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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-1551 |
| Date | 15 May 2009 |
| Creators | Zheng, Bentuo |
| Contributors | Johnson, William B. |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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