location:  Publications → journals
Search results

Search: All articles in the CMB digital archive with keyword Banach space

 Expand all        Collapse all Results 1 - 10 of 10

1. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
 Connections Between Metric Characterizations of Superreflexivity and the Radon-NikodÃ½ 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, superreflexivityCategories:46B85, 46B07, 46B22

2. CMB 2011 (vol 55 pp. 821)

Perez-Garcia, C.; Schikhof, W. H.
 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 basesCategories:46S10, 12J25

3. CMB 2011 (vol 55 pp. 410)

Service, Robert
 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 theoryCategories:46B15, 05D10

4. CMB 2011 (vol 54 pp. 726)

Ostrovskii, M. I.
 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 enlargementCategories:46B07, 52A21, 46B15

5. CMB 2008 (vol 51 pp. 604)

{\'S}liwa, Wies{\l}aw
 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 spacesCategories:47S10, 46S10, 47A15

6. CMB 2008 (vol 51 pp. 372)

Ezquerro, J. A.; Hernández, M. A.
 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 equationsCategories: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 ratioCategories: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, spaceCategories:47A15, 47D03, 20M20
 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 spaceCategories:54C20, 46B10
 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 operatorCategories:47B38, 30D55, 46E15