Polynomial daugavet centers

Abstract

Abstract A polynomial $Q: X \rightarrow Y$ is called a polynomial Daugavet center if the equality $$\begin{equation*} \|Q + P \| = \|Q\| + \|P \| \end{equation*}$$is satisfied for all rank-one polynomials $P: X \rightarrow Y$. In this paper, we present geometric characterizations of polynomial Daugavet centers. We show that if $Q$ is a polynomial Daugavet center, then every weakly compact polynomial $P$ also satisfies this equation. Finally, we prove that if $Y$ is a subspace of a Banach space $E$ and $Q: X \rightarrow Y$ is a polynomial Daugavet center, then $E$ can be equivalent renormed in such a way that the norm on $Y$ is not modified and $J \circ Q: X \rightarrow E$ is a polynomial Daugavet center. We also present some examples of polynomial Daugavet centers.