Characterizations Of Banach Space Properties Using Cone Absolutely Summing Operators PDF Download

Let X be a Banach space and $\Lambda$ a subset of the dual group of a compact metrizable abelian group G. We define that the Banach space X has type I-, (resp. II)-$\Lambda$-complete continuity property if every X-valued measure of bounded average range (resp. of bounded variation) with the property that $\\mu(\gamma)=\int\sb{G}\gamma({-}t)d\mu(t) = 0$ for $\gamma\not\in\Lambda, $ has relatively compact range. We say that the Banach space X has type III-$\Lambda$-complete continuity property if every absolutely summing operator $T:{\cal C}(G)\to X$ such that $T(\bar\gamma)$ = 0 for $\gamma\not\in\Lambda, $ is compact. We also introduce the notion of the analytic complete continuity property. We give different characterizations of this property and we establish that this property is a particular case of the types $\Lambda$-complete continuity property. We will exhibit an example of a Banach space that has the analytic complete continuity property but failing properties such as the Radon-Nikodym property or the complete continuity property. We show that the type II-$\Lambda$-complete continuity property passes from a Banach space X into the function space $L\sp1(G, X)$ for the case where G = T is the circle group and where $\Lambda$ is a Riesz set. We will also obtain some results related to the stability of these new properties under $G\sb\delta$-embeddings. We finish this thesis by a thorough investigation of the convolution operators with respect to vector measures. We characterize those convolution operators that are compact or weakly compact on $L\sp1(G)$ and on ${\cal C}(G), $ as well as those that are p-summing or nuclear on ${\cal C}(G).$