The polynomial Daugavet property for atomless \(L_{1}(\mu)\)-spaces
The (linear) Daugavet property of a Banach space \(X\) can be defined by saying that \(\|\text{Id}+T\|=1+\|T\|\) for each weakly compact linear operator \(T:X\to X\). Likewise, the polynomial Daugavet property requires that \(\|\text{Id}+P\|=1+\|P\|\) for each weakly compact polynomial \(P: X\to X\). The classical examples of Banach spaces with the Daugavet property are vector-valued \(C(K)\)- and \(L_1\)-spaces, where \(K\) is perfect and the measure is atomless, and it has been proved in [\textit{Y.\,S.\thinspace Choi, D.\,García, M.\,Maestre} and \textit{M.\,Martín}, Stud.\ Math.\ 178, No.\,1, 63--82 (2007; Zbl 1121.46039)] that those \(C(K,E)\)-spaces also have the polynomial Daugavet property. Here, the authors extend this result to atomless \(L_1(\mu,E)\)-spaces, for which a different technique of proof is needed. They also pose the problem to decide whether the Daugavet property implies the polynomial Daugavet property in general.
scientific article; zbMATH DE number 5708651