Polynomial Schur and polynomial Dunford-Pettis properties

Abstract

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 P N 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.

Polynomial properties and symmetric tensor product of Banach spaces

“…The above definition is quite similar to that given in [9] (and coincides with it under the Approximation Property; see [5] for examples and references). In [2] the definition adopted is (in our notation) N k1 w k E & dpE.…”

Properties of Q-Reflexive Banach Spaces

“…Since E is Q-reflexive it does not contain a copy of l 1 , and consequently it contains a norm-one weakly null sequence (see the proof of Proposition 3.6). Now E satisfies all conditions of Theorem 3.5 in [11] and hence there is a sequence in E which has an upper p estimate. By Proposition 4.2 this is impossible.…”

Polynomials and limited sets

“…Recall the following definition due to Farmer and Johnson [9]: A Banach space E is said to be a P N -Schur space (N ∈ N) if any sequence {a n } ⊂ E is a null sequence in E provided that {P (a n )} converges to 0 in C for all polynomials P ∈ P ( N E).…”