Banach January 2015

banach@mathdept.okstate.edu

2 participants
21 discussions

Abstract of a paper by Jan Rozendaal, Fedor Sukochev and Anna Tomskova
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Operator Lipschitz functions on Banach spaces" by Jan Rozendaal, Fedor Sukochev and Anna Tomskova. Abstract: Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form \begin{align*} \|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)} \end{align*} for a large class of functions $f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and $Y=\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also obtain results for $p\geq q$. We study the estimate above in the setting of Banach ideals in $\mathcal{L}(X,Y)$. The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix. Archive classification: math.FA math.OA Mathematics Subject Classification: Primary 47A55, 47A56, secondary 47B47 Remarks: 30 pages Submitted from: janrozendaalmath(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.03267 or http://arXiv.org/abs/1501.03267

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Abstract of a paper by Grigory Ivanov
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Convex hull deviation and contractibility" by Grigory Ivanov. Abstract: We study the Hausdorff distance between a set and its convex hull. Let $X$ be a Banach space, define the CHD-module of space $X$ as the supremum of this distance for all subset of the unit ball in $X$. In the case of finite dimensional Banach spaces we obtain the exact upper bound of the CHD-module depending on the dimension of the space. We give an upper bound for the CHD-module in $L_p$ spaces. We prove that CHD-module is not greater than the maximum of the Lipschitz constants of metric projection operator onto hyperplanes. This implies that for a Hilbert space CHD-module equals 1. We prove criterion of the Hilbert space and study the contractibility of proximally smooth sets in uniformly convex and uniformly smooth Banach spaces. Archive classification: math.FA Submitted from: grigory.ivanov(a)phystech.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.02596 or http://arXiv.org/abs/1501.02596

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Abstract of a paper by Leandro Candido and Piotr Koszmider
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "On complemented copies of $c_0(\omega_1)$ in $C(K^n)$ spaces" by Leandro Candido and Piotr Koszmider. Abstract: Given a compact Hausdorff space $K$ we consider the Banach space of real continuous functions $C(K^n)$ or equivalently the $n$-fold injective tensor product $\hat\bigotimes_{\varepsilon}C(K)$ or the Banach space of vector valued continuous functions $C(K, C(K, C(K ..., C(K)...)$. We address the question of the existence of complemented copies of $c_0(\omega_1)$ in $\hat\bigotimes_{\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains an isomorphic copy of $c_0(\omega_1)$. This is related to the results of E. Saab and P. Saab that $X\hat\otimes_\varepsilon Y$ contains a complemented copy of $c_0$, if one of the infinite dimensional Banach spaces $X$ or $Y$ contains a copy of $c_0$ and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if $C(K)$ has density $\omega_1$ and contains a copy of $c_0(\omega_1)$, then $C(K\times K)$ contains a complemented copy $c_0(\omega_1)$. The main result is that under the assumption of $\clubsuit$ for every $n\in N$ there is a compact Hausdorff space $K_n$ of weight $\omega_1$ such that $C(K)$ is Lindel\"of in the weak topology, $C(K_n)$ contains a copy of $c_0(\omega_1)$, $C(K_n^n)$ does not contain a complemented copy of $c_0(\omega_1)$ while $C(K_n^{n+1})$ does contain a complemented copy of $c_0(\omega_1)$. This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc. Archive classification: math.FA math.GN math.LO Submitted from: piotr.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.01785 or http://arXiv.org/abs/1501.01785

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Abstract of a paper by Bin Han and Zhiqiang Xu
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Robustness properties of dimensionality reduction with gaussian random matrices" by Bin Han and Zhiqiang Xu. Abstract: In this paper we study the robustness properties of dimensionality reduction with Gaussian random matrices having arbitrarily erased rows. We first study the robustness property against erasure for the almost norm preservation property of Gaussian random matrices by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate. As a consequence, we establish the robustness property of Johnson-Lindenstrauss lemma and the robustness property of restricted isometry property with corruption for Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal lower and upper bounds of norm distortion rates of Gaussian random matrices under a given erasure ratio. This allows us to establish the strong restricted isometry property with the almost optimal RIP constants, which plays a central role in the study of phaseless compressed sensing. Archive classification: cs.IT math.FA math.IT math.NA math.PR Remarks: 22 pages Submitted from: xuzq(a)lsec.cc.ac.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.01695 or http://arXiv.org/abs/1501.01695

