Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by David Alonso-Gutierrez
by Dale Alspach 04 Nov '07

04 Nov '07

This is an announcement for the paper "On an extension of the Blaschke-Santalo inequality" by David Alonso-Gutierrez. Abstract: Let $K$ be a convex body and $K^\circ$ its polar body. Call $\phi(K)=\frac{1}{|K||K^\circ|}\int_K\int_{K^\circ}\langle x,y\rangle^2 dxdy$. It is conjectured that $\phi(K)$ is maximum when $K$ is the euclidean ball. In particular this statement implies the Blaschke-Santalo inequality. We verify this conjecture when $K$ is restricted to be a $p$--ball. Archive classification: math.FA Mathematics Subject Classification: 52A20; 52A40; 46B20 Remarks: 7 pages The source file(s), p-balls5.tex: 18249 bytes, is(are) stored in gzipped form as 0710.5907.gz with size 6kb. The corresponding postcript file has gzipped size 65kb. Submitted from: 498220(a)celes.unizar.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.5907 or http://arXiv.org/abs/0710.5907 or by email in unzipped form by transmitting an empty message with subject line uget 0710.5907 or in gzipped form by using subject line get 0710.5907 to: math(a)arXiv.org.

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Abstract of a paper by Richard J. Smith
by Dale Alspach 04 Nov '07

04 Nov '07

This is an announcement for the paper "Gruenhage compacta and strictly convex dual norms" by Richard J. Smith. Abstract: We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images. Archive classification: math.FA math.GN Mathematics Subject Classification: 46B03; 46B26 The source file(s), arxiv29-10-07.tex: 67073 bytes, is(are) stored in gzipped form as 0710.5396.gz with size 19kb. The corresponding postcript file has gzipped size 112kb. Submitted from: rjs209(a)cam.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.5396 or http://arXiv.org/abs/0710.5396 or by email in unzipped form by transmitting an empty message with subject line uget 0710.5396 or in gzipped form by using subject line get 0710.5396 to: math(a)arXiv.org.

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Abstract of a paper by Richard J. Smith
by Dale Alspach 29 Oct '07

29 Oct '07

This is an announcement for the paper "Trees, linear orders and G\^ateaux smooth norms" by Richard J. Smith. Abstract: We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a G\^ateaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fr\'echet smooth norms. In addition, we prove that if C(T) admits a G\^ateaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm. Archive classification: math.FA Mathematics Subject Classification: 46B03; 46B26 Remarks: A different version of this paper is to appear in J. London Math. Soc The source file(s), arxiv12-10-07.tex: 60917 bytes, is(are) stored in gzipped form as 0710.4230.gz with size 18kb. The corresponding postcript file has gzipped size 102kb. Submitted from: rjs209(a)cam.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.4230 or http://arXiv.org/abs/0710.4230 or by email in unzipped form by transmitting an empty message with subject line uget 0710.4230 or in gzipped form by using subject line get 0710.4230 to: math(a)arXiv.org.

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Abstract of a paper by Jan van Neerven
by Dale Alspach 29 Oct '07

29 Oct '07

This is an announcement for the paper "Compactness in vector-valued Banach function spaces" by Jan van Neerven. Abstract: We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces $L_X^p$, where $X$ is a Banach space and $1\le p<\infty$, and extend the result to vector-valued Banach function spaces $E_X$, where $E$ is a Banach function space with order continuous norm. Archive classification: math.FA Mathematics Subject Classification: 46E40 Citation: Positivity 11 (2007), 461-467 Remarks: 6 pages The source file(s), compact_BFS.tex: 39718 bytes, is(are) stored in gzipped form as 0710.3241.gz with size 13kb. The corresponding postcript file has gzipped size 68kb. Submitted from: J.M.A.M.vanNeerven(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.3241 or http://arXiv.org/abs/0710.3241 or by email in unzipped form by transmitting an empty message with subject line uget 0710.3241 or in gzipped form by using subject line get 0710.3241 to: math(a)arXiv.org.

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Abstract of a paper by H.H. Bauschke, F. Deutsch and H. Hundal
by Dale Alspach 14 Oct '07

14 Oct '07

This is an announcement for the paper "Characterizing arbitrarily slow convergence in the method of alternating projections" by H.H. Bauschke, F. Deutsch and H. Hundal. Abstract: In 1997, Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can be arbitrarily slow. However, there are two errors in their proof of this theorem. In this note, we show that although one of the errors is critical, the theorem itself is correct. We give a different proof that uses the multiplicative form of the spectral theorem, and the theorem holds in any real or complex Hilbert space, not just in a real Hilbert space. Archive classification: math.FA math.OC Mathematics Subject Classification: 47B20 The source file(s), 071010.tex: 35102 bytes, is(are) stored in gzipped form as 0710.2387.gz with size 12kb. The corresponding postcript file has gzipped size 96kb. Submitted from: heinz.bauschke(a)ubc.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.2387 or http://arXiv.org/abs/0710.2387 or by email in unzipped form by transmitting an empty message with subject line uget 0710.2387 or in gzipped form by using subject line get 0710.2387 to: math(a)arXiv.org.

