Banach September 2018

banach@mathdept.okstate.edu

2 participants
23 discussions

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 Richard J. Gardner, Daniel Hug, Sudan Xing, Deping Ye
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem II” by Richard J. Gardner<https://arxiv.org/search/math?searchtype=author&query=Gardner%2C+R+J>, Daniel Hug<https://arxiv.org/search/math?searchtype=author&query=Hug%2C+D>, Sudan Xing<https://arxiv.org/search/math?searchtype=author&query=Xing%2C+S>, Deping Ye<https://arxiv.org/search/math?searchtype=author&query=Ye%2C+D>. Abstract: The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $\psi$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $\mu$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $\mu$ equals $\deV(K, \cdot)$, up to a constant. A major step in the analysis concerns discrete measures $\mu$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures $\mu$, we use an approximation argument. This approach is also applied to the case where $G$ is decreasing and $\psi$ is increasing, and hence augments our previous work. When the measures $\mu$ are even, solutions that are origin-symmetric convex bodies are also provided under some mild conditions on $G$ and $\psi$. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $\mu$ is discrete or even. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.09753

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Abstract of a paper by Leandro Candido, Marek Cúth, Michal Doucha
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Isomorphisms between spaces of Lipschitz functions” by Leandro Candido<https://arxiv.org/search/math?searchtype=author&query=Candido%2C+L>, Marek Cúth<https://arxiv.org/search/math?searchtype=author&query=C%C3%BAth%2C+M>, Michal Doucha<https://arxiv.org/search/math?searchtype=author&query=Doucha%2C+M>. Abstract: We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that $\operatorname{Lip}_0(\mathbb{Z}^d)\simeq\operatorname{Lip}_0(\mathbb{R}^d)$, for all $d\in\mathbb{N}$. More generally, we e.g. show that $\operatorname{Lip}_0(\Gamma)\simeq \operatorname{Lip}_0(G)$, where $\Gamma$ is from a large class of finitely generated nilpotent groups and $G$ is its Mal'cev closure; or that $\operatorname{Lip}_0(\ell_p)\simeq\operatorname{Lip}_0(L_p)$, for all $1\leq p<\infty$. We leave a large area for further possible research. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.09957

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Abstract of a paper by Richard Lechner, Pavlos Motakis, Paul F. X. Müller, Thomas Schlumprecht
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Strategically reproducible bases and the factorization property” by Richard Lechner<https://arxiv.org/search/math?searchtype=author&query=Lechner%2C+R>, Pavlos Motakis<https://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P>, Paul F. X. Müller<https://arxiv.org/search/math?searchtype=author&query=M%C3%BCller%2C+P+F+X>, Thomas Schlumprecht<https://arxiv.org/search/math?searchtype=author&query=Schlumprecht%2C+T>. Abstract: We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space $L^1$. Moreover, we show the strategical reproducibility is inherited by unconditional sums. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.09423

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Abstract of a paper by Saak Gabriyelyan, Jerzy Kcakol
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Dunford--Pettis type properties and the Grothendieck property for function spaces” by Saak Gabriyelyan<https://arxiv.org/search/math?searchtype=author&query=Gabriyelyan%2C+S>, Jerzy Kcakol<https://arxiv.org/search/math?searchtype=author&query=Kcakol%2C+J>. Abstract: For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. If $X$ is compact, the classic result of A.~Grothendieck states that $C_k(X)$ has the Dunford-Pettis property and the sequential Dunford--Pettis property. We extend Grothendieck's result by showing that $C_k(X)$ has both the Dunford-Pettis property and the sequential Dunford-Pettis property if $X$ satisfies one of the following conditions: (i) $X$ is a hemicompact space, (ii) $X$ is a cosmic space (=a continuous image of a separable metrizable space), (iii) $X$ is the ordinal space $[0,\kappa)$ for some ordinal $\kappa$, or (vi) $X$ is a locally compact paracompact space. We show that if $X$ is a cosmic space, then $C_k(X)$ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite. We prove that $C_p(X)$ has the Dunford--Pettis property and the sequential Dunford-Pettis property for every Tychonoff space $X$, and $C_p(X) $ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.08982

