Weighted composition operators and dynamical systems (Q2724884)

Let \(X\) be a completely regular Hausdorff space and let \(E\) be a Hausdorff locally convex topological vector space over \(K\), where \(K\) denotes either the set of real numbers \(R\) or the set of complex numbers \(C\). If \(C(X,E)\) denotes the space of continuous functions from \(X\) into \(E\) and \(V\) denotes a system of weight functions, then the spaces \(CV_b(X,E)\), \(CV_0(X, E)\) are defined by NEWLINE\[NEWLINE\begin{aligned} CV_b(X, E) &= \{f\in C(X,E):vf\text{ is bounded on \(X\) for all }v\in V\};\\ CV_0(X, E)&= \{f\in C(X,E):vf\text{ vanishes at \(\infty\) on \(X\) for all }v\in V\}.\end{aligned}NEWLINE\]NEWLINE If \(p\) is a semi-norm on \(E\), then \(\|f\|_{v,p}= \sup\{v(x) p(f(x)): x\in X\}\). If \(Y\) is a topological vector space, then a function \(\rho: R\times Y\to Y\) is said to be a dynamical system on \(Y\) ifNEWLINENEWLINENEWLINE\((^*1)\) (i) \(\rho(0,y)= y\) for \(y\in Y\);NEWLINENEWLINENEWLINE(ii) \(\rho(a+ b,y)= \rho(a,\rho(b, y))\);NEWLINENEWLINENEWLINE(iii) \(\rho\) is continuous.NEWLINENEWLINENEWLINEIf \(w: X\to{\mathcal B}(E)\) (the space of bounded operators on \(E\)), and \(T: X\to X\), then the weighted composition operator \(wG\) is defined by \(wG(f)(x)= w(x) G(T(x))\), \(x\in X\), \(f\in CV_b(X, E)\).NEWLINENEWLINENEWLINEIn an earlier paper, Rocky M. J. Math. 23, No.~3, 1107-1114 (1993; Zbl 0809.47028), the authors derive necessary and sufficient conditions that \(wG\) be a `weighted composition operator' on \(CV_b(X, E)\).NEWLINENEWLINENEWLINEIn the first main result of this paper, sufficient conditions are stated for \((w,T)\) to `induce a weighted composition operator' on \(CV_b(X, E)\). In particular, one of the conditions indicates that for \(v\in V\) and semi-norm \(p\), a weighted function \(v\in V\) and semi-norm \(q\) on \(E\) exist such that, \(v(x)p(w(x) t)\leq u(T(x)) q(t)\) for all \(x\in X\), \(t\in E\).NEWLINENEWLINENEWLINEFurther results of the paper involve the identification of dynamical system \(\rho\) with homeomorphisms on \(R\).