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1. CMB Online first
Connections between metric characterizations of superreflexivity and the Radon-NikodÃ½m property for dual Banach spaces |
Connections between metric characterizations of superreflexivity and the Radon-NikodÃ½m property for dual Banach spaces Johnson and Schechtman (2009)
characterized superreflexivity in terms of finite diamond graphs.
The present author characterized the Radon-NikodÃ½m property
(RNP) for dual spaces in terms of the infinite diamond. This
paper
is devoted to further study of relations between metric
characterizations of superreflexivity and the RNP for dual spaces.
The main result is that finite subsets of any set $M$ whose
embeddability characterizes the RNP for dual spaces, characterize
superreflexivity. It is also observed that the converse statement
does not hold, and that $M=\ell_2$ is a counterexample.
Keywords:Banach space, diamond graph, finite representability, metric characterization, Radon-NikodÃ½m property, superreflexivity Categories:46B85, 46B07, 46B22 |
2. CMB 2011 (vol 55 pp. 821)
New Examples of Non-Archimedean Banach Spaces and Applications The study carried out in this paper about some new examples of
Banach spaces, consisting of certain valued fields extensions, is
a typical non-archimedean feature. We determine whether these
extensions are of countable type, have $t$-orthogonal bases, or are
reflexive.
As an application we construct, for a class of base fields, a norm
$\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm,
without non-zero vectors that are $\|\cdot\|$-orthogonal and such
that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$
into a valued field.
Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases Categories:46S10, 12J25 |
3. CMB 2011 (vol 55 pp. 410)
A Ramsey Theorem with an Application to Sequences in Banach Spaces The notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using
Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional
basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of
Galvin's theorem is used in the proof. An alternative proof
of the dichotomy result for sequences in Banach spaces is
also sketched,
using the Galvin-Prikry theorem.
Keywords:Banach spaces, Ramsey theory Categories:46B15, 05D10 |
4. CMB 2011 (vol 54 pp. 726)
Auerbach Bases and Minimal Volume Sufficient Enlargements Let $B_Y$ denote the unit ball of a
normed linear space $Y$. A symmetric, bounded, closed, convex set
$A$ in a finite dimensional normed linear space $X$ is called a
sufficient enlargement for $X$ if, for an arbitrary
isometric embedding of $X$ into a Banach space $Y$, there exists a
linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each
finite dimensional normed space has a minimal-volume sufficient
enlargement that is a parallelepiped; some spaces have ``exotic''
minimal-volume sufficient enlargements. The main result of the
paper is a characterization of spaces having ``exotic''
minimal-volume sufficient enlargements in terms of Auerbach
bases.
Keywords:Banach space, Auerbach basis, sufficient enlargement Categories:46B07, 52A21, 46B15 |
5. CMB 2008 (vol 51 pp. 604)
The Invariant Subspace Problem for Non-Archimedean Banach Spaces It is proved that every infinite-dimensional
non-archimedean Banach space of countable type admits a linear
continuous operator without a non-trivial closed invariant
subspace. This solves a problem stated by A.~C.~M. van Rooij and
W.~H. Schikhof in 1992.
Keywords:invariant subspaces, non-archimedean Banach spaces Categories:47S10, 46S10, 47A15 |
6. CMB 2008 (vol 51 pp. 372)
Picard's Iterations for Integral Equations of Mixed Hammerstein Type A new semilocal convergence result for the Picard method is presented,
where the main required condition in the contraction mapping principle is relaxed.
Keywords:nonlinear equations in Banach spaces, successive approximations, semilocal convergence theorem, Picard's iteration, Hammerstein integral equations Categories:45G10, 47H99, 65J15 |
7. CMB 2006 (vol 49 pp. 185)
On the Inequality for Volume and Minkowskian Thickness Given a centrally symmetric convex body $B$ in $\E^d,$ we denote
by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional
Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex
body in $\M^d(B).$ The relationship between volume $V(K)$ and the
Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can
naturally be given by the sharp geometric inequality $V(K) \ge
\alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple
corollary of the Rogers--Shephard inequality we obtain that
$\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality
on the left attained if and only if $B$ is the difference body of
a simplex and on the right if $B$ is a cross-polytope. The main
result of this paper is that for $d=2$ the equality on the right
implies that $B$ is a parallelogram. The obtained results yield
the sharp upper bound for the modified Banach--Mazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio Categories:52A40, 46B20 |
8. CMB 2004 (vol 47 pp. 298)
Near Triangularizability Implies Triangularizability In this paper we consider collections of
compact operators on a real or
complex Banach space including linear operators
on finite-dimensional vector spaces. We show
that such a collection is simultaneously
triangularizable if and only if it is arbitrarily
close to a simultaneously triangularizable
collection of compact operators. As an application
of these results we obtain an invariant subspace
theorem for certain bounded operators. We
further prove that in finite dimensions near
reducibility implies reducibility whenever
the ground field is $\BR$ or $\BC$.
Keywords:Linear transformation, Compact operator,, Triangularizability, Banach space, Hilbert, space Categories:47A15, 47D03, 20M20 |
9. CMB 2000 (vol 43 pp. 208)
Extensions of Continuous and Lipschitz Functions We show a result slightly more general than the following. Let $K$
be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a
lower semi-continuous metric on $K$. Then each continuous function
$f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on
$K$ which is Lipschitz in $d$. The extension has the same supremum
norm and the same Lipschitz constant.
As a corollary we get that a Banach space $X$ is reflexive if and only
if each bounded, weakly continuous and norm Lipschitz function
defined on a weakly closed subset of $X$ admits a weakly continuous,
norm Lipschitz extension defined on the entire space $X$.
Keywords:extension, continous, Lipschitz, Banach space Categories:54C20, 46B10 |
10. CMB 1999 (vol 42 pp. 139)
Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions |
Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions Every weakly compact composition operator between weighted Banach
spaces $H_v^{\infty}$ of analytic functions with weighted sup-norms is
compact. Lower and upper estimates of the essential norm of
continuous composition operators are obtained. The norms of the point
evaluation functionals on the Banach space $H_v^{\infty}$ are also
estimated, thus permitting to get new characterizations of compact
composition operators between these spaces.
Keywords:weighted Banach spaces of holomorphic functions, composition operator, compact operator, weakly compact operator Categories:47B38, 30D55, 46E15 |