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The Quarterly Journal of Mathematics Advance Access published online on January 31, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/ham054
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© 2007. Published by Oxford University Press. All rights reserved For permissions, please email: journals.permissions@oxfordjournals.org

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

Yun Sung Choi {dagger}

Department of Mathematics, POSTECH, Pohang 790-784, South Korea

Domingo García {ddagger} and Manuel Maestre §

Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot, Valencia, Spain

Miguel Martín

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Corresponding author. E-mail: mmartins{at}ugr.es

Received 20 April 2007; revised 5 November 2007
  Abstract

We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K, X) and L{infty}(µ, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every {sigma}-finite measure µ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k ≥ slant 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X** is the least possible, namely kk/(1–k). We also give new examples of Banach spaces with the polynomial Daugavet property, namely L{infty}(µ, X) when µ is atomless, and Cw(K, X), Cw*(K, X*) when K is perfect.

{dagger} E-mail: mathchoi{at}postech.ac.kr

{ddagger} E-mail: domingo.garcia{at}uv.es

§ E-mail: manuel.maestre{at}uv.es


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