1. CMB 2009 (vol 53 pp. 278)
|Cantor-Bernstein Sextuples for Banach Spaces|
Let $X$ and $Y$ be Banach spaces isomorphic to complemented subspaces of each other with supplements $A$ and $B$. In 1996, W. T. Gowers solved the Schroeder--Bernstein (or Cantor--Bernstein) problem for Banach spaces by showing that $X$ is not necessarily isomorphic to $Y$. In this paper, we obtain a necessary and sufficient condition on the sextuples $(p, q, r, s, u, v)$ in $\mathbb N$ with $p+q \geq 1$, $r+s \geq 1$ and $u, v \in \mathbb N^*$, to provide that $X$ is isomorphic to $Y$, whenever these spaces satisfy the following decomposition scheme $$ A^u \sim X^p \oplus Y^q, \quad B^v \sim X^r \oplus Y^s. $$ Namely, $\Phi=(p-u)(s-v)-(q+u)(r+v)$ is different from zero and $\Phi$ divides $p+q$ and $r+s$. These sextuples are called Cantor--Bernstein sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$ indicates the well-known PeÅczyÅski's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder--Bernstein problem become evident.
Keywords:Pel czyÅski's decomposition method, Schroeder-Bernstein problem
2. CMB 2007 (vol 50 pp. 610)
|On Weak$^*$ Kadec--Klee Norms |
We present partial positive results supporting a conjecture that admitting an equivalent Lipschitz (or uniformly) weak$^*$ Kadec--Klee norm is a three space property.
Keywords:weak$^*$ Kadec--Klee norms, three-space problem
3. CMB 2007 (vol 50 pp. 619)
|On the Existence of Asymptotic-$l_p$ Structures in Banach Spaces |
It is shown that if a Banach space is saturated with infinite dimensional subspaces in which all ``special" $n$-tuples of vectors are equivalent with constants independent of $n$-tuples and of $n$, then the space contains asymptotic-$l_p$ subspaces for some $1 \leq p \leq \infty$. This extends a result by Figiel, Frankiewicz, Komorowski and Ryll-Nardzewski.
Categories:46B20, 46B40, 46B03
4. CMB 2005 (vol 48 pp. 69)
|Biorthogonal Systems in Weakly LindelÃ¶f Spaces |
We study countable splitting of Markushevich bases in weakly Lindel\"of Banach spaces in connection with the geometry of these spaces.
Keywords:Weak compactness, projectional resolutions,, Markushevich bases, Eberlein compacts, Va\v sÃ¡k spaces
Categories:46B03, 46B20., 46B26
5. CMB 1998 (vol 41 pp. 225)
|Mazur intersection properties for compact and weakly compact convex sets |
Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively.
Categories:46B03, 46B20, 46A55