Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Petr Hajek and Thomas Schlumprecht
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "The Szlenk index of L_p(X)" by Petr Hajek and Thomas Schlumprecht. Abstract: We find an optimal upper bound on the values of the weak$^*$-dentability index $Dz(X)$ in terms of the Szlenk index $Sz(X)$ of a Banach space $X$ with separable dual. Namely, if $\;Sz(X)=\omega^{\alpha}$, for some $\alpha<\omega_1$, and $p\in(1,\infty)$, then $$Sz(X)\le Dz(X)\le Sz(L_p(X))\le \begin{cases} \omega^{\alpha+1} &\text{ if $\alpha$ is a finite ordinal,} \omega^{\alpha} &\text{ if $\alpha$ is an infinite ordinal.} \end{cases}$$ Archive classification: math.FA Mathematics Subject Classification: 46B03 46B10 Submitted from: schlump(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.3629 or http://arXiv.org/abs/1308.3629

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Abstract of a paper by Assaf Naor
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Comparison of metric spectral gaps" by Assaf Naor. Abstract: Let $A=(a_{ij})\in M_n(\R)$ be an $n$ by $n$ symmetric stochastic matrix. For $p\in [1,\infty)$ and a metric space $(X,d_X)$, let $\gamma(A,d_X^p)$ be the infimum over those $\gamma\in (0,\infty]$ for which every $x_1,\ldots,x_n\in X$ satisfy $$ \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n d_X(x_i,x_j)^p\le \frac{\gamma}{n}\sum_{i=1}^n\sum_{j=1}^n a_{ij} d_X(x_i,x_j)^p. $$ Thus $\gamma(A,d_X^p)$ measures the magnitude of the {\em nonlinear spectral gap} of the matrix $A$ with respect to the kernel $d_X^p:X\times X\to [0,\infty)$. We study pairs of metric spaces $(X,d_X)$ and $(Y,d_Y)$ for which there exists $\Psi:(0,\infty)\to (0,\infty)$ such that $\gamma(A,d_X^p)\le \Psi\left(\gamma(A,d_Y^p)\right)$ for every symmetric stochastic $A\in M_n(\R)$ with $\gamma(A,d_Y^p)<\infty$. When $\Psi$ is linear a complete geometric characterization is obtained. Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if $n\in \N$ and $p\in (2,\infty)$ then for every $f_1,\ldots,f_n\in L_p$ there exist $x_1,\ldots,x_n\in L_2$ such that \begin{equation}\label{eq:p factor} \forall\, i,j\in \{1,\ldots,n\},\quad \|x_i-x_j\|_2\lesssim p\|f_i-f_j\|_p, \end{equation} and $$ \sum_{i=1}^n\sum_{j=1}^n \|x_i-x_j\|_2^2=\sum_{i=1}^n\sum_{j=1}^n \|f_i-f_j\|_p^2. $$ This statement is impossible for $p\in [1,2)$, and the asymptotic dependence on $p$ in~\eqref{eq:p factor} is sharp. We also obtain the best known lower bound on the $L_p$ distortion of Ramanujan graphs, improving over the work of Matou\v{s}ek. Links to Bourgain--Milman--Wolfson type and a conjectural nonlinear Maurey--Pisier theorem are studied. Archive classification: math.MG math.FA Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.2851 or http://arXiv.org/abs/1308.2851

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Abstract of a paper by Alon Dmitriyuk and Yehoram Gordon
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Large distortion dimension reduction using random variable" by Alon Dmitriyuk and Yehoram Gordon. Abstract: Consider a random matrix $H:\mathbb{R}^n\longrightarrow\mathbb{R}^m$. Let $D\geq2$ and let $\{W_l\}_{l=1}^{p}$ be a set of $k$-dimensional affine subspaces of $\mathbb{R}^n$. We ask what is the probability that for all $1\leq l\leq p$ and $x,y\in W_l$, \[ \|x-y\|_2\leq\|Hx-Hy\|_2\leq D\|x-y\|_2. \] We show that for $m=O\big(k+\frac{\ln{p}}{\ln{D}}\big)$ and a variety of different classes of random matrices $H$, which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $m$ is tight in terms of $k,p,D$. Archive classification: math.FA Remarks: 18 pages Submitted from: gordon(a)techunix.technion.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.2768 or http://arXiv.org/abs/1308.2768

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Abstract of a paper by Ludek Zajicek
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces" by Ludek Zajicek. Abstract: The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\R^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f^+_H (x,v)$, $f^-_H (x,v)$ ($v \in X$) which holds almost everywhere for an arbitrary function $f:\R^2\to \R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces. Archive classification: math.FA Mathematics Subject Classification: Primary: 46G05, Secondary: 26B05 Submitted from: zajicek(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.2415 or http://arXiv.org/abs/1308.2415

