- Banach - mathdept.okstate.edu

Abstract of a paper by Trond A. Abrahamsen<br>
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "An improvement of a theorem of Heinrich, Mankiewicz, Sims, and Yost" by Trond A. Abrahamsen. Abstract: Heinrich, Mankiewicz, Sims, and Yost proved that every separable subspace of a Banach space Y is contained in a separable ideal in Y. We improve this result by replacing the term "ideal" with the term "almost isometric ideal". As a consequence of this we obtain, in terms of subspaces, characterizations of diameter 2 properties, the Daugavet property along with the properties of being an almost square space and an octahedral space. Archive classification: math.FA Mathematics Subject Classification: 46B20 (Primary) 46B07 (Secondary) Remarks: 13 pages Submitted from: trond.a.abrahamsen(a)uia.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.0425 or http://arXiv.org/abs/1411.0425

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Abstract of a paper by Michael Cwikel and Richard Rochberg
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Lecture notes on complex interpolation of compactness" by Michael Cwikel and Richard Rochberg. Abstract: Suppose that the linear operator $T$ maps $X_0$ compactly to $Y_0$ and also maps $X_1$ boundedly to $Y_1$. We deal once again with the 51 year old question of whether $T$ also always maps the complex interpolation space $[X_0,X_1]_\theta$ compactly to $[Y_0,Y_1]_\theta$. This is a short preliminary version of our promised technical sequel to our earlier paper arXiv:1410.4527 on this topic. It contains the following two small new partial results: (i) The answer to the above question is yes, in the particular case where $Y_0$ is a UMD-space. (ii) The answer to the above question is yes for given spaces $X_0$, $X_1$, $Y_0$ and $Y_1$ if the answer to the "dualized" or "adjoint" version of the question for the duals of these particular spaces is yes. In fact we deduce (i) from (ii) and from an earlier result obtained jointly by one of us with Nigel Kalton. It is remarked that a proof of a natural converse of (ii) would answer the general form of this question completely. Archive classification: math.FA Mathematics Subject Classification: Primary 46B70, 46B50. Secondary 46E15 Remarks: 7 pages Submitted from: mcwikel(a)math.technion.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.0171 or http://arXiv.org/abs/1411.0171

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Abstract of a paper by Richard Lechner and Paul F.X. Mueller
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Localization and projections on bi--parameter BMO" by Richard Lechner and Paul F.X. Mueller. Abstract: We prove that for any operator T on bi--parameter BMO the identity factors through T or Id - T. As a consequence, bi--parameter BMO is a primary Banach space. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable Banach space bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by combinatorics of colored dyadic rectangles. Archive classification: math.FA Mathematics Subject Classification: 46B25, 60G46, 46B07, 46B26, 30H35 Submitted from: Richard.Lechner(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.8786 or http://arXiv.org/abs/1410.8786

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Abstract of a paper by E. Casini, E. Miglierina, and L. Piasecki
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Hyperplanes in the space of convergent sequences and preduals of $\ell_1$" by E. Casini, E. Miglierina, and L. Piasecki. Abstract: The main aim of the present paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of $c$ is isometric to the whole space if and only if it is $1$-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to $\ell_{1}$ and we give a complete description of the preduals of $\ell_{1}$ under the assumption that the standard basis of $\ell_{1}$ is weak$^{*}$-convergent. Archive classification: math.FA Mathematics Subject Classification: 46B45 (Primary), 46B04 (Secondary) Submitted from: enrico.miglierina(a)unicatt.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.7801 or http://arXiv.org/abs/1410.7801

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Abstract of a paper by Afonso S. Bandeira, Dustin G. Mixon, and Joel Moreira
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "A conditional construction of restricted isometries" by Afonso S. Bandeira, Dustin G. Mixon, and Joel Moreira. Abstract: We study the restricted isometry property of a matrix that is built from the discrete Fourier transform matrix by collecting rows indexed by quadratic residues. We find an $\epsilon>0$ such that, conditioned on a folklore conjecture in number theory, this matrix satisfies the restricted isometry property with sparsity parameter $K=\Omega(M^{1/2+\epsilon})$, where $M$ is the number of rows. Archive classification: math.FA cs.IT math.IT math.NT Remarks: 6 pages Submitted from: moreira(a)math.ohio-state.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.6457 or http://arXiv.org/abs/1410.6457

