Almost summing, fully absolutely summing, multilinear mapping

In this paper the classes of strongly almost summing multilinear mappings and fully almost summing multilinear mappings are introduced. We investigate the connections of these classes and other classes of absolutely summing mappings. Besides, we prove some structural properties such as a Dvoretzky-Rogers Theorem, some coincidence results and generalize a theorem of S. Kwapien which asserts that T is absolutely (1;1)-summing whenever T* is absolutely (q;q)-summing for some q>=1.