Banach February 2020

banach@mathdept.okstate.edu

2 participants
10 discussions

Announcement of a Postdoc Position at University of Memphis
by Bentuo Zheng (bzheng) 24 Mar '20

24 Mar '20

Dear All, There will be a two-year postdoc position at the Department of Mathematical Sciences of the University of Memphis which may be renewable for the third year. Please find the details in the attached ad. BR, Bentuo Zheng

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Banach spaces webinars
by Sari, Bunyamin 04 Feb '20

04 Feb '20

Dear Colleague, We are pleased to introduce webinars (online seminars) on Banach spaces and related topics via zoom<https://zoom.us/> on Fridays at 9am central US time. The idea emerged from the fact that there are relatively small number of researchers in the area scattered around the world, and so conducting a regular online seminar where anyone can participate in the convenience of their home makes a lot of sense. How will it work? To participate simply click on the link provided for the seminar and follow instructions. If it is your first time using zoom, you will likely be asked to download the app first. You can click on the link anytime before the meeting to do so. When you join the meeting your microphone is muted by default, so feel free to join anytime there won’t be any unwanted interruptions by doing so. It is interactive; you unmute the mic and ask questions or comments, or click on raise hand button to get speakers’ attention, or chat via on screen text with participants, or simply sit quietly and listen to the talk. Giving a talk is also very easy. It requires minimal tech. For instance, a computer with webcam and Ipad to write on is sufficient. To demonstrate this and to kick off seminars I will give the first talk on this Friday, February 7 at 9am central. Below is the invitation. If you like to speak please send me an email and I will schedule it and help with the set up. Please consider giving a talk and help support webinars keep going. Below is the abstract and invitation of the first talk. Best regards, Bunyamin Sari University of North Texas ____________________________________________ Title: On Sanders’ proof of inequivalence of Walsh and trigonometric systems Abstract. We will speak on Tom Sanders recent proof<https://arxiv.org/abs/1901.03109> that Walsh system in any order is not equivalent to trigonometric basis in $L_p$. The proof uses interesting ideas from additive combinatorics and discrete Fourier analysis which we will present some of the details. _____________________________________________ Bunyamin Sari is inviting you to a scheduled Zoom meeting. Topic: Banach spaces webinar Time: Feb 7, 2020 09:00 AM Central Time (US and Canada) Join Zoom Meeting https://unt.zoom.us/j/353774017 Meeting ID: 353 774 017 One tap mobile +16699006833,,353774017# US (San Jose) +19294362866,,353774017# US (New York) Dial by your location +1 669 900 6833 US (San Jose) +1 929 436 2866 US (New York) Meeting ID: 353 774 017 Find your local number: https://unt.zoom.us/u/appFn0jty Join by SIP 353774017(a)zoomcrc.com<mailto:353774017@zoomcrc.com> Join by H.323 162.255.37.11 (US West) 162.255.36.11 (US East) 221.122.88.195 (China) 115.114.131.7 (India Mumbai) 115.114.115.7 (India Hyderabad) 213.19.144.110 (EMEA) 103.122.166.55 (Australia) 209.9.211.110 (Hong Kong) 64.211.144.160 (Brazil) 69.174.57.160 (Canada) 207.226.132.110 (Japan) Meeting ID: 353 774 017 Join by Skype for Business https://unt.zoom.us/skype/353774017

