On tree characterizations of G δ-embeddings and some Banach spaces

S. Dutta, V. P. Fonf

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

We show that a one-to-one 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 T-null 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 w-null 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 (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence.

Original languageEnglish
Pages (from-to)27-48
Number of pages22
JournalIsrael Journal of Mathematics
Volume167
Issue number1
DOIs
StatePublished - 1 Oct 2008

ASJC Scopus subject areas

  • Mathematics (all)

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