Abstract Motivated by recent results on the space of compact operators between Banach spaces and by extensions of the Josefson–Nissenzweig theorem to Fréchet spaces, we investigate pairs of Fréchet spaces ( E , F ) such that every continuous linear map from E into F is Montel, i.e. it maps bounded subsets of E into relatively compact subsets of F . As a consequence of our results we characterize pairs of Köthe echelon spaces ( E , F ) such that the space of Montel operators from E into F is complemented in the space of all continuous linear maps from E into F .