\((T,E)\)-Korovkin closures in normed spaces and BKW-operators (Q1899364)

Let \(X\) and \(Y\) be normed spaces, \(S\) a subset of \(X\), \(E\) a bounded subset of the dual \(Y^*\), and \(T\in B(X,Y)\) (i.e., \(T\) is a bounded linear operator from \(X\) to \(Y\)). The author defines \(T\) to be a BKW (for Bohman, Korovkin, and Wulbert) operator (relative to \(X\), \(Y\), \(S\), and \(E\)) if for every net \(T_\lambda\in B(X,Y)\) such that \(\lim_\lambda|T_\lambda|=|T|\) and \(\sup_s \lim_\lambda\sup_{\phi\in E}|\phi(T_\lambda(s)- T(s)|=0\), it follows that \(\lim_\lambda\sup_{\phi\in E}|\phi(T_\lambda(x)- T(x)|=0\) for every \(x\in X\). The functions in \(S\) are termed test functions. Clearly, a small set of test functions means a severe restriction on the operator \(T\). The author illustrates this in Theorem 2.1, where the set \(S\) is \(\{1,x\}\) and \(\{1,x,x^2\}\), respectively (here \(x\) is the identity function on \([0,1])\). In both cases there are extremely few BKW operators, which the author describes completely.