On a difference between quantitative weak sequential completeness and the quantitative Schur property (Q2845860)

This paper is framed in an ambitious program, initiated by the authors and their coauthors, to explore appropriate quantitative versions of different classical properties of Banach spaces. In this note, the authors study quantitative versions of the Schur property and of being weakly sequentially complete. A Banach space \(X\) is said to be weakly sequentially complete if every weakly Cauchy sequence is weakly convergent; the space \(X\) has the Schur property if every weakly convergent sequence is norm convergent. Every space with the Schur property is weakly sequentially complete; \(\ell_1\) satisfies the Schur property, while \(L_1(0, 1)\) is weakly sequentially complete but does not satisfy the Schur property.NEWLINENEWLINEIf \((x_n)\) is a bounded sequence in a Banach space \(X\), then the symbol \(\text{clust}_{X^{**}} (x_n)\) stands for the set of all weak\(^*\) cluster points of the sequence \((x_n)\) in \(X^{**}\) and \(\delta (x_n)\) for the diameter of the latter set. The quantity \(\delta (x_n)\) measures how far the sequence \((x_n)\) is from being weakly Cauchy. Given \(C\geq0\), the space \(X\) is said to be \(C\)-weakly sequentially complete if NEWLINE\[NEWLINE{\widehat{d}} (\text{clust}_{X^{**}} (x_n),X)\leq C \delta (x_n),NEWLINE\]NEWLINE for every bounded sequence \((x_n)\) in \(X\), where \({\widehat{d}}\) denotes the nonsymmetrized Hausdorff distance given by \(\widehat{d}(A,B) = \sup\{ d(a,B) : a\in A\}\). A quantitative version of weak sequential completeness was studied by \textit{G. Godefroy, N. J. Kalton} and \textit{D. Li} [Indiana Univ. Math. J. 49, No. 1, 245--286 (2000; Zbl 0973.46008); J. Reine Angew. Math. 471, 43--75 (1996; Zbl 0842.46008)] and \textit{O. F. K. Kalenda, H. Pfitzner} and \textit{J. Spurný} [J. Funct. Anal. 260, No. 10, 2986--2996 (2011; Zbl 1248.46012)]. In the last mentioned paper, it was proved that any \(L\)-embedded space is \(1/2\)-weakly sequentially complete; furthermore, the constant \(1/2\) is optimal as witnessed by the space \(\ell_1\). There was also given an example of a Schur space which is not \(C\)-weakly sequentially complete for any \(C \geq 0\). Conversely, \(C\)-weak sequential completeness does not imply the Schur property.NEWLINENEWLINETo measure the how far the sequence \((x_n)\) is from being norm Cauchy, the authors in the present paper introduce the quantity NEWLINE\[NEWLINE\text{ca} (x_n)= \inf_{n} \text{diam}\{x_k: k\geq n\}.NEWLINE\]NEWLINE \(X\) has the \(C\)-Schur property if NEWLINE\[NEWLINE\text{ca} (x_n)\leq C \delta (x_n),NEWLINE\]NEWLINE for every bounded sequence \((x_n)\) in \(X\).NEWLINENEWLINEThe main results of the present paper establish the following. Every Banach space which is \(C\)-weakly sequentially complete for some \(C<1/2\) is reflexive. Motivated by a remark of M. Fabian, the authors show that \(\ell_1\) satisfies the 1-Schur property. There exists a separable \(L\)-embedded Banach space with the Schur property which fails to have the \(C\)-Schur property for every \(C\geq 0\).