Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Dong Hoon Cho and Yun Sung Choi
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Bishop-Phelps-Bolloba's theorem on bounded closed convex sets" by Dong Hoon Cho and Yun Sung Choi. Abstract: This paper deals with the \emph{Bishop-Phelps-Bollob\'as property} (\emph{BPBp} for short) on bounded closed convex subsets of a Banach space $X$, not just on its closed unit ball $B_X$. We firstly prove that the \emph{BPBp} holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces $X$ and $Y$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed convex subset $D$ of $X$, and also that for a Banach space $Y$ with property $(\beta )$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of an arbitrary Banach space $X$. For a bounded closed absorbing convex subset $D$ of $X$ with positive modulus convexity we get that the pair $(X,Y)$ has the \emph{BPBp} on $D$ for every Banach space $Y$. We further obtain that for an Asplund space $X$ and for a locally compact Hausdorff $L$, the pair $(X, C_0(L))$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of $X$. Finally we study the stability of the \emph{BPBp} on a bounded closed convex set for the $\ell_1$-sum or $\ell_{\infty}$-sum of a family of Banach spaces. Archive classification: math.FA Submitted from: meimi200(a)postech.ac.kr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1409.3008 or http://arXiv.org/abs/1409.3008

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Abstract of a paper by S Dutta and D Khurana
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Ordinal indices of small subspaces of $L_p$" by S Dutta and D Khurana. Abstract: We calculate ordinal $L_p$ index defined in "An ordinal L_p index for Banach spaces with an application to complemented subspaces of L_p" authored by J. Bourgain, H. P. Rosenthal and G. Schechtman, for Rosenthal's space $X_p$, $\ell_p$ and $\ell_2$. We show a subspace of $L_p$ $(2 < p < \infty)$ non isomorphic to $\ell_2$ embeds in $\ell_p$ if and only if its ordinal index is minimum possible. We also give a sufficient condition for a $\mathcal{L}_p$ subspace of $\ell_p\oplus\ell_2$ to be isomorphic to $X_p$. Archive classification: math.FA Submitted from: divyakhurana11(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1409.2330 or http://arXiv.org/abs/1409.2330

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Abstract of a paper by Oleg Reinov
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "O-frames for operators in Banach spaces" by Oleg Reinov. Abstract: These notes are formal. Here, in this abstract, not in the note, we should say that all that is in the text was done, essentially, by Aleksander Pe{\l}czy\'nski. BUT: Anyhow, a new notion of an O-frame for an operator is introduced. For the operators in separable spaces, it is shown that a operator has an O-frame iff it has the BAP iff it can be factored through a Banach space with a basis. Applications are given. However, looking around, I'd say that, e.g., a notion of a Banach frame (and also O-frame) was implicitely introduced by great Aleksander Pe{\l}czy\'nski. Archive classification: math.FA Mathematics Subject Classification: 46B28 Remarks: 11 pages, was as a SPb Math. Soc. preprint, in RUSSIAN! Submitted from: orein51(a)mail.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1409.0165 or http://arXiv.org/abs/1409.0165

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Abstract of a paper by Valentin Ferenczi and Christian Rosendal
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Non-unitarisable representations and maximal symmetry" by Valentin Ferenczi and Christian Rosendal. Abstract: We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of ${\rm Aut}(T)$ on $\ell_2(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of ${\rm GL}(\mathcal H)$ extending a well-known non-unitarisable representation of $\mathbb F_\infty$. As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex. Archive classification: math.FA Submitted from: rosendal.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1409.0141 or http://arXiv.org/abs/1409.0141

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Abstract of a paper by Olav Nygaard
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Uniform boundedness deciding sets, and a problem of M. Valdivia" by Olav Nygaard. Abstract: We prove that if a set $B$ in a Banach space $X$ can be written as an increasing, countable union $B=\cup_n B_n$ of sets $B_n$ such that no $B_n$ is uniform boundedness deciding, then also $B$ is not uniform boundedness deciding. From this we can give a positive answer to a question of M. Valdivia. Archive classification: math.FA Remarks: 5 pages Submitted from: olav.nygaard(a)uia.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1409.0102 or http://arXiv.org/abs/1409.0102

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Abstract of a paper by I. Asekritova, N. Kruglyak and M. Mastylo
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Interpolation of Fredholm operators" by I. Asekritova, N. Kruglyak and M. Mastylo. Abstract: We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $\theta \in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{\theta ,q}$ to $(Y_{0},Y_{1})_{\theta ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corresponding indices generated by the $K$-functional of elements from the kernel of the operator $A$ in the interpolation sum $X_{0}+X_{1}$. If in addition $q\in \lbrack 1,\infty )$ and $A$ is invertible operator on endpoint spaces, then these conditions are also necessary. We apply these results to obtain and present an affirmative solution of the famous Lions-Magenes problem on the real interpolation of closed subspaces. We also discuss some applications to the spectral theory of operators as well as to perturbation of the Hardy operator by identity on weighted $L_{p}$-spaces. Archive classification: math.FA Submitted from: mastylo(a)amu.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.7024 or http://arXiv.org/abs/1408.7024

