Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

Streszczenie

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.