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Spreading models of Banach spaces. (Modèles étalés des espaces de Banach.) (French) Zbl 0553.46012

Travaux en Cours. Paris: Hermann. iv, 210 p. FF 160.00 (1984).
[The complete version of the following abrigded review is available on demand.]
For a brief review of the preliminary version of this book see Zbl 0526.46021. The introduction contains a short “history” of the theory of spreading models; note only that this theory has its origin in papers by A. Brunel [Sémin. Maurey-Schwartz, 1973/74, Exp. No. 15 (1974; 305.46024 and Zbl 0305.46025)] and A. Brunel and L. Sucheston [Math. Syst. Theory 7, 294–299 (1974; Zbl 0323.46018)].
The first chapter is an introduction to the subject. It begins with the definition of nice sequences, spreading sequences and spreading model. The rest of the chapter deals with coordinate sequences, weak convergence of spreading and fundamental sequences, shifts of the fundamental sequence and subspaces of spreading models.
The second chapter is devoted to Banach-Saks properties (BSP), and characterizes weak and alternate BSP (WBSP and ABSP resp.) in terms of spreading models isomorphic to \(\ell^ 1\). Remember that (BSP) \(\Rightarrow\) (ABSP) \(\Rightarrow\) (WBSP), (BSP)\(=\)(ABSP) on reflexive spaces and (ABSP)\(=\)(WBSP) on spaces not containing \(\ell^ 1\).
The third chapter investigates the dual spaces in connection with spreading models. The space \(c_ 0\) is not a spreading model of \(E\) if and only if each spreading sequence \((x_ n)_{n\in N}\) in \(E\) contains a subsequence \((y_ n)_{n\in N}\) which tends to \(+\infty\) uniformly for subseries. If \(c_ 0\) is a spreading model of some quotient space of \(E\), then \(\ell^ 1\) is a spreading model of \(E'\); if \(\ell^ 1\) is a spreading model of \(E'\), then \(c_ 0\) is the spreading model of some quotient space of \(E''\). If \(E\) is separable or reflexive, then \(\ell^ 1\) is a spreading model of \(E'\) if and only if \(c_ 0\) is a spreading model of some quotient space of \(E\).
The properties of some sequence spaces \((c_ 0,\ell^ p,\ell^ p\oplus \ell^ q\), Baernstein’s, Schreier’s, Baernstein-Orlicz, Schreier-Orlicz, Tzirelson, James and Tzirelson-James spaces) are studied in detail in the fourth chapter [see, for example, B. Beauzamy, Publ. Dép. Math., Lyon 17, No. 2, 1–56 (1980; Zbl 0526.46022), B. Beauzamy and B. Maurey, Ark. Mat. 17, 193–198 (1979; Zbl 0477.46018), A. Andrew, Math. Scand. 48, 109–118 (1981; Zbl 0439.46010)].
The fifth chapter starts with the following result. There exists a Banach space \(E\), a spreading model \(F_ 1\) of \(E\), a spreading model \(F_ 2\) and \(F_ 1\) such that no spreading model of \(E\) is isomorphic to \(F_ 2\). If \(\ell^ 1\) is a spreading model of \(E\), then it is a spreading model of \(\ell^ p(E)\) \((1<p<\infty)\); but \(\ell^ 1\) is a spreading model of \(L^ 2(c_ 0)\). Note that there exists a Banach space \(B_ 1\) such that it has the (BSP), but \(L^ 2(B_ 1)\) does not, \(c_ 0\) is not a spreading model of \(B'_ 1\), but it is a spreading model of \(L^ 2(B'_ 1).\)
The sixth chapter is an introduction to local \(\ell^ p\) and \(\ell^ p\) extensions of Banach spaces. The last, seventh, chapter presents an exposition of stable Banach spaces. For example, if \(E\) is stable, then it is weakly sequentially complete and possesses the (WBSP), each extension of \(E\) is stable and each spreading model of any extension of E is isometric to a spreading model of \(E\).
The book ends by two appendices on basic sequences and on properties of an Orlicz sequence space. The material is well written, choiced and ordered. The authors do a nice job. The book may be recommended to both researchers and post-graduate students in Banach space theory.

MSC:

46B20 Geometry and structure of normed linear spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis