Quantitative Schur property and measures of weak non-compactness

TL;DR

This paper compares various versions of the quantitative Schur property in Banach spaces, establishing their equivalence, exploring their preservation under sums, and examining their relation to weak non-compactness and complex settings.

Contribution

It provides a comprehensive comparison of quantitative Schur properties, introduces a sufficient condition for the 1-Schur property, and analyzes their behavior in Lipschitz-free spaces.

Findings

Quantitative Schur properties are equivalent up to constants.

Finite and infinite sums preserve the quantitative Schur property under certain conditions.

Lipschitz-free spaces can have the quantitative Schur property without the 1-Schur property.

Abstract

We compare several versions of the quantitative Schur property of Banach spaces. We establish their equivalence up to multiplicative constants and provide examples clarifying when the change of constants is necessary. We also give exact results on preservation of the quantitative Schur property by finite or infinite direct sums. We further prove a sufficient condition for the $1$-Schur property which simplifies and generalizes previous results. We study in more detail relationship of the quantitative Schur property to quantitative weak sequential completeness and to equivalence of measures of weak non-compactness. We also illustrate the difference of real and complex settings. To this end we prove and use the optimal version of complex quantiative Rosenthal $\ell_1$-theorem. Finally, we give two examples of Lipschitz-free spaces over countable graphs which have quantitative Schur property, but not the $1$-Schur property.