Korovkin closures in cones of set-valued functions

The aim of the paper is to establish a complete characterization of the Korovkin closures in cones of set-valued functions. Korovkin closures play a central role in Korovkin approximation theory and their complete characterization is an old problem which has been considered since the beginning of the Korovkin theory's development. However, until now this characterization is well-known only in spaces of continuous single real-valued functions. In order to give a more detailed description of the Korovkin closures, it is natural to introduce the notions of onesided external and internal Korovkin closures by separating two different roles of the approximating functions in a Korovkin system. Contrary to what one might expect and differently from the single-valued case, only the role of the external Korovkin closures will be decisive in the characterization of the Korovkin systems and this will allow us to give also a more precise description together with different examples of particular interest. Further characterizations are given in the cone of set-valued functions having real compact intervals as values. In this case the study of the Korovkin closure is deepened by introducing the upper and lower Korovkin closures.