Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Earnest Akofor
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “On Lipschitz Retraction of Finite Subsets of Normed Spaces” by Earnest Akofor<https://arxiv.org/search/math?searchtype=author&query=Akofor%2C+E>. Abstract: If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset \cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space and, by Bačák and Kovalev when $X$ is a CAT(0) space, that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq 2$. We prove that when $X$ is a normed space, the above sequence admits Lipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well as concrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ and Hölder-continuous on bounded sets if $n>3$. We also prove that if $X$ is a geodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space. These results are relevant to certain questions in the aforementioned previous work which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq 2$, exist for $X$ in more general classes of Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.00603

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Abstract of a paper by Leandro CandidoApoorva Khare
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “The non-compact normed space of norms on a finite-dimensional Banach space” by Leandro Candido<https://arxiv.org/search/math?searchtype=author&query=Candido%2C+L>Apoorva Khare<https://arxiv.org/search/math?searchtype=author&query=Khare%2C+A>. Abstract: We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $\mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k \geqslant 2$ the metric subspace $\{ \| \cdot \|_p : p \in [1,\infty] \}$ maps isometrically and monotonically to $[0, \log k]$ (or $[0,1]$ by scaling the norm), again unlike in the Banach-Mazur compactum. Our analysis goes through embedding the above quotient space into a normed space, and reveals an implicit functorial construction of function spaces with diameter norms (as well as a variant of the distortion). In particular, we realize the above quotient space of norms as a normed space. We next study the parallel setting of the - also hitherto unexplored - metric space $\mathcal{S}([n])$ of all metrics on a finite set of $n$ elements, revealing the connection between log-distortion and diameter norms. In particular, we show that $\mathcal{S}([n])$ is also a normed space. We demonstrate embeddings of equivalence classes of finite metric spaces (parallel to the Gromov-Hausdorff setting), as well as of $\mathcal{S}([n-1])$, into $\mathcal{S}([n])$. We conclude by discussing extensions to norms on an arbitrary Banach space and to discrete metrics on any set, as well as some questions in both settings above. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.06188

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Abstract of a paper by Emiel Lorist
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “The $\ell^s$-boundedness of a family of integral operators on UMD Banach function spaces” by Emiel Lorist<https://arxiv.org/search/math?searchtype=author&query=Lorist%2C+E>. Abstract: We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.12833

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Abstract of a paper by Colin Petitjean, Antonín Procházka
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “On exposed points of Lipschitz free spaces” by Colin Petitjean<https://arxiv.org/search/math?searchtype=author&query=Petitjean%2C+C>, Antonín Procházka<https://arxiv.org/search/math?searchtype=author&query=Proch%C3%A1zka%2C+A>. Abstract: In this note we prove that a molecule $d(x,y)^{-1}(δ(x)-δ(y))$ is an exposed point of the unit ball of a Lispchitz free space $\mathcal F(M)$ if and only if the metric segment $[x,y]=\{z \in M \; : \; d(x,y)=d(z,x)+d(z,y) \}$ is reduced to $\{x,y\}$. This is based on a recent result due to Aliaga and Pernecká which states that the class of Lipschitz free spaces over closed subsets of M is closed under arbitrary intersections when M has finite diameter. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.12031

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Abstract of a paper by Ralph Chill, Alberto Fiorenza, Sebastian Krol
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Interpolation of nonlinear positive or order preserving operators on Banach lattices” by Ralph Chill<https://arxiv.org/search/math?searchtype=author&query=Chill%2C+R>, Alberto Fiorenza<https://arxiv.org/search/math?searchtype=author&query=Fiorenza%2C+A>, Sebastian Krol<https://arxiv.org/search/math?searchtype=author&query=Krol%2C+S>. Abstract: We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo-Peetre operators, and exact partially $K$-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple $(L^1,L^\infty )$ due to Bénilan and Crandall. https://arxiv.org/abs/1810.09684

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Abstract of a paper by Martin Mathieu, Pedro Tradacete
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Strictly singular multiplication operators on $\mathcal L(X)$” by Martin Mathieu<https://arxiv.org/search/math?searchtype=author&query=Mathieu%2C+M>, Pedro Tradacete<https://arxiv.org/search/math?searchtype=author&query=Tradacete%2C+P>. Abstract: Exploiting several $\ell_p$-factorization results for strictly singular operators, we study the strict singularity of the multiplication operator $L_A R_B\colon T\mapsto ATB$ on $\mathcal L(X)$ for various Banach spaces~$X$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.09194

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Abstract of a paper by Jose M. Calabuig, Maite Fernández Unzueta, Fernando Galaz-Fontes, Enrique A. Sánchez Pérez
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Equivalent Norms in a Banach Function Space and the Subsequence Property” by Jose M. Calabuig<https://arxiv.org/search/math?searchtype=author&query=Calabuig%2C+J+M>, Maite Fernández Unzueta<https://arxiv.org/search/math?searchtype=author&query=Unzueta%2C+M+F>, Fernando Galaz-Fontes<https://arxiv.org/search/math?searchtype=author&query=Galaz-Fontes%2C+F>, Enrique A. Sánchez Pérez<https://arxiv.org/search/math?searchtype=author&query=P%C3%A9rez%2C+E+A+S>. Abstract: Given a finite measure space $(Ω,Σ,μ)$, we show that any Banach space $X(μ)$ consisting of (equivalence classes of) real measurable functions defined on $Ω$ such that $f χ_A \in X(μ) $ and $ \|f χ_A \| \leq \|f\|, \, f \in X(μ), \ A \in Σ$, and having the subsequence property, is in fact an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05714

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Abstract of a paper by R.M. Causey
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Lipschitz subtype” by R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M> Abstract: We give necessary and sufficient conditions for a Lipschitz map, or more generally a uniformly Lipschitz family of maps, to factor the Hamming cubes. This is an extension to Lipschitz maps of a particular spatial result of Bourgain, Milman, and Wolfson. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05707

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Abstract of a paper by M. Alikhani
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “On pseudo weakly compact operators of order P” by M. Alikhani<https://arxiv.org/search/math?searchtype=author&query=Alikhani%2C+M>. Abstract: In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spaces $ X$ and $ Y $ such that projective tensor and injective tensor products between $ X $ and $ Y $ have the $ p $-sequentially Right property.\ Finally, we introduce two properties for the Banach spaces, namely $ p $-sequentially Right$ ^{\ast} $ and weak $ p $-sequentially Right$ ^{\ast} $ properties and obtain some characterizations of these properties. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05638

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Abstract of a paper by R.M. Causey
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “The ξ,ζ-Dunford Pettis property” by R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M>. Abstract: Using the hierarchy of weakly null sequences introduced by Argyros, Merkourakis, and Tsarpalias, we introduce two new families of operator classes. The first family simultaneously generalizes the completely continuous operators and the weak Banach-Saks operators. The second family generalizes the class $\mathfrak{DP}$. We study the distinctness of these classes, and prove that each class is an operator ideal. We also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. We produce a number of examples, including the higher ordinal Schreier and Baernstein spaces. We prove ordinal analogues of several known results for Banach spaces with the Dunford-Pettis, hereditary Dunford-Pettis property, and hereditary by quotients Dunford-Pettis property. For example, we prove that for any $0\leqslant ξ, ζ<ω_1$, a Banach space $X$ has the hereditary $ω^ξ, ω^ζ$-Dunford Pettis property if and only if every seminormalized, weakly null sequence either has a subsequence which is an $\ell_1^{ω^ξ}$-spreading model or a $c_0^{ω^ζ}$-spreading model. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05196

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