Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Bernardo Cascales, Rafa Chiclana, Luis García-Lirola, Miguel Martín, Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “On strongly norm attaining Lipschitz operators” by Bernardo Cascales<https://arxiv.org/search/math?searchtype=author&query=Cascales%2C+B>, Rafa Chiclana<https://arxiv.org/search/math?searchtype=author&query=Chiclana%2C+R>, Luis García-Lirola<https://arxiv.org/search/math?searchtype=author&query=Garc%C3%ADa-Lirola%2C…>, Miguel Martín<https://arxiv.org/search/math?searchtype=author&query=Mart%C3%ADn%2C+M>, Abraham Rueda Zoca<https://arxiv.org/search/math?searchtype=author&query=Zoca%2C+A+R>. Abstract: We study the set $\SA(M,Y)$ of those Lipschitz operators from a (complete pointed) metric space $M$ to a Banach space $Y$ which (strongly) attain their Lipschitz norm (i.e.\ the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when $M$ is length (or local) or when $M$ is a closed subset of $\R$ with positive Lebesgue measure, providing new example which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space $\mathcal{F}(M)$ over $M$, and show that all of them actually provide the norm density of $\SA(M,Y)$ in the space of all Lipschitz operators from $M$ to any Banach space $Y$. Next, we prove that $\SA(M,\R)$ is weak sequentially dense in the space of all Lipschitz functions for all metric spaces $M$. Finally, we show that the norm of the bidual space to $\mathcal{F}(M)$ is octahedral provided the metric space $M$ is discrete but not uniformly discrete or $M'$ is infinite. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.03363

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Abstract of a paper by Christopher Boyd, Raymond A. Ryan, Nina Snigireva
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Geometry of Spaces of Orthogonally Additive Polynomials on C(K)” by Christopher Boyd<https://arxiv.org/search/math?searchtype=author&query=Boyd%2C+C>, Raymond A. Ryan<https://arxiv.org/search/math?searchtype=author&query=Ryan%2C+R+A>, Nina Snigireva<https://arxiv.org/search/math?searchtype=author&query=Snigireva%2C+N>. Abstract: We study the space of orthogonally additive $n$-homogeneous polynomials on $C(K)$. There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive $n$-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach-Stone theorem. We conclude with a classification of the exposed points. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.02713

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Abstract of a paper by Markus Passenbrunner
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Spline Characterizations of the Radon-Nikodým property” by Markus Passenbrunner<https://arxiv.org/search/math?searchtype=author&query=Passenbrunner%2C+M>. Abstract: We give necessary and sufficient conditions for a Banach space $X$ having the Radon-Nikod\'{y}m property in terms of polynomial spline sequences. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.01861

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Abstract of a paper by L. V. Genze, S. P. Gul'ko, T. E. Khmyleva
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “Classification of spaces of Continuous Function on ordinals” by L. V. Genze<https://arxiv.org/search/math?searchtype=author&query=Genze%2C+L+V>, S. P. Gul'ko<https://arxiv.org/search/math?searchtype=author&query=Gul%27ko%2C+S+P>, T. E. Khmyleva<https://arxiv.org/search/math?searchtype=author&query=Khmyleva%2C+T+E>. Abstract: We conclude the classification of spaces of continuous functions on ordinals carried out by R. Gorak. This gives a complete topological classification of the spaces $C_p([0,\alpha])$ of all continuous real-valued functions on compact segments of ordinals endowed with the topology of pointwise convergence. Moreover, this topological classification of the spaces $C_p([0,\alpha])$ completely coincides with their uniform classification. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.09015

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Abstract of a paper by Alexandros Eskenazis
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “On extremal sections of subspaces of $L_p$” by Alexandros Eskenazis<https://arxiv.org/search/math?searchtype=author&query=Eskenazis%2C+A>. Abstract: Let $m,n\in\mathbb{N}$ and $p\in(0,\infty)$. For a finite dimensional quasi-normed space $X=(\mathbb{R}^m, \|\cdot\|_X)$, let $$B_p^n(X) = \Big\{ (x_1,\ldots,x_n)\in\big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n \|x_i\|_X^p \leq 1\Big\}.$$ We show that for every $p\in(0,2)$ and $X$ which admits an isometric embedding into $L_p$, the function $$S^{n-1} \ni \theta = (\theta_1,\ldots,\theta_n) \longmapsto \Big| B_p^n(X) \cap\Big\{(x_1,\ldots,x_n)\in \big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n \theta_i x_i=0 \Big\} \Big|$$ is a Schur convex function of $(\theta_1^2,\ldots,\theta_n^2)$, where $|\cdot|$ denotes the Lebesgue measure. In particular, it is minimized when $\theta=\big(\frac{1}{\sqrt{n}},\ldots,\frac{1}{\sqrt{n}}\big)$ and maximized when $\theta=(1,0,\ldots,0)$. This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body $\big(B_p^n(X)\big)^\circ$ if the unit ball $B_X$ of $X$ is in Lewis' position. Finally, we prove a lower bound for the volume of projections of $B_\infty^n(X)$, where $X=(\mathbb{R}^m,\|\cdot\|_X)$ is an arbitrary quasi-normed space. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.04333