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Abstract of a paper by Yin-Ting Liao, Kavita Ramanan
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper “Geometric sharp large deviations for random projections of $\ell_p^n$ spheres” by Yin-Ting Liao<https://arxiv.org/search/math?searchtype=author&query=Liao%2C+Y>, Kavita Ramanan<https://arxiv.org/search/math?searchtype=author&query=Ramanan%2C+K>. Abstract: Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics, asymptotic geometric analysis and computer science. For fixed $p \in (1,\infty)$, let $(X^n)_{n \in \mathbb{N}}$ and $(\theta^n)_{n \in \mathbb{N}}$ be independent sequences of random vectors with $X^n$ and $\theta^n$ distributed according to the normalized cone measure on the unit $\ell_p^n$ sphere and $\ell_2^n$ sphere, respectively. For almost every sequence of projection directions $(\theta^n)_{n \in \mathbb{N}}$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^n$ onto $\theta^n$. In contrast to the (quenched) large deviation rate function, the prefactor is shown to exhibit a dependence on the projection directions that encodes geometric information. Moreover, an importance sampling algorithm is developed to numerically estimate the tail probabilities, and used to illustrate the accuracy of the analytical sharp large deviation estimates for even moderate values of $n$. The results on the one hand provide quantitative estimates of tail probabilities of random projections, valid for finite $n$, generalizing previous results due to Gantert, Kim and Ramanan that characterize only logarithmic asymptotics (as the dimension $n$ tends to infinity), and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting. The proofs combine Fourier analytic and probabilistic techniques, provide a simpler representation for the large deviation rate function that shows that it is strictly convex, and entail establishing central limit theorems for random projections under a certain family of changes of measure, which may be of independent interest. https://arxiv.org/abs/2001.04053

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Abstract of a paper by Amine Marrakchi, Mikael de la Salle
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper Isometric actions on Lp-spaces: dependence on the value of p” by Amine Marrakchi<https://arxiv.org/search/math?searchtype=author&query=Marrakchi%2C+A>, Mikael de la Salle<https://arxiv.org/search/math?searchtype=author&query=de+la+Salle%2C+M>. Abstract: We prove that, for every topological group $G$, the following two sets are intervals: the set of real numbers $p > 0$ such that every continuous action of $G$ by isometries on an $L_p$ space has bounded orbits, and the set of $p > 0$ such that $G$ admits a metrically proper continuous action by isometries on an $L_p$ space. This answers a question by Chatterji--Drutu--Haglund. https://arxiv.org/abs/2001.02490

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Abstract of a paper by Francisco L. Hernández, Evgeny M. Semenov, Pedro Tradacete
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper “Strictly singular non-compact operators between $L_p$ spaces” by Francisco L. Hernández<https://arxiv.org/search/math?searchtype=author&query=Hern%C3%A1ndez%2C+F+L>, Evgeny M. Semenov<https://arxiv.org/search/math?searchtype=author&query=Semenov%2C+E+M>, Pedro Tradacete<https://arxiv.org/search/math?searchtype=author&query=Tradacete%2C+P>. Abstract: We study the structure of strictly singular non-compact operators between $L_p$ spaces. Answering a question raised in [Adv. Math. 316 (2017), 667-690], it is shown that there exist operators $T$, for which the set of points $(\frac1p,\frac1q)\in(0,1)\times (0,1)$ such that $T:L_p\rightarrow L_q$ is strictly singular but not compact contains a line segment in the triangle $\{(\frac1p,\frac1q):1<p<q<\infty\}$ of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular operators defined on subspaces of $L_p$ is also explored. https://arxiv.org/abs/2001.09677

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Abstract of a paper by M. A. Sofi
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper “Extension operators and nonlinear structure of Banach spaces” by M. A. Sofi<https://arxiv.org/search/math?searchtype=author&query=Sofi%2C+M+A>. Abstract: The problem involving the extension of functions from a certain class and defined on subdomains of the ambient space to the whole space is an old and a well investigated theme in analysis. A related question whether the extensions that result in the process may be chosen in a linear or a continuous manner between appropriate spaces of functions turns out to be highly nontrivial. That this holds for the class of continuous functions defined on metric spaces is the well-known Borsuk-Dugundji theorem which asserts that given a metric space M and a subspace S of M, each continuous function g on S can be extended to a continuous function f on X such that the resulting assignment from C(S) to C(M) is a norm-one continuous linear extension operator. The present paper is devoted to an investigation of this problem in the context of extendability of Lipschitz functions from closed subspaces of a given Banach space to the whole space such that the choice of the extended function gives rise to a bounded linear (extension) operator between appropriate spaces of Lipschitz functions. It is shown that the indicated property holds precisely when the underlying space is isomorphic to a Hilbert space. Among certain useful consequences of this theorem, we provide an isomorphic analogue of a well-known theorem of S. Reich by show ing that closed convex subsets of a Banach space X arise as Lipschitz retracts of X precisely when X is isomorphically a Hilbert space. We shall also discuss the issue of bounded linear extension operators between spaces of Lipschitz functions now defined on arbitrary subsets of Banach spaces and provide a direct proof of the known non-existence of such an extension operator by using methods which are more accessible than those initially employed by the authors. https://arxiv.org/abs/2001.09303

