We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → 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 for polynomials P: X → 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm178-1-4,
author = {Yun Sung Choi and Domingo Garc\'\i a and Manuel Maestre and Miguel Mart\'\i n},
title = {The Daugavet equation for polynomials},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {63-84},
zbl = {1121.46039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm178-1-4}
}
Yun Sung Choi; Domingo García; Manuel Maestre; Miguel Martín. The Daugavet equation for polynomials. Studia Mathematica, Tome 178 (2007) pp. 63-84. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm178-1-4/