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Abstract of a paper by Stanislaw J. Szarek, Elisabeth Werner and Karol Zyczkowski
by Dale Alspach 14 Oct '07

14 Oct '07

This is an announcement for the paper "Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive" by Stanislaw J. Szarek, Elisabeth Werner and Karol Zyczkowski. Abstract: We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard. Archive classification: quant-ph math.FA Remarks: 34 pages in Latex including 3 figures in eps The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . Submitted from: karol(a)tatry.if.uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.1571 or http://arXiv.org/abs/0710.1571 or by email in unzipped form by transmitting an empty message with subject line uget 0710.1571 or in gzipped form by using subject line get 0710.1571 to: math(a)arXiv.org.

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Abstract of a paper by Dominique Lecomte
by Dale Alspach 14 Oct '07

14 Oct '07

This is an announcement for the paper "How can we recover Baire class one functions?" by Dominique Lecomte. Abstract: Let X and Y be separable metrizable spaces, and f:X-->Y be a function. We want to recover f from its values on a small set via a simple algorithm. We show that this is possible if f is Baire class one, and in fact we get a characterization. This leads us to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space. Archive classification: math.LO math.FA math.GN Mathematics Subject Classification: 2000 MSC 26A21, 54H05, 03E15, 46A20 Citation: Mathematika 50 (2003) 171-198 The source file(s), 06.HcrB1f.tex: 108181 bytes, is(are) stored in gzipped form as 0710.0155.gz with size 29kb. The corresponding postcript file has gzipped size 132kb. Submitted from: lecomte(a)moka.ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.0155 or http://arXiv.org/abs/0710.0155 or by email in unzipped form by transmitting an empty message with subject line uget 0710.0155 or in gzipped form by using subject line get 0710.0155 to: math(a)arXiv.org.

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Abstract of a paper by Chang-Pao Chen, Chun-Yen Shen, and Kuo-Zhong Wang
by Dale Alspach 14 Oct '07

14 Oct '07

This is an announcement for the paper "Characterization of the matrix whose norm is determined by its action on decreasing sequences: The exceptional cases" by Chang-Pao Chen, Chun-Yen Shen, and Kuo-Zhong Wang. Abstract: Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{\ell_p,\ell_q}$ are determined by their actions on non-negative decreasing sequences, where one of $p$ and $q$ is 1 or $\infty$. The conditions forcing on $A$ are sufficient and they are also necessary for non-negative finite matrices. Archive classification: math.FA math.CA Mathematics Subject Classification: 15A60, 47A30, 47B37 The source file(s), shenwang9409016.tex: 25759 bytes, is(are) stored in gzipped form as 0710.0038.gz with size 8kb. The corresponding postcript file has gzipped size 79kb. Submitted from: shenc(a)indiana.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.0038 or http://arXiv.org/abs/0710.0038 or by email in unzipped form by transmitting an empty message with subject line uget 0710.0038 or in gzipped form by using subject line get 0710.0038 to: math(a)arXiv.org.

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Abstract of a paper by Gitta Kutyniok, Ali Pezeshki, Robert Calderbank, and Taotao Liu
by Dale Alspach 14 Oct '07

14 Oct '07

This is an announcement for the paper "Robust dimension reduction, fusion frames, and Grassmannian packings" by Gitta Kutyniok, Ali Pezeshki, Robert Calderbank, and Taotao Liu. Abstract: We consider estimating a random vector from its noisy projections onto low dimensional subspaces constituting a fusion frame. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first determine the minimum mean-squared error (MSE) in linearly estimating the random vector of interest from its fusion frame projections, in the presence of white noise. We show that MSE assumes its minimum value when the fusion frame is tight. We then analyze the robustness of the constructed linear minimum MSE (LMMSE) estimator to erasures of the fusion frame subspaces. We prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures. We call such fusion frames equi-distance tight fusion frames, and prove that the chordal distance between subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for construction of equi-distance tight fusion frames. Archive classification: math.FA Mathematics Subject Classification: 94A12; 42C15; 68P30; 93E10 Remarks: 21 pages The source file(s), fusionframe_final_arxiv.bbl: 2844 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.2340 or http://arXiv.org/abs/0709.2340 or by email in unzipped form by transmitting an empty message with subject line uget 0709.2340 or in gzipped form by using subject line get 0709.2340 to: math(a)arXiv.org.

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Abstract of a paper by Dongni Tan
by Dale Alspach 11 Oct '07

11 Oct '07

This is an announcement for the paper "On maps which preserve equality of distance in F-spaces" by Dongni Tan. Abstract: In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then , as a special case linear isometries are characterized through a simple property of their range. Archive classification: math.FA math.MG Mathematics Subject Classification: 46A16 Remarks: 11 pages, 385 figures The source file(s), DongniTan.tex: 17852 bytes, is(are) stored in gzipped form as 0709.3620.gz with size 6kb. The corresponding postcript file has gzipped size 66kb. Submitted from: 0110127(a)mail.nankai.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0709.3620 or http://arXiv.org/abs/0709.3620 or by email in unzipped form by transmitting an empty message with subject line uget 0709.3620 or in gzipped form by using subject line get 0709.3620 to: math(a)arXiv.org.

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