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Abstract of a paper by Carlos H. Jimenez and Ignacio Villanueva
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Characterization of dual mixed volumes via polymeasures" by Carlos H. Jimenez and Ignacio Villanueva. Abstract: We prove a characterization of the dual mixed volume in terms of functional properties of the polynomial associated to it. To do this, we use tools from the theory of multilinear operators on spaces of continuos functions. Along the way we reprove, with these same techniques, a recently found characterization of the dual mixed volume. Archive classification: math.FA Submitted from: ignaciov(a)mat.ucm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.6796 or http://arXiv.org/abs/1408.6796

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Abstract of a paper by S. J. Dilworth, Denka Kutzarova, and N. Lovasoa Randrianarivony
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "The transfer of property $(\beta)$ of Rolewicz by a uniform quotient" by S. J. Dilworth, Denka Kutzarova, and N. Lovasoa Randrianarivony. Abstract: We provide a Laakso construction to prove that the property of having an equivalent norm with the property $(\beta)$ of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces. On the other hand, we show that the $(\beta)$-modulus is not quantitatively preserved via such a map by exhibiting two uniformly homeomorphic Banach spaces that do not have $(\beta)$-moduli of the same power-type even under renorming. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B20 (Primary), 46B80, 46T99, 51F99 (Secondary) Submitted from: nrandria(a)slu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.6424 or http://arXiv.org/abs/1408.6424

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Abstract of a paper by Afonso S. Bandeira and Ramon van Handel
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Sharp nonasymptotic bounds on the norm of random matrices with independent entries" by Afonso S. Bandeira and Ramon van Handel. Abstract: We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^2)$, we show that $$\mathbf{E}\|X\|\lesssim \max_i\sqrt{\sum_{j}b_{ij}^2} +\max_{ij}|b_{ij}|\sqrt{\log n}. $$ This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general subgaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices. Archive classification: math.PR math.FA Mathematics Subject Classification: 60B20, 46B09, 60F10 Remarks: 23 pages Submitted from: rvan(a)princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.6185 or http://arXiv.org/abs/1408.6185

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Abstract of a paper by Assaf Naor and Gideon Schechtman
by alspach＠math.okstate.edu 12 Sep '14

12 Sep '14

This is an announcement for the paper "Metric ${X}_p$ inequalities" by Assaf Naor and Gideon Schechtman. Abstract: We show that if $m,n\in \mathbb{N}$ and $k\in \{1,\ldots, n\}$ satisfy $m\ge \frac{n^{3/2}}{\sqrt{k}}$ then for every $p\in [2,\infty)$ and $f:\mathbb{Z}_{4m}^n\to \mathbb{R}$ we have \begin{equation} \frac{1}{\binom{n}{k}}\sum_{\substack{S\subseteq \{1,\ldots,n\}\\|S|= k}}\frac{\mathbb{E}\left[\big|f\big(x+2m\sum_{j\in S} \varepsilon_j e_j\big)-f(x)\big|^p\right]}{m^p}\lesssim_p \frac{k}{n}\sum_{j=1}^n\mathbb{E}\big[\left| f(x+e_j)-f(x)\right|^p\big]+\left(\frac{k}{n}\right)^{\frac{p}{2}} \mathbb{E}\big[\left|f\left(x+ \varepsilon{e}\right)-f(x)\right|^p\big], \end{equation} where the expectation is with respect to $(x,\varepsilon)\in \mathbb{Z}_{4m}^n\times \{-1,1\}^n$ chosen uniformly at random and $e_1,\ldots e_n$ is the standard basis of $\mathbb{Z}_{4m}^n$. The above inequality is a nonlinear extension of a linear inequality for Rademacher sums that was proved by Johnson, Maurey, Schechtman and Tzafriri in 1979. We show that for the above statement to hold true it is necessary that $m$ tends to infinity with $n$. The formulation (and proof) of the above inequality completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the nonembeddability of $L_p$ spaces into each other, the previously understood cases of which were metric notions of type and cotype, which fail to certify the nonembeddability of $L_q$ into $L_p$ when $2<q<p$. Among the consequences of the above inequality are new quantitative restrictions on the bi-Lipschitz embeddability into $L_p$ of snowflakes of $L_q$ and integer grids in $\ell_q^n$, for $2<q<p$. As a byproduct of our investigations, we also obtain results on the geometry of the Schatten $p$ trace class $S_p$ that are new even in the linear setting. Archive classification: math.FA math.MG math.OA 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/1408.5819 or http://arXiv.org/abs/1408.5819

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