Banach September 2014

banach@mathdept.okstate.edu

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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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Banach September 2014