The Daugavet equation for polynomials

Yun Sung Choi¹ ; Domingo García² ; Manuel Maestre³ ; Miguel Martín⁴

¹ Department of Mathematics POSTECH Pohang 790-784, Korea
² Departamento de Análisis Matemático Universidad de Valencia Doctor Moliner 50 46100 Burjasot (Valencia), Spain
³ Departamento de Análisis Matemático Universidad de Valencia Doctor Moliner 50 $46100$ Burjasot (Valencia), Spain
⁴ Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain

Studia Mathematica, Tome 178 (2007) no. 1, pp. 63-84

Résumé

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.

Détail

DOI : 10.4064/sm178-1-4

Keywords: study daugavet equation satisfied weakly compact polynomials banach space equality mathop satisfied weakly compact polynomials real complex space continuous functions compact space without isolated points study alternative daugavet equation max omega mathop omega polynomials rightarrow equation holds every polynomial complex space arbitrary values result real finally study daugavet alternative daugavet equations k homogeneous polynomials