Banach February 2014

banach@mathdept.okstate.edu

3 participants
26 discussions

Abstract of a paper by Duanxu Dai
by alspach＠math.okstate.edu 21 Feb '14

21 Feb '14

This is an announcement for the paper "Quantifying (weak) injectivity of a Banach space and its second dual" by Duanxu Dai. Abstract: Let $X$, $Y$ be two Banach spaces. Let $\varepsilon\geq 0$. A mapping $f: X\rightarrow Y$ is said a standard $\varepsilon-$ isometry if $f(0)=0$ and $\|f(x)-f(y)\|-\|x-y\|\leq \eps$. In this paper, we first show that if $X$ is a separable Banach space and $Y^*$ has the point of $w^*$-norm continuity property(in short,$w^*$-PCP), then for every standard $\varepsilon-$ isometry $f:X\rightarrow Y$ there exists a $w^*$-dense $G_\delta$ subset $\Omega$ of $ExtB_{X^*}$ such that there is a bounded linear operator $T: Y\rightarrow C(\Omega,\tau_{w^*})$ with $\|T\|=1$ such that $Tf-Id$ is uniformly bounded by $4\eps$ on $X$. More general results are also given. As a corollary, we obtain quantitative characterizations of injectivity, cardinality injectivity and separably injectivity of a Banach space and its second dual which turn out to give a positive answer to Qian's problem of 1995 in the sense of universality. We also discuss Qian's problem in a $\mathcal{L}_{\infty,\lambda}$-space, $C(K)$-space for a compact Hausdorff space $K$. Moreover, by using some results from Avil$\acute{e}$s-S$\acute{a}$nchez-Castillo-Gonz$\acute{a}$lez- Moreno, Cheng-Dong-Zhang, Johnson-Oikhberg, Rosenthal and Lindenstrauss, estimates for several separably injective Banach spaces are given. Finally, we show a more sharp quantitative and generalized Sobczyk 's theorem. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04, 46B20, 47A58, Secondary 26E25, 54C60, 54C65, 46A20 Remarks: 21 page Submitted from: dduanxu(a)163.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.2123 or http://arXiv.org/abs/1402.2123

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Abstract of a paper by Goulnara Arzhantseva and Romain Tessera
by alspach＠math.okstate.edu 21 Feb '14

21 Feb '14

This is an announcement for the paper "Relatively expanding box spaces with no expansion" by Goulnara Arzhantseva and Romain Tessera. Abstract: We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n\trianglelefteq G$ such that for every finite generating subset $S\subseteq G$, the sequence of finite Cayley graphs $(G/N_n, S)$ does not coarsely embed into any $L^p$-space for $1\leqslant p<\infty$ (moreover, into any uniformly curved Banach space), and yet admits no weakly embedded expander. Archive classification: math.GR math.FA math.MG Mathematics Subject Classification: 46B85, 20F69, 22D10, 20E22 Remarks: 20 pages Submitted from: goulnara.arjantseva(a)univie.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.1481 or http://arXiv.org/abs/1402.1481

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Abstract of a paper by Szymon Draga
by alspach＠math.okstate.edu 21 Feb '14

21 Feb '14

This is an announcement for the paper "On weakly locally uniformly rotund norms which are not locally rotund" by Szymon Draga. Abstract: We show that every infinite-dimensional Banach space with separable dual admits an equivalent norm which is weakly locally uniformly rotund but not locally uniformly rotund. Archive classification: math.FA Submitted from: szymon.draga(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.1097 or http://arXiv.org/abs/1402.1097

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Abstract of a paper by Trond A. Abrahamsen, Johann Langemets, Vegard Lima and Olav Nygaard
by alspach＠math.okstate.edu 21 Feb '14

21 Feb '14

This is an announcement for the paper "Observations on thickness and thinness of Banach spaces" by Trond A. Abrahamsen, Johann Langemets, Vegard Lima and Olav Nygaard. Abstract: The aim of this note is to complement and extend some recent results on Whitley's indices of thinness and thickness. As an example we prove that every Banach space $X$ containing a copy of $c_0$ can be equivalently renormed so that we at the same time have that $c_0$ becomes an M-ideal and both the thickness and thinness index of $X$ equal 1. Archive classification: math.FA Remarks: 8 pages Submitted from: trond.a.abrahamsen(a)uia.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.0996 or http://arXiv.org/abs/1402.0996

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Abstract of a paper by Apostolos Giannopoulos and Emanuel Milman
by alspach＠math.okstate.edu 21 Feb '14

21 Feb '14

This is an announcement for the paper "$M$-estimates for isotropic convex bodies and their $L_q$-centroid bodies" by Apostolos Giannopoulos and Emanuel Milman. Abstract: Let $K$ be a centrally-symmetric convex body in $\mathbb{R}^n$ and let $\|\cdot\|$ be its induced norm on ${\mathbb R}^n$. We show that if $K \supseteq r B_2^n$ then: \[ \sqrt{n} M(K) \leqslant C \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \min\left(\frac{1}{r} , \frac{n}{k} \log\Big(e + \frac{n}{k}\Big) \frac{1}{v_{k}^{-}(K)}\right) . \] where $M(K)=\int_{S^{n-1}} \|x\|\, d\sigma(x)$ is the mean-norm, $C>0$ is a universal constant, and $v^{-}_k(K)$ denotes the minimal volume-radius of a $k$-dimensional orthogonal projection of $K$. We apply this result to the study of the mean-norm of an isotropic convex body $K$ in ${\mathbb R}^n$ and its $L_q$-centroid bodies. In particular, we show that if $K$ has isotropic constant $L_K$ then: \[ M(K) \leqslant \frac{C\log^{2/5}(e+ n)}{\sqrt[10]{n}L_K} . \] Archive classification: math.FA Remarks: 19 pages Submitted from: emanuel.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.0904 or http://arXiv.org/abs/1402.0904

