Embeddings and factorizations of Banach spaces



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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.