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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## The Daugavet equation for polynomials

### Tom 178 / 2007

Studia Mathematica 178 (2007), 63-84 MSC: Primary 46G25; Secondary 46B20, 47A12. DOI: 10.4064/sm178-1-4

#### Streszczenie

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.

#### Autorzy

• Yun Sung ChoiDepartment of Mathematics
POSTECH
Pohang 790-784, Korea
e-mail
• Domingo GarcíaDepartamento de Análisis Matemático
Doctor Moliner 50
46100 Burjasot (Valencia), Spain
e-mail
• Manuel MaestreDepartamento de Análisis Matemático
Doctor Moliner 50
$46100$ Burjasot (Valencia), Spain
e-mail
• Miguel MartínDepartamento de Análisis Matemático