A result in best approximation theory (Q6371698)

Let T, I be two commuting operators on a subset c of a Banach space X, and let F(T) (respectively F(I)) be the set of fixed points of T (respectively I). For \(\bar x\in F(T)\cap F(I)\), D denotes the set of best approximation elements of \(\bar x\) in C. The authors prove that \(D\cap F(T)\cap F(I)\) is non-empty if C, F and I satisfies certain conditions. If the operator I is the identity on X, one obtains a result of \textit{S. P. Singh} [J. Approximation Theory 25, 89-90 (1979; Zbl 0399.41032)].