On tree characterizations of G _{δ}embeddings and some Banach spaces
 S. Dutta,
 V. P. Fonf
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We show that a onetoone bounded linear operator T from a separable Banach space E to a Banach space X is a G _{δ}embedding if and only if every Tnull tree in S _{ E } has a branch which is a boundedly complete basic sequence. We then consider the notions of regulators and skipped blocking decompositions of Banach spaces and show, in a fairly general set up, that the existence of a regulator is equivalent to that of special skipped blocking decomposition. As applications, the following results are obtained.
(a) A separable Banach space E has separable dual if and only if every w*null tree in S _{ E }* has a branch which is a boundedly complete basic sequence.
(b) A Banach space E with separable dual has the point of continuity property if and only if every wnull tree in S _{ E } has a branch which is a boundedly complete basic sequence.
We also give examples to show that the tree hypothesis in both the cases above cannot be replaced in general with the assumption that every normalized w*null (wnull in (b)) sequence has a subsequence which is a boundedly complete basic sequence.
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 Title
 On tree characterizations of G _{δ}embeddings and some Banach spaces
 Journal

Israel Journal of Mathematics
Volume 167, Issue 1 , pp 2748
 Cover Date
 20081001
 DOI
 10.1007/s1185600810395
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 The Hebrew University Magnes Press
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 Authors

 S. Dutta ^{(1)}
 V. P. Fonf ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Ben Gurion University of the Negev, P. O. B. 653, BeerSheva, 84105, Israel