July 2018 - Banach - mathdept.okstate.edu

Informal Analysis Seminar February 24-25, 2018
by Artem Zvavitch 05 Sep '19

05 Sep '19

Dear Colleagues, The Analysis group at Kent State University is happy to announce a meeting of the Informal Analysis Seminar, which will be held at the Department of Mathematical Sciences at Kent State University, February 24-25. The seminar will feature plenary speakers Robert Connelly (Cornell University), and Peter Sternberg (Indiana University) Each speaker will deliver a four hour lecture series designed to be accessible for graduate students. Funding is available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Posters can be submitted electronically in PDF format. Further information, and an online registration form, can be found online http://www.math.kent.edu/informal We encourage you to register as soon as possible, but to receive support and/or help with hotel reservation, please register before January 29, 2018. Finally, please feel free to forward this email to any colleagues or students who you think may be interested in attending. Best regards, The Kent State Analysis Group

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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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Abstract of a paper by Antonio Avilés, Pedro Tradacete, Ignacio Villanueva
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “The free Banach lattices generated by $\ell_p$ and $c_0$” by Antonio Avilés<https://arxiv.org/search/math?searchtype=author&query=Avil%C3%A9s%2C+A>, Pedro Tradacete<https://arxiv.org/search/math?searchtype=author&query=Tradacete%2C+P>, Ignacio Villanueva<https://arxiv.org/search/math?searchtype=author&query=Villanueva%2C+I>. Abstract: We prove that, when $2<p<\infty$, in the free Banach lattice generated by $\ell_p$ (respectively by $c_0$), the absolute values of the canonical basis form an $\ell_r$-sequence, where $\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$ (respectively an $\ell_2$-sequence). In particular, in any Banach lattice, the absolute values of any $\ell_p$ sequence always have an upper $\ell_r$-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable $\ell_p(\Gamma)$ for $2<p<\infty$, as well as $c_0(\Gamma)$, are weakly compactly generated whereas this is not the case for $1\leq p\leq 2$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.02553

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Abstract of a paper by Ben Wallis
by Bentuo Zheng (bzheng) 31 Jul '18

31 Jul '18

This is an announcement for the paper “Closed ideals of operators acting on some families of sequence spaces” by Ben Wallis<https://arxiv.org/search/math?searchtype=author&query=Wallis%2C+B>. Abstract: We study the lattice of closed ideals in the algebra of continuous linear operators acting on $p$th Tandori and $p'$th Ces\`{a}ro sequence spaces, $1\leqslant p<\infty$, which we show are isomorphic to the classical sequence spaces $(\oplus_{n=1}^\infty\ell_\infty^n)_p$ and $(\oplus_{n=1}^\infty\ell_1^n)_{p'}$, respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1806.00382

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