1Department of Mathematics POSTECH Pohang 790-784, Korea 2Departamento de Análisis Matemático Universidad de Valencia Doctor Moliner 50 46100 Burjasot (Valencia), Spain 3Departamento de Análisis Matemático Universidad de Valencia Doctor Moliner 50 $46100$ Burjasot (Valencia), Spain 4Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
Studia Mathematica, Tome 178 (2007) no. 1, pp. 63-84
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.
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
Affiliations des auteurs :
Yun Sung Choi
1
;
Domingo García
2
;
Manuel Maestre
3
;
Miguel Martín
4
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
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author = {Yun Sung Choi and Domingo Garc{\'\i}a and Manuel Maestre and Miguel Mart{\'\i}n},
title = {The {Daugavet} equation for polynomials},
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AU - Manuel Maestre
AU - Miguel Martín
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Yun Sung Choi; Domingo García; Manuel Maestre; Miguel Martín. The Daugavet equation for polynomials. Studia Mathematica, Tome 178 (2007) no. 1, pp. 63-84. doi: 10.4064/sm178-1-4
