Dunford-Pettis and Dieudonné polynomials on Banach spaces (Q5954258)

The work is a generalization of Dunford-Pettis and Dieudonné operator to the polynomial context: the generalization is done by introducing two classes of \(m\)-homogeneous polynomials, defined on Banach spaces; the initiated generalization is based on the general behavior of \(m\)-homogeneous polynomials with respect to the polynomial and weak topologies; the behavior shows that every \(m\)-homogeneous polynomial transforms sequences which are Cauchy in the \(m\)-polynomial topology (the \(\tau_n\)-Cauchy sequences) into weak Cauchy sequences; the generalization comes through those polynomials which transform \(\tau_n\)-Cauchy sequences into norm and weakly convergent sequences respectively. These extensions allow us to prove that several characterization theorems related to Dunford-Pettis, Schur, and reciprocal Dunford-Pettis properties, are also valid in the more general case of homogeneous polynomials of any degree \(m\in\mathbb{N}\).