Quantitative Schur property and measures of weak non-compactness
Ondřej F. K. Kalenda
TL;DR
The paper develops a comprehensive quantitative framework for the Schur property in Banach spaces, relating multiple versions of quantitative Schur properties to several measures of weak non-compactness. It establishes equivalences among these properties up to multiplicative constants, analyzes preservation under finite and infinite direct sums, and provides a new sufficient condition for the 1-Schur property. The work also delves into the relationship between quantitative Schur properties and quantitative weak sequential completeness, clarifies real versus complex settings (including Rosenthal-type results), and applies these ideas to Lipschitz-free spaces, presenting explicit graph-based examples illustrating nuanced Schur behavior. The results advance understanding of when quantitative measures align and how structural operations affect Schur-type properties, with several open problems highlighted for future study.
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.