Banach空间结构理论的重大进展 ──关于Gowers-Maurey系列成果

The Significant Development in the Structural Theory of Banach Spaces ──On a Series of the Gowers-Maurey's Results

最近，Ｇｏｗｅｒｓ Ｗ． Ｔ．和 Ｍａｕｒｅｙ Ｂ．构造出第一例遗传不可分解空间，否定地解决了无条件基序列问题，由此导致了 Ｂａｎａｃｈ空间结构理论研究中系列问题的解决．本综述介绍这一新动向，反映Ｇ－Ｍ系列成果．全文分为七个部分：１．历史回顾与问题沿革；２． Ｇ－Ｍ空间ＸＧ１及其遗传不可分解性质；３．关于空间 ＸＧ１上的算子构成； ４．关于共轭空间 ＸＧ１；５．关于 Ｇ－Ｍ的系列成果； ６． Ｇ－Ｍ型空间构造的新视角探讨；７．ＯＰｅｎ问题总汇．

Recently Gowers W.T. and Maurey B. constructed the first example XG1 of the hereditarily indecomposable Banach spaces containing no unconditional basic sequence. Some further studies and reconstructions of this space result in some satisfactory answers of a series of open questions in the Banach spaces theory. In this paper, recent developments about this direction have been introduced. It composes of the following seven sections: 1. A review of the history and the evolution of the problems, 2. The G-M space and its hereditarily indecomposable property; 3. On the operators on the space XG1;4. On the conjugate space XG1;5. On a series of G-M's results; 6. A new viewpotint of construction of G-M-type space; 7. A list of open problems.