Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Florent Pierre Baudier
by alspach＠math.okstate.edu 26 Nov '14

26 Nov '14

This is an announcement for the paper "On the $(\beta)$-distortion of countably branching hyperbolic trees" by Florent Pierre Baudier. Abstract: In this note we show that the distortion incurred by a bi-Lipschitz embedding of the countably branching hyperbolic tree of height $N$ into a Banach space admitting a norm satisfying Rolewicz property $(\beta)$ with power type $p>1$ is at least of the order of $\log(N)^{1/p}$. An application of our result gives a quantitative version of the non-embeddability of countably branching hyperbolic trees into reflexive Banach spaces admitting an equivalent asymptotically uniformly smooth norm and an equivalent asymptotically uniformly convex norm from Baudier, Kalton and Lancien. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B20, 46B85 Remarks: 5 pages Submitted from: florent(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.3915 or http://arXiv.org/abs/1411.3915

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Abstract of a paper by Tengiz Kopaliani, Nino Samashvili and Shalva Zviadadze
by alspach＠math.okstate.edu 26 Nov '14

26 Nov '14

This is an announcement for the paper "On the upper and lower estimates of norms in variable exponent spaces" by Tengiz Kopaliani, Nino Samashvili and Shalva Zviadadze. Abstract: In the present paper we investigate some geometrical properties of the norms in Banach function spaces. Particularly there is shown that if exponent $1/p(\cdot)$ belongs to $BLO^{1/\log}$ then for the norm of corresponding variable exponent Lebesgue space we have the following lower estimate $$\left\|\sum \chi_{Q}\|f\chi_{Q}\|_{p(\cdot)}/\|\chi_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}\leq C\|f\|_{p(\cdot)}$$ where $\{Q\}$ defines disjoint partition of $[0;1]$. Also we have constructed variable exponent Lebesgue space with above property which does not possess following upper estimation $$\|f\|_{p(\cdot)}\leq C\left\|\sum \chi_{Q}\|f\chi_{Q}\|_{p(\cdot)}/\|\chi_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}. $$ Archive classification: math.FA Mathematics Subject Classification: 42B35, 42B20, 46B45, 42B25 Remarks: 13 pages Submitted from: sh.zviadadze(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.3461 or http://arXiv.org/abs/1411.3461

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 26 Nov '14

26 Nov '14

This is an announcement for the paper "Metric characterizations of some classes of Banach spaces" by Mikhail I. Ostrovskii. Abstract: The main purpose of the paper is to present some recent results on metric characterizations of superreflexivity and the Radon-Nikod\'ym property. Archive classification: math.FA math.MG Mathematics Subject Classification: Primary 46B85, Secondary: 05C12, 20F67, 30L05, 46B07, 46B22 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.3366 or http://arXiv.org/abs/1411.3366

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Abstract of a paper by Marek Cuth and Michal Doucha
by alspach＠math.okstate.edu 26 Nov '14

26 Nov '14

This is an announcement for the paper "Lipschitz-free spaces over ultrametric spaces" by Marek Cuth and Michal Doucha. Abstract: We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to $\ell_1$. This extends results of A. Dalet using an alternative approach. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B15, 54E35 Submitted from: marek.cuth(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.2434 or http://arXiv.org/abs/1411.2434

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Abstract of a paper by Kasper Green Larsen and Jelani Nelson
by alspach＠math.okstate.edu 26 Nov '14

26 Nov '14

This is an announcement for the paper "The Johnson-Lindenstrauss lemma is optimal for linear dimensionality reduction" by Kasper Green Larsen and Jelani Nelson. Abstract: For any $n>1$ and $0<\varepsilon<1/2$, we show the existence of an $n^{O(1)}$-point subset $X$ of $\mathbb{R}^n$ such that any linear map from $(X,\ell_2)$ to $\ell_2^m$ with distortion at most $1+\varepsilon$ must have $m = \Omega(\min\{n, \varepsilon^{-2}\log n\})$. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma, improving the previous lower bound of Alon by a $\log(1/\varepsilon)$ factor. Archive classification: cs.IT cs.CG cs.DS math.FA math.IT Submitted from: minilek(a)seas.harvard.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.2404 or http://arXiv.org/abs/1411.2404

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Abstract of a paper by Daher Mohammad
by alspach＠math.okstate.edu 26 Nov '14

