Polynomial Schur and Polynomial Dunford-Pettis Properties
Functional Analysis
2016-09-06 v1
Abstract
A Banach space is {\it polynomially Schur} if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford--Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space will has the Dunford--Pettis property if and only if every weakly compact homogeneous polynomial (in the sense of Ryan) on the space is completely continuous. A certain geometric condition, involving estimates on spreading models and implied by nontrivial type, is shown to be sufficient to imply that a space is polynomially Schur.
Cite
@article{arxiv.math/9211210,
title = {Polynomial Schur and Polynomial Dunford-Pettis Properties},
author = {Jeff Farmer and William B. Johnson},
journal= {arXiv preprint arXiv:math/9211210},
year = {2016}
}