P}\), \(v\) \(P\circ g\leq u\) \(Q\circ f\), for some \(u\in V\), \(Q\in\mathbb{P}\), \(f\in coz(F)\). The results provide adequate conditions for earlier considerations of \textit{R. K. Singh} and \textit{J. S. Manhas} [J. Aust. Math. Soc., Ser. A 53, 91-102 (1992; Zbl 0762.34020); corrig. ibid. 58, 142 (1995; Zbl 0822.34044)].
scientific article; zbMATH DE number 1787285