26 Nov '14

This is an announcement for the paper "K(X,Y) as subspace complemented of L(X,Y)" by Daher Mohammad. Abstract: Let X,Y be two Banach spaces ; in the first part of this work, we show that K(X,Y) contains a complemented copy of c0 if Y contains a copy of c0 and each bounded sequence in Y has a subsequece which is w* convergente. Afterward we obtain some results of M.Feder and G.Emmanuele: Finally in this part we study the relation between the existence of projection from L(X,Y) on K(X,Y) and the existence of pro- jection from K(X,Y ) on K(X,Y) if Y has the approximation property. In the second part we study the Radon-Nikodym property in L(X,Y): Archive classification: math.FA Mathematics Subject Classification: 46EXX Remarks: 21 pages Submitted from: m.daher(a)orange.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.2217 or http://arXiv.org/abs/1411.2217

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Abstract of a paper by Debmalya Sain, Kallol Paul and Kanhaiya Jha
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Strictly convex space : Strong orthogonality and conjugate diameters" by Debmalya Sain, Kallol Paul and Kanhaiya Jha. Abstract: In a normed linear space X an element x is said to be orthogonal to another element y in the sense of Birkhoff-James, written as $ x \perp_{B}y, $ iff $ \| x \| \leq \| x + \lambda y \| $ for all scalars $ \lambda.$ We prove that a normed linear space X is strictly convex iff for any two elements x, y of the unit sphere $ S_X$, $ x \perp_{B}y $ implies $ \| x + \lambda y \| > 1~ \forall~ \lambda \neq 0. $ We apply this result to find a necessary and sufficient condition for a Hamel basis to be a strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real strictly convex space X. Applying the result we give an estimation for lower bounds of $ \| tx+(1-t)y\|, t \in [0,1] $ and $ \| y + \lambda x \|, ~\forall ~\lambda $ for all elements $ x,y \in S_X $ with $ x \perp_B y. $ We find a necessary and sufficient condition for the existence of conjugate diameters through the points $ e_1,e_2 \in ~S_X $ in a real strictly convex space of dimension 2. The concept of generalized conjuagte diameters is then developed for a real strictly convex smooth space of finite dimension. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 47A30 Submitted from: kalloldada(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.1464 or http://arXiv.org/abs/1411.1464

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Abstract of a paper by Daniel Pellegrino and Joedson Santos
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Lineability and uniformly dominated sets of summing nonlinear" by Daniel Pellegrino and Joedson Santos. Abstract: In this note we prove an abstract version of a result from 2002 due to Delgado and Pi\~{n}ero on absolutely summing operators. Several applications are presented; some of them in the multilinear framework and some in a completely nonlinear setting. In a final section we investigate the size of the set of non uniformly dominated sets of linear operators under the point of view of lineability. Archive classification: math.FA Submitted from: pellegrino(a)pq.cnpq.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.1100 or http://arXiv.org/abs/1411.1100

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Abstract of a paper by Nigel Kalton and Lutz Weis
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "The $H^{\infty}$--functional calculus and square function estimates" by Nigel Kalton and Lutz Weis. Abstract: Using notions from the geometry of Banach spaces we introduce square functions $\gamma(\Omega,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(\Omega,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $\gamma(\Omega,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e.~g., that a $c_0$--group of operators $T_s$ on a Banach space with finite cotype has an $H^{\infty}$--calculus on a strip if and only if $e^{-a|s|}T_s$ is $R$--bounded for some $a > 0$. Similarly, a sectorial operator $A$ has an $H^{\infty}$--calculus on a sector if and only if $A$ has $R$--bounded imaginary powers. We also consider vector valued Paley--Littlewood $g$--functions on $UMD$--spaces. Archive classification: math.FA Submitted from: Lutz.weis(a)kit.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.0472 or http://arXiv.org/abs/1411.0472

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Abstract of a paper by Daniel Azagra and Carlos Mudarra
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Global approximation of convex functions by differentiable convex functions on Banach spaces" by Daniel Azagra and Carlos Mudarra. Abstract: We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by $C^k$ smooth convex functions. Archive classification: math.FA Mathematics Subject Classification: 46B20, 52A99, 26B25, 41A30 Remarks: 8 pages Submitted from: dazagra(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.0471 or http://arXiv.org/abs/1411.0471

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