Expand all Collapse all | Results 1 - 5 of 5 |
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 Categories:46B03, 46B20 |
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 Categories:46B03, 46B2 |
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 |