Banach

banach@mathdept.okstate.edu

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Abstract of a paper by S.Waleed Noor and Dan Timotin
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Embeddings of M\"{u}ntz spaces: the Hilbertian case" by S.Waleed Noor and Dan Timotin. Abstract: Given a strictly increasing sequence $\Lambda=(\lambda_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty$, the M\"untz spaces $M_\Lambda^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{\lambda_n}$. We discuss properties of the embedding $M_\Lambda^p\subset L^p(\mu)$, where $\mu$ is a finite positive Borel measure on the interval $[0,1]$. Most of the results are obtained for the Hilbertian case $p=2$, in which we give conditions for the embedding to be bounded, compact, or to belong to the Schatten--von Neumann ideals. Archive classification: math.FA math.CA Mathematics Subject Classification: 46E15, 46E20, 46E35 Submitted from: dtimotin(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.5422 or http://arXiv.org/abs/1110.5422

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Abstract of a paper by Ohad Giladi, Assaf Naor, and Gideon Schechtman
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Bourgain's discretization theorem" by Ohad Giladi, Assaf Naor, and Gideon Schechtman. Abstract: Bourgain's discretization theorem asserts that there exists a universal constant $C\in (0,\infty)$ with the following property. Let $X,Y$ be Banach spaces with $\dim X=n$. Fix $D\in (1,\infty)$ and set $\d= e^{-n^{Cn}}$. Assume that $\mathcal N$ is a $\d$-net in the unit ball of $X$ and that $\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem. We also obtain an improvement of Bourgain's theorem in the important case when $Y=L_p$ for some $p\in [1,\infty)$: in this case it suffices to take $\delta= C^{-1}n^{-5/2}$ for the same conclusion to hold true. The case $p=1$ of this improved discretization result has the following consequence. For arbitrarily large $n\in \N$ there exists a family $\mathscr Y$ of $n$-point subsets of $\{1,\ldots,n\}^2\subseteq \R^2$ such that if we write $|\mathscr Y|= N$ then any $L_1$ embedding of $\mathscr Y$, equipped with the Earthmover metric (a.k.a. transportation cost metric or minimumum weight matching metric) incurs distortion at least a constant multiple of $\sqrt{\log\log N}$; the previously best known lower bound for this problem was a constant multiple of $\sqrt{\log\log \log N}$. Archive classification: math.FA math.MG 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/1110.5368 or http://arXiv.org/abs/1110.5368

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Abstract of a paper by Jakub Olejnik
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "On a complete characterization of a.s.\ convergence of multiple orthogonal series" by Jakub Olejnik. Abstract: We present a relation between convergence of multiple and single orthogonal series. This relation implies a complete characterization of all multiple sequences $(a_{n_1\ldots n_d})_{n_1,\ldots,n_d\in\bb N}$ such that for all orthonormal $(\Phi_{n_1\ldots n_d})$ multiple orthogonal series $\sum_{n_1,\ldots,n_d\in\bb N}a_{n_1\ldots n_d}\Phi_{n_1\ldots n_d}$ are a.s.\ convergent. Archive classification: math.FA math.PR Mathematics Subject Classification: 60G60, 60G17 (MSC2010) Submitted from: jakubo(a)math.uni.lodz.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.3942 or http://arXiv.org/abs/1110.3942

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Abstract of a paper by Daniel Galicer, Silvia Lassalle and Pablo Turco
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "The ideal of p-compact operators: a tensor product approach" by Daniel Galicer, Silvia Lassalle and Pablo Turco. Abstract: We study the space of $p$-compact operators $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$ (which is equal to $g_{p'}'$). This allows us to relate the theory of $p$-summing operators with that of $p$-compact operators. With the results known for the former class and appropriate hypothesis on $E$ and $F$ we prove that $\mathcal K_p(E;F)$ is equal to $\mathcal K_q(E;F)$ for a wide range of values of $p$ and $q$, and show that our results are sharp. We also exhibit several structural properties of $\mathcal K_p$. For instance, we obtain that $\mathcal K_p$ is regular, surjective, totally accessible and characterize its maximal hull $\mathcal K_p^{max}$ as the dual ideal of the $p$-summing operators, $\Pi_p^{dual}$. Furthermore, we prove that $\mathcal K_p$ coincides isometrically with $\mathcal {QN}_p^{dual}$, the dual ideal of the quasi $p$-nuclear operators. Archive classification: math.FA Mathematics Subject Classification: 47L20, 46A32, 47B07, 47B10 Remarks: 18 pages Submitted from: paturco(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.3251 or http://arXiv.org/abs/1110.3251

