On some operator functional equations related to evaluation functionals and weighted composition operators on function spaces

The paper is concerned with some operator functional equations which have been recently considered in several papers in developing some extensions and generalizations of Korovkin-type convergence theorems. It is shown that such operator functional equations although arising from different contexts, have a common root. Starting from an unifying setting, it is shown that the solutions of such operator functional equations are necessarily evaluation functionals and weighted composition operators, respectively. Several special cases are discussed in detail, including the one which concerns spaces of \(2\pi -\) $2\pi -$ periodic functions on the real line.