## 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 language | English |
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Pages (from-to) | 27-48 |

Number of pages | 22 |

Journal | Israel Journal of Mathematics |

Volume | 167 |

Issue number | 1 |

DOIs | |

State | Published - 1 Oct 2008 |

## ASJC Scopus subject areas

- General Mathematics

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