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Abstract of a paper by Trond A. Abrahamsen, Johann Langemets, and Vegard Lima
by alspach＠math.okstate.edu 05 Feb '14

05 Feb '14

This is an announcement for the paper "Almost square Banach spaces" by Trond A. Abrahamsen, Johann Langemets, and Vegard Lima. Abstract: We single out and study a natural class of Banach spaces -- almost square Banach spaces. These spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2. We provide several examples and characterizations of almost square spaces. In an almost square space we can find, given a finite set $x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that $\|x_i+y\|$ is almost one. We prove that non-reflexive spaces which are M-ideals in their biduals are almost square. We show that every space containing a copy of $c_0$ can be renormed to be almost square. A local and a weak version of almost square spaces are also studied. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B04, 46B07 Remarks: 22 pages Submitted from: veli(a)hials.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.0818 or http://arXiv.org/abs/1402.0818

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Abstract of a paper by Jarno Talponen
by alspach＠math.okstate.edu 05 Feb '14

05 Feb '14

This is an announcement for the paper "ODE representation for varying exponent $L^p$ norm" by Jarno Talponen. Abstract: We will construct Banach function space norms arising as weak solutions to ordinary differential equations of first order. This provides as a special case a new way of defining varying exponent $L^p$ spaces, different from the Orlicz type approach. It turns out that the duality of these spaces behaves in an anticipated way, same as the uniform convexity and uniform smoothness. Archive classification: math.FA math.CA Mathematics Subject Classification: 46E30, 46B10, 34A12, 31B10 Submitted from: talponen(a)iki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.0528 or http://arXiv.org/abs/1402.0528

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Abstract of a paper by Emanuel Milman
by alspach＠math.okstate.edu 05 Feb '14

05 Feb '14

This is an announcement for the paper "On the mean-width of isotropic convex bodies and their associated $L_p$-centroid bodies" by Emanuel Milman. Abstract: For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropic position, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \] where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropic constant of $K$, and $C>0$ is a universal constant. This improves the previous best-known estimate $M^*(K) \leq C n^{3/4} L_K$. Up to the power of the $\log(n)$ term and the $L_K$ one, the improved bound is best possible, and implies that the isotropic position is (up to the $L_K$ term) an almost $2$-regular $M$-position. The bound extends to any arbitrary position, depending on a certain weighted average of the eigenvalues of the covariance matrix. Furthermore, the bound applies to the mean-width of $L_p$-centroid bodies, extending a sharp upper bound of Paouris for $1 \leq p \leq \sqrt{n}$ to an almost-sharp bound for an arbitrary $p \geq \sqrt{n}$. The question of whether it is possible to remove the $L_K$ term from the new bound is essentially equivalent to the Slicing Problem, to within logarithmic factors in $n$. Archive classification: math.FA Remarks: 14 pages Submitted from: emanuel.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.0209 or http://arXiv.org/abs/1402.0209

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Abstract of a paper by Asuman G. Aksoy and Grzegorz Lewicki
by alspach＠math.okstate.edu 05 Feb '14

05 Feb '14

This is an announcement for the paper "Minimal projections with respect to numerical radius" by Asuman G. Aksoy and Grzegorz Lewicki. Abstract: In this paper we survey some results on minimality of projections with respect to numerical radius. We note that in the cases $L^p$, $p=1,2,\infty$, there is no difference between the minimality of projections measured either with respect to operator norm or with respect to numerical radius. However, we give an example of a projection from $l^p_3$ onto a two-dimensional subspace which is minimal with respect to norm, but not with respect to numerical radius for $p\neq 1,2,\infty$. Furthermore, utilizing a theorem of Rudin and motivated by Fourier projections, we give a criterion for minimal projections, measured in numerical radius. Additionally, some results concerning strong unicity of minimal projections with respect to numerical radius are given. Archive classification: math.FA Mathematics Subject Classification: Primary 41A35, 41A65, Secondary 47A12 Remarks: 15 pages Submitted from: aaksoy(a)cmc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.0032 or http://arXiv.org/abs/1402.0032

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Abstract of a paper by Daniel Galicer and Roman Villafane
by alspach＠math.okstate.edu 05 Feb '14

05 Feb '14

This is an announcement for the paper "Coincidence of extendible vector-valued ideals with their minimal" by Daniel Galicer and Roman Villafane. Abstract: We provide coincidence results for vector-valued ideals of multilinear operators. More precisely, if $\mathfrak A$ is an ideal of $n$-linear mappings we give conditions for which the following equality $\mathfrak A(E_1,\dots,E_n;F) = {\mathfrak A}^{min}(E_1,\dots,E_n;F)$ holds isometrically. As an application, we obtain in many cases that the monomials form a Schauder basis on the space $\mathfrak A(E_1,\dots,E_n;F)$. Several structural and geometric properties are also derived using this equality. We apply our results to the particular case where $\mathfrak A$ is the classical ideal of extendible or Pietsch-integral multilinear operators. Similar statements are given for ideals of vector-valued homogeneous polynomials. For our purposes we also establish a vector-valued version of the Littlewood-Bogdanowicz-Pe{\l}czy\'nski theorem, which we believe is interesting in its own right. Archive classification: math.FA Mathematics Subject Classification: 46G25, 46B22, 46M05, 47H60 Remarks: 25 pages Submitted from: dgalicer(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1401.7896 or http://arXiv.org/abs/1401.7896

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Banach February 2014