Israel Journal of Mathematics

, Volume 167, Issue 1, pp 27–48 | Cite as

On tree characterizations of Gδ-embeddings and some Banach spaces

Article

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 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.

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Copyright information

© Hebrew University Magnes Press 2008

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael

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