Embeddings and factorizations of Banach spaces

One problem, considered important in Banach space theory since at least the 1970’s, asks for intrinsic characterizations of subspaces of a Banach space with an unconditional basis. A more general question is to give necessary and sufficient conditions for operators from Lp (2 < p < 1) to factor through `p. In this dissertaion, solutions for the above problems are provided. More precisely, I prove that for a reflexive Banach space, being a subspace of a reflexive space with an unconditional basis or being a quotient of such a space, is equivalent to having the unconditional tree property. I also show that a bounded linear operator from Lp (2 < p < 1) factors through `p if and only it satisfies an upper-(C, p)-tree estimate. Results are then extended to operators from asymptotic lp spaces