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Abstract of a paper by Enrico Boasso
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "On the Moore-Penrose inverse, EP Banach space operators and EP Banach algebra elements" by Enrico Boasso. Abstract: The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and $C^*$-algebra elements. Archive classification: math.FA Mathematics Subject Classification: Primary 15A09, Secondary 47A05 Citation: J. Math. Anal. Appl. 339(2) (2008), 1003-1014 Remarks: 20 pages, original research article Submitted from: eboasso(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.1897 or http://arXiv.org/abs/1308.1897

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Abstract of a paper by Marek Cuth and Ondrej F.K. Kalenda
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Rich families and elementary submodels" by Marek Cuth and Ondrej F.K. Kalenda. Abstract: We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system an in spaces of density $\aleph_1$. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality indexed by ranges of its projections. Archive classification: math.FA Mathematics Subject Classification: 46B26, 03C30 Submitted from: cuthm5am(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.1818 or http://arXiv.org/abs/1308.1818

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Abstract of a paper by Sun Kwang Kim and Han Ju Lee
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Simultaneously continuous retraction and its application" by Sun Kwang Kim and Han Ju Lee. Abstract: We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ topology. Such a retraction is called a uniformly simultaneously continuous retraction. It is shown that if $X$ has a normalized unconditional Schauder basis with unconditional basis constant 1 and $X^*$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$ exists. It is also shown that if $\{X_i\}$ is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity $\delta_i(\eps)$ such that $\inf_i \delta_i(\eps)>0$ and $X= \left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^*$ onto $B_{X^*}$. The relation between the existence of a uniformly simultaneously continuous retraction and the Bishsop-Phelps-Bollob\'as property for operators is investigated and it is proved that the existence of a uniformly simultaneously continuous retraction from $X^*$ onto its unit ball implies that a pair $(X, C_0(K))$ has the Bishop-Phelps-Bollob\'as property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the Bishop-Phelps-Bollob\'as property if $C_0(S)$ and $C_0(K)$ are the spaces of all real-valued continuous functions vanishing at infinity on locally compact metric space $S$ and locally compact Hausdorff space $K$ respectively. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 46B04, 46B22 Remarks: 15 pages Submitted from: hanjulee(a)dongguk.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.1638 or http://arXiv.org/abs/1308.1638

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Abstract of a paper by Horst Martini, Pier Luigi Papini, and Margarita Spirova
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Complete sets and completion of sets in Banach spaces" by Horst Martini, Pier Luigi Papini, and Margarita Spirova. Abstract: In this paper we study properties of complete sets and of completions of sets in Banach spaces. We consider the family of completions of a given set and its size; we also study in detail the relationships concerning diameters, radii, and centers. The results are illustrated by several examples. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B20, 46B99, 52A05, 52A20, 52A21 Submitted from: margarita.spirova(a)mathematik.tu-chemnitz.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.0789 or http://arXiv.org/abs/1308.0789

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Abstract of a paper by Sonia Berrios and Geraldo Botelho
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "A general abstract approach to approximation properties in Banach" by Sonia Berrios and Geraldo Botelho. Abstract: We propose a unifying approach to many approximation properties studied in the literature from the 1930s up to our days. To do so, we say that a Banach space E has the (I,J,{\tau})-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the topology {\tau}, by operators belonging to the operator ideal J. Restricting {\tau} to a class of linear topologies, which we call ideal topologies, this concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general results that are valid in this abstract framework. Archive classification: math.FA Submitted from: botelho(a)ufu.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1307.8073 or http://arXiv.org/abs/1307.8073

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Abstract of a paper by Charles John Read
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Banach spaces with no proximinal subspaces of codimension 2" by Charles John Read. Abstract: The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving functionals have to be: if (x,f) is in X x X* with ||x||=||f||=1 and |1-f(x)|< h^2/4 then there are (x',f') in X x X* with ||x'||= ||f'||=1, ||x-x'||, ||f-f'||< h and f'(x')=1. This means that there are always "proximinal" hyperplanes H in X (a nonempty subset E of a metric space is said to be "proximinal" if, for x not in E, the distance d(x,E) is always achieved - there is always an e in E with d(x,E)=d(x,e)); for if H= ker f (f in X*) then it is easy to see that H is proximinal if and only if f is norm-achieving. Indeed the set of proximinal hyperplanes H is, in the appropriate sense, dense in the set of all closed hyperplanes H in X. Quite a long time ago [Problem 2.1 in his monograph "The Theory of Best approximation and Functional Analysis" Regional Conference series in Applied Mathematics, SIAM, 1974], Ivan Singer asked if this result generalized to closed subspaces of finite codimension - if every Banach space has a proximinal subspace of codimension 2, for example. In this paper I show that there is a Banach space X such that X has no proximinal subspace of finite codimension n>1. So we have a converse to Bishop-Phelps-Bollobas: a dense set of proximinal hyperplanes can always be found, but proximinal subspaces of larger, finite codimension need not be. Archive classification: math.FA Mathematics Subject Classification: 46B04 (Primary), 46B45, 46B25 (Secondary) Remarks: The paper has been submitted for publication to the Israel Journal of The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1307.7958 or http://arXiv.org/abs/1307.7958

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