Quantification of the reciprocal Dunford-Pettis property (Q2919699)

There is a growing interest in the study of appropriate quantitative versions of classical properties. The main idea behind quantitative versions is an attempt to replace the implication defining the property by an inequality. For example, we recall that a Banach space \(X\) is said to have the Dunford-Pettis property if, for any Banach space \(Y\), and every bounded operator \(T : X \to Y\), NEWLINE\[NEWLINET \text{ weakly compact } \Rightarrow T \text{ completely continuous.} NEWLINE\]NEWLINE Further, \(X\) is said to have the reciprocal Dunford-Pettis property if, for any Banach space \(Y\), and every bounded operator \(T : X \to Y\), NEWLINE\[NEWLINET \text{ completely continuous } \Rightarrow T \text{ weakly compact.} NEWLINE\]NEWLINE Roughly speaking, the quantitative versions of the Dunford-Pettis and the reciprocal Dunford-Pettis properties are defined by the existence of a constant \(C>0\) satisfying an inequality of the form: NEWLINE\[NEWLINE\text{measure of non-complete continuity of \(T\)} NEWLINE\]NEWLINE NEWLINE\[NEWLINE\leq C \cdot \text{measure of weak non-compactness of \(T\),}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\text{measure of weak non-compactness of \(T\)} NEWLINE\]NEWLINE NEWLINE\[NEWLINE\leq C \cdot \text{measure of non-complete continuity of \(T\),}NEWLINE\]NEWLINE respectively. Of course, the above definitions make sense when we have quantifiers to measure the weak non-compactness and the non-complete-continuity. The necessary quantifiers were essentially defined by \textit{M. Kačena} and the authors [Adv. Math. 234, 488--527 (2013; Zbl 1266.46007)], while appropriate reformulations for the reciprocal Dunford-Pettis property are made in the note under review. More precisely, if \(A,B\) are two non-empty subsets of a metric space \(X\), their non-symmetrized Hausdorff distance is defined by \(\widehat{\text{d}}(A,B) = \sup \{ \text{dist} (x,B) : x \in A\}\). The Hausdorff measure of non-compactness of a non-empty set \(A \subseteq X\) is defined by NEWLINE\[NEWLINE\chi (A) = \inf \{ \widehat{\text{d}} (A, F) : F \subset X \text{ finite} \}.NEWLINE\]NEWLINE When \(X\) is a Banach space, there are two measures of weak non-compactness of a subset \(A\subset X\): the De Blasi measure of weak non-compactness NEWLINE\[NEWLINE\omega (A) = \inf \{ \widehat{\text{d}} (A,K) : K \subset X \text{ weakly compact }\},NEWLINE\]NEWLINE and a smaller quantity defined by NEWLINE\[NEWLINE\text{wk}_X (A) = \widehat{\text{d}} (\overline{A} ^{w^*}, X), \text{ obviously in } X^{**}.NEWLINE\]NEWLINE The note under review begins with the study of the quantitative reciprocal Dunford-Pettis property. The quantitative version of the Dunford-Pettis property was studied by M. Kačena and the authors of this note in [loc. cit.], where it was proved, among other things, that both \(L^1(\mu)\) spaces and \(C_0(\Omega)\) spaces enjoy the strongest possible version of the quantitative Dunford-Pettis property, where \(\Omega\) is a locally compact Hausdorff space. The present paper complements this line of study by establishing the following quantitative version of the reciprocal Dunford-Pettis property for \(C_0(\Omega)\), where \(\Omega\) is a locally compact Hausdorff space: Let \(Y\) be a Banach space, then for every bounded linear operator \(T : C_0 (\Omega)\to Y\) we have NEWLINE\[NEWLINE \frac{1}{4\pi} \text{wk}_Y (T) \leq \text{cc} (T) \leq 4 \text{wk}_Y (T),NEWLINE\]NEWLINE where \(\text{wk}_Y (T) = \text{wk}_Y (T(B_{C_0 (\Omega)}))\) and cc\((T)\) is the measure of non-complete continuity of \(T\). Some other quantitative versions of results proved by Grothendieck are also established with the help of a quantifier to measure Mackey non-compactness of a subset \(A\) of the dual of a Banach space \(X\).NEWLINENEWLINEThe paper under review and [loc. cit.] are intrinsically related and open a very interesting line of problems to explore.NEWLINENEWLINEIt should be remarked that quantitative versions of the Krein theorem were studied by \textit{M. Fabian, P. Hájek, V. Montesinos} and \textit{V. Zizler} [Rev. Mat. Iberoam. 21, No. 1, 237--248 (2005; Zbl 1083.46012)], \textit{A. S. Granero} [Rev. Mat. Iberoam. 22, No. 1, 93--110 (2006; Zbl 1117.46002)], \textit{A. S. Granero, P. Hájek} and \textit{V. Montesinos Santalucía} [Math. Ann. 328, No. 4, 625--631 (2004; Zbl 1059.46015)], and \textit{B. Cascales, W. Marciszewski} and \textit{M. Raja} [Topology Appl. 153, No. 13, 2303--2319 (2006; Zbl 1118.46012)], quantitative versions of the Eberlein-Šmulyan and the Gantmacher theorem were investigated by \textit{C. Angosto} and \textit{B. Cascales} [Topology Appl. 156, No. 7, 1412--1421 (2009; Zbl 1176.46012)], a quantitative version of James's compactness theorem was proved by \textit{B. Cascales, O. F. K. Kalenda} and \textit{J. Spurný} [Proc. Edinb. Math. Soc., II. Ser. 55, No. 2, 369--386 (2012; Zbl 1255.46009)] and \textit{A. S. Granero, J. M. Hernández} and \textit{H. Pfitzner} [Proc. Am. Math. Soc. 139, No. 3, 1095--1098 (2011; Zbl 1225.46011)], a quantification of weak sequential completeness and of the Schur property was addressed by \textit{O. F. K. Kalenda, H. Pfitzner} and \textit{J. Spurný} [J. Funct. Anal. 260, No. 10, 2986--2996 (2011; Zbl 1248.46012)] and \textit{O. F. K. Kalenda} and \textit{J. Spurný} [Proc. Am. Math. Soc. 140, 3435--3444 (2012; Zbl 1283.46009)].