ar X iv : m at h / 92 11 21 0 v 1 [ m at h . FA ] 1 7 N ov 1 99 2 POLYNOMIAL SCHUR AND POLYNOMIAL DUNFORD - PETTIS PROPERTIES

A Banach space is 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 PN Dunford–Pettis property if and only if every weakly compact N−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.