1992/11/17 by Jeff D. Farmer, Farmer, Jeff, William B. Johnson +1
Mathematics · #46B #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.math/9211210
openalex publication_date 1992/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.