On quantitative Schur and Dunford-Pettis properties

TL;DR

This paper investigates the quantitative versions of the Schur and Dunford-Pettis properties in Banach spaces, establishing new relationships and applying these to subspaces of compact operators on ll_p spaces.

Contribution

It proves that duals of subspaces of c_0(mma) have the strongest quantitative Schur property and links this to quantitative Dunford-Pettis properties, with applications to operator spaces.

Findings

Duals of subspaces of c_0(mma) have the strongest quantitative Schur property.

Subspaces of compact operators on ll_p with Dunford-Pettis property satisfy their quantitative versions.

Established relationships between quantitative Schur and Dunford-Pettis properties.

Abstract

We show that the dual to any subspace of $c_0(\Gamma)$ has the strongest possible quantitative version of the Schur property. Further, we establish relationship between the quantitative Schur property and quantitative versions of the Dunford-Pettis property. Finally, we apply these results to show, in particular, that any subspace of the space of compact operators on $\ell_p$ ($1<p<\infty$) with Dunford-Pettis property satisfies automatically both its quantitative versions.