# On tree characterizations of *G*_{δ}-embeddings and some Banach spaces

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## 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.

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