Banach January 2013

banach@mathdept.okstate.edu

2 participants
28 discussions

Abstract of a paper by Anna Kaminska and Yves Raynaud
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "New formulas for decreasing rearrangements and a class of spaces" by Anna Kaminska and Yves Raynaud. Abstract: Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{\varphi,w}$ and $m_{\varphi,w}$ respectively, introduced earlier in \cite{HKM} for describing the K\"othe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($\varphi$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{\varphi,w}$ classes in the most general setting, where they may even not be linear, and identify their K\"othe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces $\mathcal{M}_{\varphi,w}$ which proves to be the K\"othe biduals of the $M_{\varphi,w}$ classes. In the case when the class $M_{\varphi,w}$ is a separable quasi-Banach space, $\mathcal{M}_{\varphi,w}$ is its Banach envelope. Archive classification: math.FA Mathematics Subject Classification: 26D07, 39B62, 42B25, 46B10, 46E30 Remarks: 25 pages Submitted from: kaminska(a)memphis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4465 or http://arXiv.org/abs/1301.4465

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Abstract of a paper by William B. Johnson and Gideon Schechtman
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Subspaces of $L_p$ that embed into $L_p(\mu)$ with $\mu$ finite" by William B. Johnson and Gideon Schechtman. Abstract: Enflo and Rosenthal proved that $\ell_p(\aleph_1)$, $1 < p < 2$, does not (isomorphically) embed into $L_p(\mu)$ with $\mu$ a finite measure. We prove that if $X$ is a subspace of an $L_p$ space, $1< p < 2$, and $\ell_p(\aleph_1)$ does not embed into $X$, then $X$ embeds into $L_p(\mu)$ for some finite measure $\mu$. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B26, 46B03 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4086 or http://arXiv.org/abs/1301.4086

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Abstract of a paper by Manor Mendel and Assaf Naor
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Spectral calculus and Lipschitz extension for barycentric metric spaces" by Manor Mendel and Assaf Naor. Abstract: The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for $CAT(0)$ targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp. 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/1301.3963 or http://arXiv.org/abs/1301.3963

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Abstract of a paper by Lixin Cheng, Duanxu Dai, Yunbai Dong and Yu Zhou
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "On $\eps$-isometry, isometry and linear isometry" by Lixin Cheng, Duanxu Dai, Yunbai Dong and Yu Zhou. Abstract: Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $\gamma>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$ such that $\|Tf(x)-x\|\leq\gamma\eps$ for all $x\in X$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show first that if such an $\eps$-isometry $f$ exists, then there is a linear isometry $U:X^{**}\rightarrow Y^{**}$. Then we prove that universally- right-stable spaces are just Hilbert spaces; every injective space is universally-left-stable; Finally, we verify that a Banach space $X$ which is linear isomorphic to a subspace of $\ell_\infty$ is universally-left-stable if and only if it is linearly isomorphic to $\ell_\infty$; and a separable space $X$ satisfying that $(X,Y)$ is stable for every separable $Y$ if and only if $X$ is linearly isomorphic to $c_0$. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 47A58 (Primary) 26E25, 46A20, 46A24 (Secondary) Remarks: 14 pages, submitted to Israel Journal of Mathematics Submitted from: dduanxu(a)163.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.3374 or http://arXiv.org/abs/1301.3374

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Abstract of a paper by Duanxu Dai
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "A note on the Cheng-Dong-Zhang Theorem and its applications" by Duanxu Dai. Abstract: In this paper, we first give a short introduction to recent development on the stability of Banach spaces via $\eps$-isometry and then present an application of the Cheng-Dong-Zhang Theorem to the continuous selections of a set valued map via $\eps-$ isometries. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 54C60 (Primary) 26E25, 46A20, 54C65 (Secondary) Remarks: 7 pages Submitted from: dduanxu(a)163.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.3396 or http://arXiv.org/abs/1301.3396

