Banach April 2020

banach@mathdept.okstate.edu

3 participants
16 discussions

Abstract of a paper by Fernando Albiac, Jose L. Ansorena, Przemyslaw Wojtaszczyk
by Bentuo Zheng (bzheng) 01 Apr '20

01 Apr '20

This is an announcement for the paper “On a 'philosophical' question about Banach envelopes” 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>, Przemyslaw Wojtaszczyk<https://arxiv.org/search/math?searchtype=author&query=Wojtaszczyk%2C+P>. Abstract: We show how to construct nonlocally convex quasi-Banach spaces $X$ whose dual separates the points of a dense subspace of $X$ but does not separate the points of $X$. Our examples connect with a question raised by Pietsch [About the Banach envelope of $l_{1,\infty}$, Rev. Mat. Complut. 22 (2009), 209-226] and shed light into the unexplored class of quasi-Banach spaces with nontrivial dual which do not have sufficiently many functionals to separate the points of the space. https://arxiv.org/abs/2003.07742

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Abstract of a paper by J. Rodríguez, E.A. Sánchez-Pérez
by Bentuo Zheng (bzheng) 01 Apr '20

01 Apr '20

This is an announcement for the paper “A class of summing operators acting in spaces of operators” by J. Rodríguez<https://arxiv.org/search/math?searchtype=author&query=Rodr%C3%ADguez%2C+J>, E.A. Sánchez-Pérez<https://arxiv.org/search/math?searchtype=author&query=S%C3%A1nchez-P%C3%A9r…>. Abstract: Let $X$, $Y$ and $Z$ be Banach spaces and let $U$ be a subspace of $\mathcal{L}(X^*,Y)$, the Banach space of all operators from $X^*$ to $Y$. An operator $S: U \to Z$ is said to be $(\ell^s_p,\ell_p)$-summing (where $1\leq p <\infty$) if there is a constant $K\geq 0$ such that $$ \Big( \sum_{i=1}^n \|S(T_i)\|_Z^p \Big)^{1/p} \le K \sup_{x^* \in B_{X^*}} \Big(\sum_{i=1}^n \|T_i(x^*)\|_Y^p\Big)^{1/p} $$ for every $n\in \mathbb{N}$ and every $T_1,\dots,T_n \in U$. In this paper we study this class of operators, introduced by Blasco and Signes as a natural generalization of the $(p,Y)$-summing operators of Kislyakov. On one hand, we discuss Pietsch-type domination results for $(\ell^s_p,\ell_p)$-summing operators. In this direction, we provide a negative answer to a question raised by Blasco and Signes, and we also give new insight on a result by Botelho and Santos. On the other hand, we extend to this setting the classical theorem of Kwapień characterizing those operators which factor as $S_1\circ S_2$, where $S_2$ is absolutely $p$-summing and $S_1^*$ is absolutely $q$-summing ($1<p,q<\infty$ and $1/p+1/q \leq 1$). https://arxiv.org/abs/2003.07252

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Abstract of a paper by Paata Ivanisvili, Ramon van Handel, Alexander Volberg
by Bentuo Zheng (bzheng) 01 Apr '20

01 Apr '20

This is an announcement for the paper “Rademacher type and Enflo type coincide” by Paata Ivanisvili<https://arxiv.org/search/math?searchtype=author&query=Ivanisvili%2C+P>, Ramon van Handel<https://arxiv.org/search/math?searchtype=author&query=van+Handel%2C+R>, Alexander Volberg<https://arxiv.org/search/math?searchtype=author&query=Volberg%2C+A>. Abstract: A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2003.06345

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Abstract of a paper by Ramón J. Aliaga, Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 01 Apr '20

01 Apr '20

This is an announcement for the paper “Points of differentiability of the norm in Lipschitz-free spaces” by Ramón J. Aliaga<https://arxiv.org/search/math?searchtype=author&query=Aliaga%2C+R+J>, Abraham Rueda Zoca<https://arxiv.org/search/math?searchtype=author&query=Zoca%2C+A+R>. Abstract: We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form $\mu=\sum_n \lambda_n \frac{\delta_{x_n}-\delta_{y_n}}{d(x_n,y_n)}$ such that $\|\mu\|=\sum_n |\lambda_n |$. We characterise these elements in terms of geometric conditions on the points $x_n$, $y_n$ of the underlying metric space, and determine when they are points of Gâteaux differentiability of the norm. In particular, we show that Gâteaux and Fréchet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and that their tensor products with Gâteaux (resp. Fréchet) differentiable elements of a Banach space are Gâteaux (resp. Fréchet) differentiable in the corresponding projective tensor product. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2003.01439

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Abstract of a paper by Gilles Lancien, Matias Raja
by Bentuo Zheng (bzheng) 01 Apr '20

01 Apr '20

This is an announcement for the paper “Nonlinear aspects of super weakly compact sets” by Gilles Lancien<https://arxiv.org/search/math?searchtype=author&query=Lancien%2C+G>, Matias Raja<https://arxiv.org/search/math?searchtype=author&query=Raja%2C+M>. Abstract: We study the notion of super weakly compact subsets of a Banach space, which can be described as a local version of super-reflexivity. Our first result is that the closed convex hull of a super weakly compact set is super weakly compact. This allows us to extend to the non convex setting the main properties of these sets. In particular, we give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2003.01030

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Abstract of a paper by Anna Kamińska, Han Ju Lee, Hyung-Joon Tag
by Bentuo Zheng (bzheng) 01 Apr '20

01 Apr '20

This is an announcement for the paper “Diameter two properties and the Radon-Nikodým property in Orlicz spaces” by Anna Kamińska<https://arxiv.org/search/math?searchtype=author&query=Kami%C5%84ska%2C+A>, Han Ju Lee<https://arxiv.org/search/math?searchtype=author&query=Lee%2C+H+J>, Hyung-Joon Tag<https://arxiv.org/search/math?searchtype=author&query=Tag%2C+H>. Abstract: Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikodým property. Consequently it is shown that an Orlicz space $L_\varphi$ over a non-atomic $\sigma$-finite measure space $(\Omega, \Sigma,\mu)$, not necessarily separable, has the Radon-Nikodým property if and only if $\varphi$ is an $N$-function at infinity and satisfies the appropriate $\Delta_2$ condition. For an Orlicz sequence space $\ell_\varphi$, it has the Radon-Nikodým property if and only if $\varphi$ satisfies condition $\Delta_2^0$. In the second part the relationships between uniformly $\ell_1^2$ points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space $L_\varphi$ has the Daugavet property only if $\varphi$ is linear, so when $L_\varphi$ is isometric to $L_1$. The other consequence is that the Orlicz spaces equipped with the Orlicz norm generated by $N$-functions never have local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, the strong diameter two property are equivalent in function and sequence Orlicz spaces with the Luxemburg norm under appropriate conditions on $\varphi$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/2003.00396

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Banach April 2020