Szlenk indices and uniform homeomorphisms (Q2723456)

The Szlenk index \(Sz(X)\) of a given Banach space \(X\) is an ordinal number that was defined by \textit{W. Szlenk} [Studia Math. 30, 53-61 (1968; Zbl 0169.15303)]. Its uncountability implies that the conjugate space \(X^{\ast}\) is no separable. NEWLINENEWLINENEWLINEIn the paper is studied those Banach spaces \(X\) for which the Szlenk index \(Sz(X)\) and its modification \(Cz(X)\) , defined in the reviewed paper, are not large then the first infinite ordinal \(\omega \). \(Sz(X)\) and \(Cz(X)\) are defined by using ordinal functions \(Sz(X,\varepsilon)\) and \(Cz(X,\varepsilon)\) where \(\varepsilon >0\) (their definitions are in original Szlenk's paper and in reviewed one) which are both of finite in this case. It is shown that if \(Sz(X)=\omega \) then \(Sz(X,\varepsilon)\) is of power type: \(Sz(X,\varepsilon)\leq C\varepsilon ^{p}\) for suitable constants \(p\) and \(C\) and the same is true for \(Cz(X,\varepsilon)\): if \(\varepsilon ^{p}Sz(X,\varepsilon)\) is bounded (\(\varepsilon \to 0\)) then \(\varepsilon ^{p}Cz(X,\varepsilon)\) is bounded too. NEWLINENEWLINENEWLINEIn general case it is not known whether the functions \(Sz(X,\varepsilon)\) and \(Cz(X,\varepsilon)\) are equivalent? In the paper their equivalence was proved under the additional assumption \(Sz(X)=Sz(X^*)=\omega \). This question is of importance because in the paper was proved (among others) the following theorem. NEWLINENEWLINENEWLINELet \(X\) be a separable Banach space with \(Sz(X)=\omega \). Then there exists an absolute constant \(C<19200\) such that for any \(\tau \in (0,1)\) there is a 2-equivalent norm \(|\cdot |\) on \(X\) satisfy \(|x^*|=1\) , \(|x_{n}^*|=\tau \) and \(\lim_{n\to \infty }x_{n}^*=0\) weak\(^*\) , then \(\liminf \) \(|x^*-x_{n}^*|\geq 1+[Cz(X,\tau /C)]^{-1}\).NEWLINENEWLINENEWLINEThis is in a sense a best possible result in the direction of so called UKK\(^*\) renorming of a Banach space \(X\) (UKK\(^*\) is the dual version of uniform Kadec-Klee renorming; for definitions see \textit{S. J. Dilworth, M. Girardi} and \textit{D. Kutzarova} [Studia Math. 112, 266-297 (1995; Zbl 0824.46024)]. NEWLINENEWLINENEWLINEThe part of the paper is devoted to applications of Szlenk indices to uniform homeomorphisms. E.g., it was shown that the class of Banach spaces with \(Sz(X)=\omega \) is stable under uniform homeomorphisms. For the class of all Banach spaces with a separable dual this is not true as Ribe showed it in 1984. Besides above mentioned results in this excellent work are presented many other ones on properties and applications of Szlenk indices.