  A Banach space $X$ has Pe{\l}czy\' nski's property (V) if for every Banach
space $Y$ every unconditionally converging operator $T\colon X\to Y$ is weakly
compact. In 1962, Aleksander Pe{\l}czy\' nski showed that $C(K)$ spaces for a
compact Hausdorff space $K$ 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 $C(K)$ 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.