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Abstract of a paper by Vladimir G. Troitsky and Foivos Xanthos
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Spaces of regular abstract martingales" by Vladimir G. Troitsky and Foivos Xanthos. Abstract: In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm. Archive classification: math.FA Submitted from: foivos(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.01685 or http://arXiv.org/abs/1501.01685

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Abstract of a paper by Sergo A. Episkoposian
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "On the divergence of greedy algorithms with respect to Walsh subsystems in $L$" by Sergo A. Episkoposian. Abstract: In this paper we prove that there exists a function which $f(x)$ belongs to $L^1[0,1]$ such that a greedy algorithm with regard to the Walsh subsystem does not converge to $f(x)$ in $L^1[0,1]$ norm, i.e. the Walsh subsystem $\{W_{n_k}\}$ is not a quasi-greedy basis in its linear span in $L^1$ Archive classification: math.FA Citation: Journal of Nonlinear Analysis Series A: Theory, Methods & The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.00832 or http://arXiv.org/abs/1501.00832

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Abstract of a paper by Erhan Caliskan and Pilar Rueda
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "s-Numbers sequences for homogeneous polynomials" by Erhan Caliskan and Pilar Rueda. Abstract: We extend the well known theory of $s$-numbers of linear operators to homogeneous polynomials defined between Banach spaces. Approximation, Kolmogorov and Gelfand numbers of polynomials are introduced and some well-known results of the linear and multilinear settings are obtained for homogeneous polynomials. Archive classification: math.FA Submitted from: pilar.rueda(a)uv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.00785 or http://arXiv.org/abs/1501.00785

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Abstract of a paper by Daniel Pellegrino
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "On the optimal constants of the Bohnenblust--Hille and inequalities" by Daniel Pellegrino. Abstract: We find the optimal constants of the generalized Bohnenblust--Hille inequality for $m$-linear forms over $\mathbb{R}$ and with multiple exponents $ \left( 1,2,...,2\right)$, sometimes called mixed $\left( \ell _{1},\ell _{2}\right) $-Littlewood inequality. We show that these optimal constants are precisely $\left( \sqrt{2}\right) ^{m-1}$ and this is somewhat surprising since a series of recent papers have shown that the constants of the Bohnenblust--Hille inequality have a sublinear growth, and in several cases the same growth was obtained for the constants of the generalized Bohnenblust--Hille inequality. This result answers a question raised by Albuquerque et al. (2013) in a paper published in 2014 in the Journal of Functional Analysis. We also improve the best known constants of the generalized Hardy--Littlewood inequality in such a way that an unnatural behavior of the old estimates (that will be clear along the paper) does not happen anymore. Archive classification: math.FA Submitted from: pellegrino(a)pq.cnpq.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.00965 or http://arXiv.org/abs/1501.00965

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Abstract of a paper by Anthony Weston
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "An application of virtual degeneracy to two-valued subsets of $L_{p}$-spaces" by Anthony Weston. Abstract: Suppose $0 < p < 2$ and that $(\Omega, \mu)$ is a measure space for which $L_{p}(\Omega, \mu)$ is at least two-dimensional. Kelleher, Miller, Osborn and Weston have shown that if a subset $B$ of $L_{p}(\Omega, \mu)$ does not have strict $p$-negative type, then $B$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space). Examples show that the converse of this statement is not true in general. In this note we describe a class of subsets of $L_{p}(\Omega, \mu)$ for which the converse statement holds. We prove that if a two-valued set $B \subset L_{p}(\Omega, \mu)$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space), then $B$ does not have strict $p$-negative type. This result is peculiar to two-valued subsets of $L_{p}(\Omega, \mu)$ and generalizes an elegant theorem of Murugan. It follows, moreover, that of certain types of isometry with range in $L_{p}(\Omega, \mu)$ cannot exist. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B85 Remarks: 3 page note Submitted from: westona(a)canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.8481 or http://arXiv.org/abs/1412.8481

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Abstract of a paper by Alexander Koldobsky
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Slicing inequalities for measures of convex bodies" by Alexander Koldobsky. Abstract: We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $\lambda n$ for some $\lambda\in (0,1).$ The constant depends only on $\lambda.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A20 Submitted from: koldobskiya(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.8550 or http://arXiv.org/abs/1412.8550

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Banach January 2015