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Abstract of a paper by Valentin Ferenczi and Christian Rosendal
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Displaying Polish groups on separable Banach spaces" by Valentin Ferenczi and Christian Rosendal. Abstract: A display of a topological group G on a Banach space X is a topological isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm ||.|| on X, where the latter group is equipped with the strong operator topology. Displays of Polish groups on separable real spaces are studied. It is proved that any closed subgroup of the infinite symmetric group S_\infty containing a non-trivial central involution admits a display on any of the classical spaces c0, C([0,1]), lp and Lp for 1 <=p <\infty. Also, for any Polsih group G, there exists a separable space X on which {-1,1} x G has a display. Archive classification: math.GR math.FA math.LO Mathematics Subject Classification: 20E08, 03E15, 46B03 Remarks: 27 pages Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.2970 or http://arXiv.org/abs/1110.2970

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Abstract of a paper by Miroslav Kacena, Ondrej F.K. Kalenda and Jiri Spurny
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "Quantitative Dunford-Pettis property" by Miroslav Kacena, Ondrej F.K. Kalenda and Jiri Spurny. Abstract: We investigate possible quantifications of the Dunford-Pettis property. We show, in particular, that the Dunford-Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford-Pettis property. We investigate their relationship with a quantitative Schur property and prove that $L^1$ spaces and $C(K)$ spaces posses both of them. We also show that several natural measures of weak non-compactness are equal in $L^1$ spaces. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B20, 47B07, 47B10 Remarks: 47 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.1243 or http://arXiv.org/abs/1110.1243

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Abstract of a paper by Wieslaw Kubis and Slawomir Solecki
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "A proof of uniqueness of the Gurarii space" by Wieslaw Kubis and Slawomir Solecki. Abstract: We present a short and elementary proof of isometric uniqueness of the Gurarii space. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20 Remarks: 6 pages Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.0903 or http://arXiv.org/abs/1110.0903

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Abstract of a paper by Geraldo Botelho, Daniel Cariello, Vinicius Favaro and Daniel Pellegrino
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "Maximal spaceability in topological vector spaces" by Geraldo Botelho, Daniel Cariello, Vinicius Favaro and Daniel Pellegrino. Abstract: In this paper we introduce a new technique to prove the existence of closed subspaces of maximal dimension inside sets of topological vector sequence spaces. The results we prove cover some sequence spaces not studied before in the context of spaceability and settle some questions on classical sequence spaces that remained open. Archive classification: math.FA Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.6863 or http://arXiv.org/abs/1109.6863

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Abstract of a paper by Jipu Ma
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "A geometry characteristic for Banach space with $c^1$-norm" by Jipu Ma. Abstract: Let $E$ be a Banach space with the $c^1$-norm $\|\cdot\|$ in $ E \backslash \{0\}$ and $S(E)=\{e\in E: \|e\|=1\}.$ In this paper, a geometry characteristic for $E$ is presented by using a geometrical construct of $S(E).$ That is, the following theorem holds : the norm of $E$ is of $c^1$ in $ E \backslash \{0\}$ if and only if $S(E)$ is a $c^1$-submanifold of $E,$ with ${\rm codim}S(E)=1.$ The theorem is very clear, however, its proof is non-trivial, which shows an intrinsic connection between the continuous differentiability of the norm $\|\cdot\|$ in $ E \backslash \{0\}$ and differential structure of $S(E).$ Archive classification: math.FA Mathematics Subject Classification: 54Exx, 46Txx, 58B20 Submitted from: huangql(a)yzu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.6823 or http://arXiv.org/abs/1109.6823

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Abstract of a paper by Anna Pelczar-Barwacz
by alspach＠math.okstate.edu 07 Oct '11

07 Oct '11

This is an announcement for the paper "Strictly singular operators in asymptotic $\ell_p$ Banach spaces" by Anna Pelczar-Barwacz. Abstract: We present condition on higher order asymptotic behaviour of basic sequences in a Banach space ensuring the existence of bounded non-compact strictly singular operator on a subspace. We apply it in asymptotic $\ell_p$ spaces, $1\leq p<\infty$, in particular in convexified mixed Tsirelson spaces and related asymptotic $\ell_p$ HI spaces. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B06 Remarks: 19 pages Submitted from: anna.pelczar(a)im.uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.5874 or http://arXiv.org/abs/1109.5874

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