Banach October 2016

banach@mathdept.okstate.edu

2 participants
22 discussions

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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Abstract of a paper by Andrei Sipos
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “Proof mining in $L_p$ spaces” by Andrei Sipos<https://arxiv.org/find/math/1/au:+Sipos_A/0/1/0/all/0/1>. Abstract: We obtain an equivalent implicit characterization of $L_p$ Banach spaces that is amenable to a logical treatment. Using that, we obtain an axiomatization for such spaces into a higher-order logical system, the kind of which is used in proof mining, a research program that aims to obtain the hidden computational content of mathematical proofs using tools from mathematical logic. The axiomatization is followed by a corresponding metatheorem in the style of proof mining. We illustrate its use with an application, namely the derivation of the standard modulus of uniform convexity. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.02080

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Abstract of a paper by James Kilbane
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “On embeddings of finite subsets of $\ell_2$” by James Kilbane<https://arxiv.org/find/math/1/au:+Kilbane_J/0/1/0/all/0/1>. Abstract: We study finite subsets of $\ell_2$, and more generally any metric space, and consider whether these isometrically embed into a Banach space. Our results partially answer a question of Ostrovskii, on whether every infinite-dimensional Banach space contains every finite subset of $\ell_2$ isometrically. The updated version contains acknowledgement that Theorem 3.1 has been proven previously in a paper of Shkarin. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.08971

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Abstract of a paper by Eusebio Gardella, Hannes Thiel
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “Representations of $p$-convolution algebras on $L_q$-spaces” by Eusebio Gardella<https://arxiv.org/find/math/1/au:+Gardella_E/0/1/0/all/0/1>, Hannes Thiel<https://arxiv.org/find/math/1/au:+Thiel_H/0/1/0/all/0/1>. Abstract: For a nontrivial locally compact group $G$, and $p\in [1, \infty)$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L_p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\in [1, \infty)$, these Banach algebras can be represented on an $L_q$-space if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $|\frac{1}{p}-\frac{1}{2}|=|\frac{1}{q}-\frac{1}{2}|$. This result can be interpreted as follows: for $p, q\in [1, \infty)$, the $L_p$- and $L_q$-representation theories of a group are incomparable, except in the trivial cases when they are equivalent. As an application, we show that, for distinct $p, q\in [1, \infty)$, if the $L_p$ and $L_q$ crossed products of a topological dynamical system are isomorphic, then $\frac{1}{p}+\frac{1}{q}=1$. In order to prove this, we study the following relevant aspects of $L_p$-crossed products: existence of approximate identities, duality with respect to $p$, and existence of canonical isometric maps from group algebras into their multiplier algebras. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.08612

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Abstract of a paper by S. Cobzaş
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “Lipschitz properties of convex mappings” by S. Cobzaş<https://arxiv.org/find/math/1/au:+Cobzas_S/0/1/0/all/0/1>. Abstract: The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset $\Omega$ of a locally convex space $X$ and taking values in a locally convex space $Y$ ordered by a normal cone. One proves also equi-Lipschitz properties for pointwise bounded families of continuous convex mappings, provided the source space $X$ is barrelled. Some results on Lipschitz properties of continuous convex functions defined on metrizable topological vector spaces are included as well. The paper has a methodological character - its aim is to show that some geometric properties (monotonicity of the slope, the normality of the seminorms) allow to extend the proofs from the scalar case to the vector one. In this way the proofs become more transparent and natural. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.07839

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Abstract of a paper by Torrey M. Gallagher
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “Fixed point results for a new mapping related to mean nonexpansive mappings” by Torrey M. Gallagher<https://arxiv.org/find/math/1/au:+Gallagher_T/0/1/0/all/0/1>. Abstract: Mean nonexpansive mappings were first introduced in 2007 by Goebel and Japon Pineda and advances have been made by several authors toward understanding their fixed point properties in various contexts. For any given $(a_1, a_2)$-nonexpansive mapping $T$ of a Banach space, many of the positive results have been derived from properties of the mapping $T_a=a_1T+a_2T^2=(a_1I+a_2T)\circ T$ which is nonexpansive. However, the related mapping $T\circ (a_1I +a_2T)$ has not yet been studied. In this paper, we investigate some fixed point properties of this new mapping and discuss relationships between $(a_1I+a_2T)\circ T$ and $T\circ (a_1I +a_2T)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.07163

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Abstract of a paper by Jin Xi Chen, Anton R. Schep
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “Two-sided multiplication operators on the space of regular operators” by Jin Xi Chen<https://arxiv.org/find/math/1/au:+Chen_J/0/1/0/all/0/1>, Anton R. Schep<https://arxiv.org/find/math/1/au:+Schep_A/0/1/0/all/0/1>. Abstract: Let $W, X, Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\in L^r(Y, Z)$ and $B\in L^r(W, X)$ let $M_{A, B}$ be the two-sided multiplication operator from $L^r(X, Y)$ into $L^r(W, Z)$ defined by $M_{A, B}(T)=ATB$. We show that for every $0\leq A_0\in L^{rn}(Y, Z), |M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^r(W, X)$ and all $T\in L^{rn}(X, Y)$. Furthermore, if $W, X, Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A, B}|=M_{|A|, |B|}$ for all $A\in L^r(Y, Z)$and all $B\in L^r(W, X)$. Our results generalize the related results of Synnatzschke and Wickstead, respectively. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.06913

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Abstract of a paper by Jin Xi Chen, Lei Li
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “On a Question of Bouras concerning weak compactness of almost Dunford-Pettis sets” by Jin Xi Chen<https://arxiv.org/find/math/1/au:+Chen_J/0/1/0/all/0/1>, Lei Li<https://arxiv.org/find/math/1/au:+Li_L/0/1/0/all/0/1>. Abstract: We give a positive answer to the question of K. Bouras [`Almost Dunford-Pettis sets in Banach lattices', \textit{Rend. Circ. Mat. Palermo (2)} \textbf{ 62} (2013), 227--236] concerning weak compactness of almost Dunford-Pettis sets in Banach lattices. That is, every almost Dunford-Pettis set in a Banach lattice $E$ is relatively weakly compact if, and only if, $E$ is a $KB$-space. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.06906

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Abstract of a paper by S. H. Kulkarni, G. Ramesh
by Bentuo Zheng (bzheng) 01 Oct '16

01 Oct '16

This is an announcement for the paper “On the denseness of minimum attaining operators” by S. H. Kulkarni<https://arxiv.org/find/math/1/au:+Kulkarni_S/0/1/0/all/0/1>, G. Ramesh<https://arxiv.org/find/math/1/au:+Ramesh_G/0/1/0/all/0/1>. Abstract: Let $H_1, H_2$ be complex Hilbert spaces and $T$ be a densely defined closed linear operator (not necessarily bounded). It is proved that for each $\epsilon>0$, there exists a bounded operator $S$ with $\|S\|<\epsilson$ such that $T+S$ is minimum attaining. Further, if $T$ is bounded below, then $S$ can be chosen to be rank one. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.06869

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Banach October 2016