Banach

banach@mathdept.okstate.edu

2549 discussions

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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Announcement of a paper by M. Brixey, R.M. Causey, P. Frankart
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper Uniform subsequential estimates on weakly null sequences” by M. Brixey<https://arxiv.org/search/math?searchtype=author&query=Brixey%2C+M>, R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M>, P. Frankart<https://arxiv.org/search/math?searchtype=author&query=Frankart%2C+P>. Abstract: We provide a generalization of two results of Knaust and Odell from \cite{KO2} and \cite{KO}. We prove that if $X$ is a Banach space and $(g_n)_{n=1}^\infty$ is a right dominant Schauder basis such that every normalized, weakly null sequence in $X$ admits a subsequence dominated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists a constant $C$ such that every normalized, weakly null sequence in $X$ admits a subsequence $C$-dominated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove that if every spreading model generated by a normalized, weakly null sequence in $X$ is dominated by some spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists $C$ such that every spreading model generated by a normalized, weakly null sequence in $X$ is $C$-dominated by every spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove a single, ordinal-quantified result which unifies and interpolates between these two results. https://arxiv.org/abs/1912.13443

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Announcement of a paper by R.M. Causey
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “$Sz(\cdot)\leqslant ω^ξ$ is rarely a three space property” by R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M>. Abstract: We prove that for any non-zero, countable ordinal $\xi$ which is not additively indecomposable, the property of having Szlenk index not exceeding $\omega^\xi$ is not a three space property. This complements a result of Brooker and Lancien, which states that if $\xi$ is additively indecomposable, then having Szlenk index not exceeding $\omega^\xi$ is a three space property. https://arxiv.org/abs/1912.13429

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Announcement of a paper by Joanna Garbulińska-Wegrzyn, Wieslaw Kubiś
by Bentuo Zheng (bzheng) 01 Jan '20

01 Jan '20

This is an announcement for the paper “A note on universal operators between separable Banach spaces” by Joanna Garbulińska-Wegrzyn<https://arxiv.org/search/math?searchtype=author&query=Garbuli%C5%84ska-Wegr…>, Wieslaw Kubiś<https://arxiv.org/search/math?searchtype=author&query=Kubi%C5%9B%2C+W>. Abstract: We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator $\Omega$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $\Omega$ is generic in the sense of a natural infinite game. https://arxiv.org/abs/1912.13312

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