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Many characterizations of fragmentability of topological spaces have been investigated. In this paper we deal with some properties of weak-fragmentability of Banach spaces.
Recently, H’michane et al. [‘On the class of limited operators’, Acta Math. Sci. (submitted)] introduced the class of weak$^*$ Dunford–Pettis operators on Banach spaces, that is, operators which send weakly compact sets onto limited sets. In this paper, the domination problem for weak$^*$ Dunford–Pettis operators is considered. Let $S, T:E\to F$ be two positive operators between Banach lattices $E$ and $F$ such that $0\leq S\leq T$. We show that if $T$ is a weak$^{*}$ Dunford–Pettis operator and $F$ is $\sigma $-Dedekind complete, then $S$ itself is weak$^*$ Dunford–Pettis.
We study the metric entropy of the metric space ${\mathcal{B}}_{n}$ of all $n$-dimensional Banach spaces (the so-called Banach–Mazur compactum) equipped with the Banach–Mazur (multiplicative) “distance” $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({\mathcal{B}}_{n},d,1+{\it\varepsilon})$ is the smallest number of “balls” of “radius” $1+{\it\varepsilon}$ that cover ${\mathcal{B}}_{n}$, then for any ${\it\varepsilon}>0$ we have
We also prove an analogous result for the metric entropy of the set of $n$-dimensional operator spaces equipped with the distance $d_{N}$ naturally associated with $N\times N$ matrices with operator entries. In that case $N$ is arbitrary but our estimates are valid independently of $N$. In the Banach space case (i.e. $N=1$) the above upper bound is part of the folklore, and the lower bound is at least partially known (but apparently has not appeared in print). While we follow the same approach in both cases, the matricial case requires more delicate ingredients, namely estimates (from our previous work) on certain $n$-tuples of $N\times N$ unitary matrices known as “quantum expanders”.
We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$, $2\le p<\infty $.
In this paper, it is proved that every isometry between the unit spheres of two real Banach spaces preserves the frames of the unit balls. As a consequence, if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ and $Y$ are $n$-dimensional Banach spaces and $T_0$ is an isometry from the unit sphere of $X$ onto that of $Y$ then it maps the set of all $(n-1)$-extreme points of the unit ball of $X$ onto that of $Y$.
Let $X$ be an infinite-dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exist a constant $\lambda \gt 0$ and an infinite sequence $\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$ such that $\Vert {x}_{i} - {x}_{j} \Vert = \lambda $ for all $i\not = j$.
We study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number $n\geq 3$ a set of $n$ closed subspaces of ${\ell }^{2} $ which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces.
We prove a normalized version of the restricted invertibility principle obtained by Spielman and Srivastava in [An elementary proof of the restricted invertibility theorem. Israel J. Math.190 (2012), 83–91]. Applying this result, we get a new proof of the proportional Dvoretzky–Rogers factorization theorem recovering the best current estimate in the symmetric setting while we improve the best known result in the non-symmetric case. As a consequence, we slightly improve the estimate for the Banach–Mazur distance to the cube: the distance of every $n$-dimensional normed space from ${ \ell }_{\infty }^{n} $ is at most $\mathop{(2n)}\nolimits ^{5/ 6} $. Finally, using tools from the work of Batson et al in [Twice-Ramanujan sparsifiers. In STOC’09 – Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255–262], we give a new proof for a theorem of Kashin and Tzafriri [Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993] on the norm of restricted matrices.
Every bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\in (2, \infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\infty } $ to bounded mean oscillation ($\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\sharp $-maximal operator.
We show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.
For an arbitrary subset $X$ of a finite-dimensional real Banach space $E$, the ball intersection with parameter $\lambda \gt 0$ is defined as the intersection of all balls of radius $\lambda $ whose centers are in $X$. On the other hand, the intersection of all balls of radius $\lambda $ that contain $X$ is said to be the respective ball hull. We present new results on these two notions and use them to get new insights into complete sets and (pairs of) sets of constant width, e.g., their representation as vector sums of suitable ball intersections and ball hulls. Also in this framework, we give partial answers to the known question, in what finite-dimensional real Banach spaces any complete set is of constant width. For polyhedral norms we obtain characterizations of monotypic balls via constant width properties of pairs formed by the ball intersection and ball hull of the same bounded and non-empty set. Finally, we present some new results on Borsuk numbers of sets of constant width in normed spaces, closely related to (unique) completions of compact sets. For example, the lower estimate on Borsuk numbers of bodies of constant width due to Lenz is extended to arbitrary normed spaces. Furthermore, we also derive the Borsuk number of the normed space with maximum norm.
We prove that any surjective isometry between unit spheres of the ${\ell }^{\infty } $-sum of strictly convex normed spaces can be extended to a linear isometry on the whole space, and we solve the isometric extension problem affirmatively in this case.
Every Banach space with separable second dual can be equivalently renormed to have weakly uniformly rotund dual. Under certain embedding conditions a Banach space with weakly uniformly rotund dual is reflexive.
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New and straightforward proofs of these theorems are given. A number of necessary and sufficient conditions for the second representation theorem to hold are obtained. A new simple and explicit example of a self-adjoint operator for which the second representation theorem fails to hold is also provided.
In this paper we give a lower bound on the waist of the unit sphere of a uniformly convex normed space by using the localization technique in codimension greater than one and a strong version of the Borsuk–Ulam theorem. The tools used in this paper follow ideas of Gromov in [Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178–215] and we also include an independent proof of our main theorem which does not rely on Gromov’s waist of the sphere. Our waist inequality in codimension one recovers a version of the Gromov–Milman inequality in [Generalisation of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263–282].
The self-affine measure μM, D corresponding to M = diag[p1, p2, p3] (pj ∈ ℤ \ {0, ± 1}, j = 1, 2, 3) and D = {0, e1, e2, e3} in the space ℝ3 is supported on the three-dimensional Sierpinski gasket T(M, D), where e1, e2, e3 are the standard basis of unit column vectors in ℝ3. We shall determine the spectrality and non-spectrality of μM, D, and show that if pj ∈ 2ℤ \ {0, 2} for j = 1, 2, 3, then μM, D is a spectral measure, and if pj ∈ (2ℤ + 1) \ {±1} for j = 1, 2, 3, then μM, D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM, D), where the number 4 is the best possible. This generalizes the known results on the spectrality of self-affine measures.
We introduce two measures of weak non-compactness JaE and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x* ∈ E*, how far from E or C one needs to go to find x** ∈ ⊂ E** with x**(x*) = sup x*(C). A quantitative version of James's Compactness Theorem is proved using JaE and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0. If there is an elementinwhose distance to C is greater than r, then there is x* ∈ E* such that each x** ∈at which sup x*(C) is attained has distance to E greater than ½r. We indeed establish that JaE and Ja are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp.
In this work, we study and investigate the ultrapowers of ℓ1-Munn algebras. Then we show that the class of ℓ1-Munn algebras is stable under ultrapowers. Finally, applying this result to semigroup algebras, we show that for a semigroup S, ultra-amenability of ℓ1(S) and amenability of the second dual ℓ1(S)′′ are equivalent.
We present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’.
In this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.