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Abstract of a paper by Rainis Haller, Katriin Pirk, Triinu Veeorg
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper “Daugavet- and Delta-points in absolute sums of Banach spaces” by Rainis Haller<https://arxiv.org/search/math?searchtype=author&query=Haller%2C+R>, Katriin Pirk<https://arxiv.org/search/math?searchtype=author&query=Pirk%2C+K>, Triinu Veeorg<https://arxiv.org/search/math?searchtype=author&query=Veeorg%2C+T>. Abstract: A Daugavet-point (resp.~$\Delta$-point) of a Banach space is a norm one element $x$ for which every point in the unit ball (resp.~element $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from $x$. A Banach space has the well-known Daugavet property (resp.~diametral local diameter 2 property) if and only if every norm one element is a Daugavet-point (resp.~$\Delta$-point). This paper complements the article "Delta- and Daugavet-points in Banach spaces" by T. A. Abrahamsen, R. Haller, V. Lima, and K. Pirk, where the study of the existence of Daugavet- and $\Delta$-points in absolute sums of Banach spaces was started. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2001.06197

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Abstract of a paper by Robert Deville, Miguel García-Bravo
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper “Normal tilings of a Banach space and its ball” by Robert Deville<https://arxiv.org/search/math?searchtype=author&query=Deville%2C+R>, Miguel García-Bravo<https://arxiv.org/search/math?searchtype=author&query=Garc%C3%ADa-Bravo%2C+M>. Abstract: We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal. In 2010 Preiss constructed a convex normal tiling of the separable Hilbert space. For Banach spaces with Schauder basis we will show that Preiss' result is still true with starshaped tiles instead of convex ones. Also, whenever $X$ is uniformly convex we give precise constructions of convex normal tilings of the unit sphere, the unit ball or in general of any convex body. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2001.04372

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Abstract of a paper by Alexandros Margaris
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper “Almost bi--Lipschitz embeddings using covers of balls centred at the origin” by Alexandros Margaris<https://arxiv.org/search/math?searchtype=author&query=Margaris%2C+A>. Abstract: In 2010, Olson \& Robinson [Transactions of the American Mathematical Society, 362(1), 145-168] introduced the notion of an almost homogeneous metric space and showed that if $X$ is a subset of a Hilbert space such that $X-X$ is almost homogeneous, then $X$ admits almost bi--Lipschitz embeddings into Euclidean spaces. In this paper, we extend this result and we show that if $X$ is a subset of a Banach space such that $X-X$ is almost homogeneous at the origin, then $X$ can be embedded in a Euclidean space in an almost bi--Lipschitz way. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2001.02607

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Announcement of a paper by Miguel Berasategui, Pablo M. Berná, Silvia Lasalle
by Bentuo Zheng (bzheng) 01 Feb '20

01 Feb '20

This is an announcement for the paper “Strong-partially greedy bases and Lebesgue-type inequalities” by Miguel Berasategui<https://arxiv.org/search/math?searchtype=author&query=Berasategui%2C+M>, Pablo M. Berná<https://arxiv.org/search/math?searchtype=author&query=Bern%C3%A1%2C+P+M>, Silvia Lasalle<https://arxiv.org/search/math?searchtype=author&query=Lasalle%2C+S>. Abstract: In this paper we continue the study of Lebsgue-type inequalities for the greedy algorithm. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesguey-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in [9] for Schauder bases. We also give the characterization of 1-strong partial greediness, following the study started in [3,1]. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2001.01226

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Banach February 2020