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Abstract of a paper by Florent Baudier, Gilles Lancien, Pavlos Motakis, Thomas Schlumprecht
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “A new coarsely rigid class of Banach spaces” by Florent Baudier<https://arxiv.org/search/math?searchtype=author&query=Baudier%2C+F>, Gilles Lancien<https://arxiv.org/search/math?searchtype=author&query=Lancien%2C+G>, Pavlos Motakis<https://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P>, Thomas Schlumprecht<https://arxiv.org/search/math?searchtype=author&query=Schlumprecht%2C+T>. Abstract: We prove that the class of reflexive asymptotic-$c_0$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_0$ space $Y$, then $X$ is also reflexive and asymptotic-$c_0$. In order to achieve this result we provide a purely metric characterization of this class of Banach spaces which is rigid under coarse embeddings. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.00702

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Abstract of a paper by Trond Arnold Abrahamsen, Julio Becerra Guerrero, Rainis Haller, Vegard Lima, Märt Põldvere
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open” by Trond Arnold Abrahamsen<https://arxiv.org/search/math?searchtype=author&query=Abrahamsen%2C+T+A>, Julio Becerra Guerrero<https://arxiv.org/search/math?searchtype=author&query=Guerrero%2C+J+B>, Rainis Haller<https://arxiv.org/search/math?searchtype=author&query=Haller%2C+R>, Vegard Lima<https://arxiv.org/search/math?searchtype=author&query=Lima%2C+V>, Märt Põldvere<https://arxiv.org/search/math?searchtype=author&query=P%C3%B5ldvere%2C+M>. Abstract: We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has property CWO. Several Banach spaces are proved to possess this geometric property; among others: 2-dimensional real spaces, finite dimensional strictly convex spaces, finite dimensional polyhedral spaces, and the complex space $\ell_1^n$. In contrast to this, we provide an example of a $3$-dimensional real Banach space $X$ for which $C_0(K,X)$ fails to have property CWO. We also show that $c_0$-sums of finite dimensional Banach spaces with property (co) have property CWO. In particular, this provides examples of such spaces outside the class of $C_0(K,X)$-spaces. https://arxiv.org/abs/1806.10693

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Abstract of a paper by José Rodríguez
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “On integration in Banach spaces and total sets” by José Rodríguez<https://arxiv.org/search/math?searchtype=author&query=Rodr%C3%ADguez%2C+J>. Abstract: Let $X$ be a Banach space and $\Gamma \subseteq X^*$ a total linear subspace. We study the concept of $\Gamma$-integrability for $X$-valued functions $f$ defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions $\langle x^*,f \rangle$ for $x^*\in \Gamma$. We show that $\Gamma$-integrability and Pettis integrability are equivalent whenever $X$ has Plichko's property ($\mathcal{D}'$) (meaning that every $w^*$-sequentially closed subspace of $X^*$ is $w^*$-closed). This property is enjoyed by many Banach spaces including all spaces with $w^*$-angelic dual as well as all spaces which are $w^*$-sequentially dense in their bidual. A particular case of special interest arises when considering $\Gamma=T^*(Y^*)$ for some injective operator $T:X \to Y$. Within this framework, we show that if $T:X \to Y$ is a semi-embedding, $X$ has property ($\mathcal{D}'$) and $Y$ has the Radon-Nikod\'{y}m property, then $X$ has the weak Radon-Nikod\'{y}m property. This extends earlier results by Delbaen (for separable $X$) and Diestel and Uhl (for weakly $\mathcal{K}$-analytic $X$). The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.10049

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Abstract of a paper by Yun Sung Choi, Sheldon Dantas, Mingu Jung, Miguel Martín
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “The Bishop-Phelps-Bollobás property and absolute sums” by Yun Sung Choi<https://arxiv.org/search/math?searchtype=author&query=Choi%2C+Y+S>, Sheldon Dantas<https://arxiv.org/search/math?searchtype=author&query=Dantas%2C+S>, Mingu Jung<https://arxiv.org/search/math?searchtype=author&query=Jung%2C+M>, Miguel Martín<https://arxiv.org/search/math?searchtype=author&query=Mart%C3%ADn%2C+M>. Abstract: In this paper we study conditions assuring that the Bishop-Phelps-Bollob\'as property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X, Y) of Banach spaces having the BPBp, (a) if Y1 is an absolute summand of Y, then (X, Y1) has the BPBp; (b) if X1 is an absolute summand of X of type 1 or \infty, then (X1, Y) has the BPBp. Besides, analogous results for the BPBp for compact operators and for the density of norm attaining operators are also given. We also show that the Bishop-Phelps-Bollob\'as property for numerical radius is inherited by absolute summands of type 1 or \infty. Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.09366

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Abstract of a paper by Vinicius Coelho, Joilson Ribeiro, Luciana Salgado
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “On Schauder basis in normed spaces” by Vinicius Coelho<https://arxiv.org/search/math?searchtype=author&query=Coelho%2C+V>, Joilson Ribeiro<https://arxiv.org/search/math?searchtype=author&query=Ribeiro%2C+J>, Luciana Salgado<https://arxiv.org/search/math?searchtype=author&query=Salgado%2C+L>. Abstract: In this work, we prove the criterion of Banach-Grunblum and the principle of selection of Bessaga-Pe\l{}czy\'nski for normed spaces. Given these results, it can be shown that in infinite dimension spaces (not necessarily complete) there is an infinite-dimension subspaces with an essential Schauder basis. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.07943

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