Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Bernd Carl, Aicke Hinrichs, and Philipp Rudolph
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Entropy numbers of convex hulls in Banach spaces and applications" by Bernd Carl, Aicke Hinrichs, and Philipp Rudolph. Abstract: Entropy numbers and Kolmogorov numbers of convex hulls in Banach spaces are studied. Applications are given. Archive classification: math.FA Mathematics Subject Classification: 41A46, 46B20, 47B06, 46B50 Submitted from: a.hinrichs(a)uni-jena.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.1559 or http://arXiv.org/abs/1211.1559

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Abstract of a paper by Gilles Pisier
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Quantum expanders and geometry of operator spaces II" by Gilles Pisier. Abstract: In this appendix to our paper with the same title posted on arxiv we give a quick proof of an inequality that can be substituted to Hastings's result, quoted as Lemma 1.9 in our previous paper. Our inequality is less sharp but also appears to apply with more general (and even matricial) coefficients. It shows that up to a universal constant all moments of the norm of a linear combination of the form $$S=\sum\nolimits_j a_j U_j \otimes \bar U_j (1-P)$$ are dominated by those of the corresponding Gaussian sum $$S'=\sum\nolimits_j a_j Y_j \otimes \bar Y'_j .$$ The advantage is that $S'$ is now simply separately a Gaussian random variable with respect to the independent Gaussian random matrices $(Y_j)$ and $(Y'_j)$, and hence is much easier to majorize. Note we plan to incorporate this appendix into our future publication. Archive classification: math.OA Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.1055 or http://arXiv.org/abs/1211.1055

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Abstract of a paper by Constantinos Kardaras
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Uniform integrability and local convexity in $L^0$" by Constantinos Kardaras. Abstract: Let $L^0$ be the vector space of all (equivalence classes of) real-valued random variables built over a probability space $(\Omega, \mathcal{F}, P)$, equipped with a metric topology compatible with convergence in probability. In this work, we provide a necessary and sufficient structural condition that a set $X \subseteq L^0$ should satisfy in order to infer the existence of a probability $Q$ that is equivalent to $P$ and such that $X$ is uniformly $Q$-integrable. Furthermore, we connect the previous essentially measure-free version of uniform integrability with local convexity of the $L^0$-topology when restricted on convex, solid and bounded subsets of $L^0$. Archive classification: math.FA math.PR Remarks: 14 pages Submitted from: langostas(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.0475 or http://arXiv.org/abs/1211.0475

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Abstract of a paper by Shanwen Hu
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Least squares problems in orthornormalization" by Shanwen Hu. Abstract: For any $n$-tuple $(\alpha_1,\cdots,\alpha_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $(\epsilon_1,\cdots,\epsilon_n)$ of $span\{\alpha_1,\cdots,\alpha_n\}$ satisfying: $$\sum_{i=1}^n\|\epsilon_i-\alpha_i\|^2\le\sum_{i=1}^n\|\beta_i-\alpha_i\|^2$$ for all orthonormal basis $(\beta_1,\cdots,\beta_n)$ of $span\{\alpha_1,\cdots,\alpha_n\}$. We study the stability of the orthornormalization and give some applications and examples. Archive classification: math.FA Remarks: 10 pages Submitted from: swhu(a)math.ecnu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1210.7400 or http://arXiv.org/abs/1210.7400

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Abstract of a paper by Dongni Tan, Xujian Huang, and Rui Liu
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Generalized-Lush Spaces and the Mazur-Ulam Property" by Dongni Tan, Xujian Huang, and Rui Liu. Abstract: We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (specially, separable $C$-rich subspaces of $C(K)$), and even the two-dimensional space with hexagonal norm. We obtain that the space $C(K,E)$ of the vector-valued continuous functions is a GL-space whenever $E$ is, and show that the GL-space is stable under $c_0$-, $l_1$- and $l_\infty$-sums. As an application, we prove that the Mazur-Ulam property holds for a larger class of Banach spaces, called local-GL-spaces, including all lush spaces and GL-spaces. Furthermore, we generalize the stability properties of GL-spaces to local-GL-spaces. From this, we can obtain many examples of Banach spaces having the Mazur-Ulam property. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04, Secondary 46B20, 46A22 Remarks: 16 pages Submitted from: ruiliu(a)nankai.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1210.7324 or http://arXiv.org/abs/1210.7324