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Abstract of a paper by Gerard Buskes and Chris Schwanke
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Functional completions of archimedean vector lattices" by Gerard Buskes and Chris Schwanke. Abstract: We study completions of Archimedean vector lattices relative to any nonempty set of positively-homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper. Archive classification: math.FA Mathematics Subject Classification: 06F20, 46A40 Submitted from: mmbuskes(a)olemiss.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5878 or http://arXiv.org/abs/1410.5878

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Abstract of a paper by Pandelis Dodos, Vassilis Kanellopoulos and Thodoris Karageorgos
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Szemer\'{e}di's regularity lemma via martingales" by Pandelis Dodos, Vassilis Kanellopoulos and Thodoris Karageorgos. Abstract: We prove a variant of the abstract probabilistic version of Szemer\'{e}di's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences. Archive classification: math.CO math.FA math.PR Remarks: 24 pages, no figures Submitted from: pdodos(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5966 or http://arXiv.org/abs/1410.5966

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Abstract of a paper by Pandelis Dodos, Vassilis Kanellopoulos and Konstantinos Tyros
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "A concentration inequality for product spaces" by Pandelis Dodos, Vassilis Kanellopoulos and Konstantinos Tyros. Abstract: We prove a concentration inequality which asserts that, under some mild regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits pseudorandom behavior. Archive classification: math.PR math.CO math.FA Remarks: 11 pages, no figures Submitted from: pdodos(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5965 or http://arXiv.org/abs/1410.5965

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Abstract of a paper by Silvia Lassalle and Pablo Turco
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "The weak bounded approximation property for $\mathcal A$" by Silvia Lassalle and Pablo Turco. Abstract: Fixed a Banach operator ideal $\mathcal A$, we introduce and investigate the weak bounded approximation property for $\mathcal A$, which is strictly weaker than the bounded approximation property for $\mathcal A$ of Lima, Lima and Oja (2010). We relate the weak BAP for $\mathcal A$ with approximation properties given by tensor norms and show that the metric approximation property of order $p$ of Saphar is the weak BAP for the ideal of $p'$-summing operators, $1<p<\infty$, $\frac 1p + \frac 1{p'}=1$. Under this framework, we address the question of approximation properties passing from $X'$ to $X$ or from $X''$ to $X'$. Archive classification: math.FA Mathematics Subject Classification: 47B10, 46A32, 46B28 Remarks: 15 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/1410.5670 or http://arXiv.org/abs/1410.5670

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Abstract of a paper by Mario Chica, Vladimir Kadets, Miguel Martin, Javier Meri, and Soloviova
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Two refinements of the Bishop-Phelps-Bollob\'as modulus" by Mario Chica, Vladimir Kadets, Miguel Martin, Javier Meri, and Soloviova. Abstract: Extending the celebrated result by Bishop and Phelps that the set of norm attaining functionals is always dense in the topological dual of a Banach space, Bollob\'as proved the nowadays known as the Bishop-Phelps-Bollob\'as theorem, which allows to approximate at the same time a functional and a vector in which it almost attains the norm. Very recently, two Bishop-Phelps-Bollob\'as moduli of a Banach space have been introduced [J. Math. Anal. Appl. 412 (2014), 697--719] to measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. In this paper we present two refinements of the results of that paper. On the one hand, we get a sharp general estimation of the Bishop-Phelps-Bollob\'as modulus as a function of the norms of the point and the functional, and we also calculate it in some examples, including Hilbert spaces. On the other hand, we relate the modulus of uniform non-squareness with the Bishop-Phelps-Bollob\'as modulus obtaining, in particular, a simpler and quantitative proof of the fact that a uniformly non-square Banach space cannot have the maximum value of the Bishop-Phelps-Bollob\'as modulus. Archive classification: math.FA Mathematics Subject Classification: 46B04 Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5570 or http://arXiv.org/abs/1410.5570

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