Komleva-type expansions and asymptotics for linear operators

In this paper, we prove asymptotic Komleva-type expansions valid for sequences of linear operators {Tn(·)} approximating the identity in supremum norm over the space of the continuous functions. In particular, under suitable mild conditions on the sequence {Tn(·)}, we obtain rational expansions for {Tn(f)} that are of special interest in a numerical analysis context. As special cases of these results, we find asymptotic expansions for expotential-type and De La Vallée Poussin polynomial operators. The case of the Cesaro sums is discussed in connection with the Komleva theory, but the main asymptotic results are proved by using other tools coming from a context of structured linear algebra. Some numerical applications of the theoretical part are then discussed.