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Abstract of a paper by Lingxin Bao, Lixin Cheng, Qingjin Cheng and Duanxu Dai
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "On universally-left-stability of Banach spaces for $\varepsilon$-isometries" by Lingxin Bao, Lixin Cheng, Qingjin Cheng and Duanxu Dai. Abstract: Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $\gamma>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$ such that $\|Tf(x)-x\|\leq\gamma\eps$ for all $x\in X$. $X (Y)$ is said to be left (right)-universally stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that if a dual Banach space $X$ is universally-left-stable, then it is isometric to a complemented $w^*$-closed subspace of $\ell_\infty(\Gamma)$ for some set $\Gamma$, hence, an injective space; and that a Banach space is universally-left-stable if and only if it is a cardinality injective space. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 47A58 (Primary) 26E25, 46A20, 46A24 (Secondary) Remarks: 10 pages, submitted to Acta Mathematica Sinica, English Series Submitted from: dduanxu(a)163.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.3656 or http://arXiv.org/abs/1301.3656

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Abstract of a paper by Patrick J. Rabier
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Differentiability of quasiconvex functions on separable Banach spaces" by Patrick J. Rabier. Abstract: We investigate the differentiability properties of real-valued quasiconvex functions f defined on a separable Banach space X. Continuity is only assumed to hold at the points of a dense subset. If so, this subset is automatically residual. Sample results that can be quoted without involving any new concept or nomenclature are as follows: (i) If f is usc or strictly quasiconvex, then f is Hadamard differentiable at the points of a dense subset of X (ii) If f is even, then f is continuous and Gateaux differentiable at the points of a dense subset of X. In (i) or (ii), the dense subset need not be residual but, if X is also reflexive, it contains the complement of a Haar null set. Furthermore, (ii) remains true without the evenness requirement if the definition of Gateaux differentiability is generalized in an unusual, but ultimately natural, way. The full results are much more general and substantially stronger. In particular, they incorporate the well known theorem of Crouzeix, to the effect that every real-valued quasiconvex function on R^N is Frechet differentiable a.e. Archive classification: math.OC math.FA Submitted from: rabier(a)imap.pitt.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.2852 or http://arXiv.org/abs/1301.2852

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Abstract of a paper by Mathieu Meyer, Carsten Schuett, and Elisabeth M. Werner
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Affine invariant points" by Mathieu Meyer, Carsten Schuett, and Elisabeth M. Werner. Abstract: We answer in the negative a question by Gruenbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Gruenbaum about the "size" of the set of all affine invariant points. Related, we show that the set of all convex bodies K, for which the set of affine invariant points is all of n-dimensional Euclidean space, is dense in the set of convex bodies. Crucial to establish these results, are new affine invariant points, not previously considered in the literature. Archive classification: math.FA Submitted from: elisabeth.werner(a)case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.2606 or http://arXiv.org/abs/1301.2606

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Abstract of a paper by Mark Rudelson
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Lecture notes on non-asymptotic theory of random matrices" by Mark Rudelson. Abstract: We discuss recent developments in the study of the spectral properties of random matrices of a large fixed size, concentrating on the extreme singular values. Bounds for the extreme singular values were crucial in establishing several limit laws of random matrix theory. Besides the random matrix theory itself, these bounds have applications in geometric functional analysis and computer science. Archive classification: math.PR math.FA Mathematics Subject Classification: 60B20 Remarks: Lecture notes from the AMS short course on random matrices, 44 Submitted from: rudelson(a)umich.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.2382 or http://arXiv.org/abs/1301.2382

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Abstract of a paper by Hossein Dehghan
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "A characterization of inner product spaces related to the distance" by Hossein Dehghan. Abstract: A new refinement of the triangle inequality is presented in normed linear spaces. Moreover, a simple characterization of inner product spaces is obtained by using the skew-angular distance. Archive classification: math.FA Remarks: To appear in Math. Notes Submitted from: h_dehghan(a)iasbs.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.1001 or http://arXiv.org/abs/1301.1001

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Banach January 2013