The Daugavet equation for polynomials
Volume 178 / 2007
Abstract
We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space $X$, i.e. when the equality $$ \|\mathop{\rm Id}+P\|=1+\|P\| $$ is satisfied for all weakly compact polynomials $P:X\to X$. We show that this is the case when $X=C(K)$, the real or complex space of continuous functions on a compact space $K$ without isolated points. We also study the alternative Daugavet equation $$ \max_{|\omega|=1} \|\mathop{\rm Id} +\omega P\| = 1 + \|P\| $$ for polynomials $P:X\rightarrow X$. We show that this equation holds for every polynomial on the complex space $X=C(K)$ ($K$ arbitrary) with values in $X$. This result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for $k$-homogeneous polynomials.
