Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by C. S. Barroso, V. Ferreira
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps” by C. S. Barroso<https://arxiv.org/find/math/1/au:+Barroso_C/0/1/0/all/0/1>, V. Ferreira<https://arxiv.org/find/math/1/au:+Ferreira_V/0/1/0/all/0/1>. Abstract: In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-(s) subsequence $(x_n)$ which admits an equivalent convex basic sequence. This fact is used to characterize weak-compactness of bounded, closed convex sets in terms of the generic fixed point property ($\mathcal{G}$-FPP) for the class of affine bi-Lipschitz maps. This result generalizes a theorem by Benavides, Jap\'on Pineda and Prus previously proved for the class of continuous maps. We also introduce a relaxation of this notion ($\mathcal{WG}$-FPP) and observe that a closed convex bounded subset of a Banach space is weakly compact iff it has the $\matcal{WG}$-FPP for affine $1$-Lipschitz maps. Related results are also proved. For example, a complete convex bounded subset $C$ of a Hlcs $X$ is weakly compact iff it has the $\mathcal{G}$-FPP for the class of affine continuous maps $f: C\rightarrow X$ with weak-approximate fixed point nets. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.05642

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Abstract of a paper by Antonio Avilés, Gonzalo Martínez-Cervantes, Grzegorz Plebanek
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “Weakly Radon-Nikodým Boolean algebras and independent sequences” by Antonio Avilés<https://arxiv.org/find/math/1/au:+Aviles_A/0/1/0/all/0/1>, Gonzalo Martínez-Cervantes<https://arxiv.org/find/math/1/au:+Martinez_Cervantes_G/0/1/0/all/0/1>, Grzegorz Plebanek<https://arxiv.org/find/math/1/au:+Plebanek_G/0/1/0/all/0/1>. Abstract: A compact space is said to be weakly Radon-Nikod\'{y}m (WRN) if it can be weak$^*$-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, we construct a minimally generated nonWRN Boolean algebra whose Stone space is a separable Rosenthal compactum, answering in this way a question of W. Marciszewski. We also study questions of J. Rodr\'{i}guez and R. Haydon concerning measures and the existence of nontrivial convergent sequences on WRN compacta, obtaining partial results on some natural subclasses. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.04257

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Abstract of a paper by Marek Cúth, Ondřej F.K. Kalenda, Petr Kaplický
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces” by Marek Cúth<https://arxiv.org/find/math/1/au:+Cuth_M/0/1/0/all/0/1>, Ondřej F.K. Kalenda<https://arxiv.org/find/math/1/au:+Kalenda_O/0/1/0/all/0/1>, Petr Kaplický<https://arxiv.org/find/math/1/au:+Kaplicky_P/0/1/0/all/0/1>. Abstract: Let $E$ be a finite-dimensional normed space and $\Omega$ a nonempty convex open set in $E$. We show that the Lipschitz-free space of $\Omega$ is canonically isometric to the quotient of $L_1(\Omega, E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.03966

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Abstract of a paper by Marek Cúth, Ondřej F.K. Kalenda, Petr Kaplický
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

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Abstract of a paper by Satish K. Pandey
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “A Spectral Characterization of Absolutely Norming Operators on S.N. Ideals” by Satish K. Pandey<https://arxiv.org/find/math/1/au:+Pandey_S/0/1/0/all/0/1>. Abstract: The class of absolutely norming operators on complex Hilbert spaces of arbitrary dimensions was introduced in [6] and a spectral characterization theorem for these operators was established in [11]. In this paper we extend the concept of absolutely norming operators to various symmetric norms. We establish a few spectral characterization theorems for operators on complex Hilbert spaces to be absolutely norming with respect to various symmetric norms. It is also shown that for many symmetric norms the absolutely norming operators have the same spectral characterization as proven earlier in [11] for the class of operators that are absolutely norming with respect to the usual operator norm. Finally, we prove the existence of a symmetric norm on the algebra $B(H)$ with respect to which the even the identity operator does not attain its norm. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.02095

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Abstract of a paper by Satish K. Pandey
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

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Abstract of a paper by Christian Rosendal
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “Equivariant geometry of Banach spaces and topological groups” by Christian Rosendal<https://arxiv.org/find/math/1/au:+Rosendal_C/0/1/0/all/0/1>. Abstract: We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, i.e., continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.01070

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Abstract of a paper by Maysam Maysami Sadr
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “Decomposition of functions between Banach spaces in the orthogonality equation” by Maysam Maysami Sadr<https://arxiv.org/find/math/1/au:+Sadr_M/0/1/0/all/0/1>. Abstract: Let $E, F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f, g)$ of the Banach-orthogonality equation $$\langle f(x), g(\alpha) \rangle=\langle x, \alpha \rangle, \forall x\i E, \alpha\in E^*$$, where $f: E\rightarrow F, g: E^*\rightarrow F^*$ are two maps. Our result generalizes the recent result of {\L}ukasik and W\'{o}jcik in the case that $E$ and $F$ are Hilbert spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.00423

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Advances in Operator Theory
by Dale Alspach 01 Oct '16

01 Oct '16

Dear colleague, Having a professional editorial board, the newly launched and peer reviewed journal "*Advances in Operator Theory (AOT) *publishes papers with deep results, new ideas, profound impact and significant implications in operator theory and related topics. It is published (free of charges for authors and readers) by the Tusi Math. Research Group, which is the publisher of *Banach J. Math. Anal.* and *Ann. Funct. Anal.* The website of journal for online submission is *http://aot-math.org/ <http://aot-math.org/>* Best wishes, M. S. Moslehian Editor-in-chief

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Abstract of a paper by Kasper Green Larsen, Jelani Nelson
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “Optimality of the Johnson-Lindenstrauss Lemma” by Kasper Green Larsen<https://arxiv.org/find/cs/1/au:+Larsen_K/0/1/0/all/0/1>, Jelani Nelson<https://arxiv.org/find/cs/1/au:+Nelson_J/0/1/0/all/0/1>. Abstract: For any integers $d, n \geq 2$ and $1/({\min\{n,d\}})^{0.4999} <\eps<1$, we show the existence of a set of $n$ vectors $X\subset \R^d$ such that any embedding $f:X\rightarrow \R^m$ satisfying $$\forall x,y\in X,\ (1-\eps)\|x-y\|_2^2\le \|f(x)-f(y)\|_2^2 \le (1+\eps)\|x-y\|_2^2$$ must have $$m = \Omega(\eps^{-2} \lg n).$$ This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma \cite{JL84}. Furthermore, our lower bound holds for nearly the full range of $\eps$ of interest, since there is always an isometric embedding into dimension $\min\{d, n\}$ (either the identity map, or projection onto $\mathop{span}(X)$). Previously such a lower bound was only known to hold against {\em linear} maps $f$, and not for such a wide range of parameters $\eps, n, d$ \cite{LarsenN16}. The best previously known lower bound for general $f$ was $m = \Omega(\eps^{-2}\lg n/\lg(1/\eps))$ \cite{Welch74,Alon03}, which is suboptimal for any $\eps = o(1)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.02094

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