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Abstract of a paper by Enrique A. Sanchez Perez and Dirk Werner
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Slice continuity for operators and the Daugavet property for bilinear maps" by Enrique A. Sanchez Perez and Dirk Werner. Abstract: We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04, secondary 46B25 Submitted from: werner(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1210.7099 or http://arXiv.org/abs/1210.7099

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Abstract of a paper by J. Lopez-Abad
by alspach＠math.okstate.edu 26 Oct '12

26 Oct '12

This is an announcement for the paper "A Bourgain-Pisier construction for general Banach spaces" by J. Lopez-Abad. Abstract: We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $\mathcal L_{\infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain $\mathcal L_\infty$ spaces of arbitrary large densities with the Schur and the Radon-Nikodym properties. This extents the a classical result by J. Bourgain and G. Pisier. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B26 Submitted from: abad(a)icmat.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1210.5728 or http://arXiv.org/abs/1210.5728

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Abstract of a paper by Rafa Espinola and Miguel Lacruz
by alspach＠math.okstate.edu 26 Oct '12

26 Oct '12

This is an announcement for the paper "Applications of fixed point theorems in the theory of invariant subspaces" by Rafa Espinola and Miguel Lacruz. Abstract: We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space. Archive classification: math.OA Mathematics Subject Classification: 47A15, 47H10 Remarks: 13 pages, to appear in Fixed Point Theory and Applications Submitted from: lacruz(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1210.5557 or http://arXiv.org/abs/1210.5557

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Abstract of a paper by Ludek Zajicek
by alspach＠math.okstate.edu 26 Oct '12

26 Oct '12

This is an announcement for the paper "Hadamard differentiability via Gateaux differentiability" by Ludek Zajicek. Abstract: Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ a mapping. We prove that there exists a $\sigma$-directionally porous set $A\subset X$ such that if $x\in X \setminus A$, $f$ is Lipschitz at $x$, and $f$ is G\^ateaux differentiable at $x$, then $f$ is Hadamard differentiable at $x$. If $f$ is Borel measurable (or has the Baire property) and is G\^ ateaux differentiable at all points, then $f$ is Hadamard differentiable at all points except a set which is $\sigma$-directionally porous set (and so is Aronszajn null, Haar null and $\Gamma$-null). Consequently, an everywhere G\^ ateaux differentiable $f: \R^n \to Y$ is Fr\' echet differentiable except a nowhere dense $\sigma$-porous set. Archive classification: math.FA Mathematics Subject Classification: Primary: 46G05, Secondary: 26B05, 49J50 Remarks: 9 pages 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/1210.4715 or http://arXiv.org/abs/1210.4715

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Abstract of a paper by Vitali Milman and Liran Rotem
by alspach＠math.okstate.edu 26 Oct '12

26 Oct '12

This is an announcement for the paper "Mixed integrals and related inequalities" by Vitali Milman and Liran Rotem. Abstract: In this paper we define an addition operation on the class of quasi-concave functions. While the new operation is similar to the well-known sup-convolution, it has the property that it polarizes the Lebesgue integral. This allows us to define mixed integrals, which are the functional analogs of the classic mixed volumes. We extend various classic inequalities, such as the Brunn-Minkowski and the Alexandrov-Fenchel inequality, to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to log-concave functions, we prove generalizations of the Alexandrov inequalities in a more familiar form. Archive classification: math.FA math.MG Mathematics Subject Classification: 52A39, 26B25 Remarks: 30 pages Submitted from: liranro1(a)post.tau.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1210.4346 or http://arXiv.org/abs/1210.4346

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