On tree characterizations of Gδ-embeddings and some Banach spaces

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(delta)-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.