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Abstract of a paper by Jose L. Ansorena
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Primarity of direct sums of Orlicz spaces and Marcinkiewicz spaces” by Jose L. Ansorena<https://arxiv.org/search/math?searchtype=author&query=Ansorena%2C+J+L>. Abstract: Let $\mathbb{Y}$ be either an Orlicz sequence space or a Marcinkiewicz sequence space. We take advantage of the recent advances in the theory of factorization of the identity carried on in [R. Lechner, Subsymmetric weak* Schauder bases and factorization of the identity, arXiv:1804.01372<https://arxiv.org/abs/1804.01372> [math.FA]] to provide conditions on $\mathbb{Y}$ that ensure that, for any $1\le p\le\infty$, the infinite direct sum of $\mathbb{Y}$ in the sense of $\ell_p$ is a primary Banach space, enlarging this way the list of Banach spaces that are known to be primary. https://arxiv.org/abs/1809.05749

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Abstract of a paper by Weight-partially greedy bases and weight-Property $(A)$
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Weight-partially greedy bases and weight-Property $(A)$” by Divya Khurana<https://arxiv.org/search/math?searchtype=author&query=Khurana%2C+D>. Abstract: In this paper, motivated by the notion of $w$-Property $(A)$ defined in [2], we introduce the notions of $w$-left Property $(A)$ and $w$-right Property $(A)$. We also introduce the notions of $w$-partially greedy basis (using a characterization of partially greedy basis from [4]) and $w$-reverse partially greedy basis. The main aim of this paper is to study $(i)$ some characterizations of $w$-partially greedy and $w$-reverse partially greedy basis $(ii)$ conditions on the weight sequences when $w$-left Property $(A)$ and (or) $w$-right Property $(A)$ implies $w$-Property $(A)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.04890

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Abstract of a paper by Debmalya Sain, Kallol Paul, Pintu Bhunia, Santanu Bag
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “On the numerical index of polyhedral Banach spaces” by Debmalya Sain<https://arxiv.org/search/math?searchtype=author&query=Sain%2C+D>, Kallol Paul<https://arxiv.org/search/math?searchtype=author&query=Paul%2C+K>, Pintu Bhunia<https://arxiv.org/search/math?searchtype=author&query=Bhunia%2C+P>, Santanu Bag<https://arxiv.org/search/math?searchtype=author&query=Bag%2C+S>. Abstract: We present a general method to estimate the numerical index of any finite-dimensional real polyhedral Banach space, by considering the action of only finitely many functionals on the unit sphere of the space. As an application of our study, we explicitly compute the exact numerical index of the family of $3$-dimensional polyhedral Banach spaces whose unit balls are prisms with regular polygons as its base. Our results generalize some of the earlier results regarding the computation of the exact numerical index of certain $2$-dimensional polyhedral Banach spaces having regular polygons as the unit balls. We further estimate the numerical index of two particular families of $3$-dimensional polyhedral Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.04778

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Abstract of a paper by Fernando Albiac, Jose L. Ansorena, Stephen J. Dilworth, Denka Kutzarova
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases” by Fernando Albiac<https://arxiv.org/search/math?searchtype=author&query=Albiac%2C+F>, Jose L. Ansorena<https://arxiv.org/search/math?searchtype=author&query=Ansorena%2C+J+L>, Stephen J. Dilworth<https://arxiv.org/search/math?searchtype=author&query=Dilworth%2C+S+J>, Denka Kutzarova<https://arxiv.org/search/math?searchtype=author&query=Kutzarova%2C+D>. Abstract: In this paper we settle in the negative the problem of the superreflexivity of Garling sequence spaces by showing that they contain a complemented subspace isomorphic to a non superreflexive mixed-norm sequence space. As a by-product of our work, we give applications of this result to the study of conditional Schauder bases and conditional almost greedy bases in this new class of Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.03013

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Abstract of a paper by Niushan Gao, Denny H. Leung, Foivos Xanthos
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “A Local Hahn-Banach Theorem and Its Applications” by Niushan Gao<https://arxiv.org/search/math?searchtype=author&query=Gao%2C+N>, Denny H. Leung<https://arxiv.org/search/math?searchtype=author&query=Leung%2C+D+H>, Foivos Xanthos<https://arxiv.org/search/math?searchtype=author&query=Xanthos%2C+F>. Abstract: An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local' version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8]. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1809.01795

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Banach September 2018