On tree characterizations of Gδ-embeddings and some Banach spaces
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- Dutta, S. & Fonf, V.P. Isr. J. Math. (2008) 167: 27. doi:10.1007/s11856-008-1039-5
We show that a one-to-one bounded linear operator T from a separable Banach space E to a Banach space X is a Gδ-embedding if and only if every T-null tree in SE 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 SE* 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 w-null tree in SE 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 (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence.