A Banach space has Pełczyński's property (V) if for every Banach space every unconditionally converging operator ∶ → is weakly compact. In 1962, Aleksander Pełczyński showed that ( ) spaces for a compact Hausdorff space enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about ( ) spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces. K E Y W O R D S Measures of weak non-compactness, Pełczyński's property (V), quantitative Pełczyński's property (V), unconditionally converging operators M S C ( 2 0 1 0 ) 46B04, 47B10, 46E15

A Banach space X has Pe lczyński's property (V) if for every Banach space Y every unconditionally converging operator T : X → Y is weakly compact. H. Pfitzner proved that C * -algebras have Pe lczyński's property (V). In the preprint [8] the author explores possible quantifications of the property (V) and shows that C(K) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner's theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.

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