An alternative polynomial Daugavet property

Elisa R. Santos ¹

¹ Faculdade de Matemática Universidade Federal de Uberlândia Av. João Naves de Ávila 2121 Uberlândia, MG, 38408-100, Brazil

Studia Mathematica, Tome 224 (2014) no. 3, pp. 265-276

Résumé

We introduce a weaker version of the polynomial Daugavet property: a Banach space $X$ has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial $P: X \rightarrow X$ satisfies $$ \max_{\omega \in \mathbb T} \|{\rm Id} + \omega P\| = 1+\|P\|. $$ We study the stability of the APDP by $c_0$-, $\ell_\infty$- and $\ell_1$-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_\infty(\mu, X)$ and $C(K, X)$, where $X$ has the APDP.

Détail

DOI : 10.4064/sm224-3-4

Keywords: introduce weaker version polynomial daugavet property banach space has alternative polynomial daugavet property apdp every weakly compact polynomial rightarrow satisfies max omega mathbb omega study stability apdp ell infty ell sums banach spaces consequence obtain examples banach spaces apdp namely infty where has apdp