The Daugavet equation for polynomials

Yun Sung Choi; Domingo García; Manuel Maestre; Miguel Martín

Studia Mathematica (2007)

  • Volume: 178, Issue: 1, page 63-84
  • ISSN: 0039-3223

Abstract

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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 m a x | ω | = 1 | | I d + ω P | | = 1 + | | P | | 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.

How to cite

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Yun Sung Choi, et al. "The Daugavet equation for polynomials." Studia Mathematica 178.1 (2007): 63-84. <http://eudml.org/doc/285199>.

@article{YunSungChoi2007,
abstract = {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 $max_\{|ω|=1\} ||Id + ωP|| = 1 + ||P||$ 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.},
author = {Yun Sung Choi, Domingo García, Manuel Maestre, Miguel Martín},
journal = {Studia Mathematica},
keywords = {Daugavet equation; alternative Daugavet equation; Daugavet property; alternative Daugavet property},
language = {eng},
number = {1},
pages = {63-84},
title = {The Daugavet equation for polynomials},
url = {http://eudml.org/doc/285199},
volume = {178},
year = {2007},
}

TY - JOUR
AU - Yun Sung Choi
AU - Domingo García
AU - Manuel Maestre
AU - Miguel Martín
TI - The Daugavet equation for polynomials
JO - Studia Mathematica
PY - 2007
VL - 178
IS - 1
SP - 63
EP - 84
AB - 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 $max_{|ω|=1} ||Id + ωP|| = 1 + ||P||$ 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.
LA - eng
KW - Daugavet equation; alternative Daugavet equation; Daugavet property; alternative Daugavet property
UR - http://eudml.org/doc/285199
ER -

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