Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Petr Hajek and Richard Haydon
by Dale Alspach 16 Oct '06

16 Oct '06

This is an announcement for the paper "Smooth norms and approximation in Banach spaces of the type C(K)" by Petr Hajek and Richard Haydon. Abstract: We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class C^m. (ii) If C(K) admits an equivalent norm with locally uniformly convex dual norm, then C(K) admits an equivalent norm which is of class C^m (except at 0). Archive classification: Functional Analysis Mathematics Subject Classification: 46B03; 46B26 The source file(s), SmoothNormsAndApprox.tex: 25237 bytes, is(are) stored in gzipped form as 0610421.gz with size 9kb. The corresponding postcript file has gzipped size 46kb. Submitted from: richard.haydon(a)bnc.ox.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610421 or http://arXiv.org/abs/math.FA/0610421 or by email in unzipped form by transmitting an empty message with subject line uget 0610421 or in gzipped form by using subject line get 0610421 to: math(a)arXiv.org.

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Abstract of a paper by Luis Rademacher, Santosh Vempala
by Dale Alspach 13 Oct '06

13 Oct '06

This is an announcement for the paper "Dispersion of mass and the complexity of randomized geometric algorithms" by Luis Rademacher, Santosh Vempala. Abstract: How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing. Archive classification: Computational Complexity; Computational Geometry; Data Structures; Functional Analysis The paper may be downloaded from the archive by web browser from URL http://arXiv.org/abs/cs.CC/0608054 or by email in unzipped form by transmitting an empty message with subject line uget 0608054 or in gzipped form by using subject line get 0608054 to: cs(a)arXiv.org.

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Abstract of a paper by Richard Haydon
by Dale Alspach 13 Oct '06

13 Oct '06

This is an announcement for the paper "Locally uniformly convex norms in Banach spaces and their duals" by Richard Haydon. Abstract: It is shown that a Banach space with locally uniformly convex norm admits an equivalent norm which is itself locally uniformly convex. It follows that on any such space all continuous real-valued functions may be uniformly approximated by C^1 functions. Archive classification: Functional Analysis; General Topology Mathematics Subject Classification: 46B20 The source file(s), LURnormsAndDuals.tex: 50635 bytes, is(are) stored in gzipped form as 0610420.gz with size 15kb. The corresponding postcript file has gzipped size 65kb. Submitted from: richard.haydon(a)bnc.ox.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610420 or http://arXiv.org/abs/math.FA/0610420 or by email in unzipped form by transmitting an empty message with subject line uget 0610420 or in gzipped form by using subject line get 0610420 to: math(a)arXiv.org.

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Abstract of a paper by Rajesh Mahadevan
by Dale Alspach 12 Oct '06

12 Oct '06

This is an announcement for the paper "A note on a non-linear Krein-Rutman theorem" by Rajesh Mahadevan. Abstract: In this note we will present an extension of the Krein-Rutman theorem for an abstract nonlinear, compact, positively 1-homogeneous, monotone non-decreasing operators on a Banach space and apply the result to many nonlinear elliptic partial differential operators. Archive classification: Functional Analysis; Analysis of PDEs Mathematics Subject Classification: 47H12,47H11 The source file(s), nlKRt-rev1.tex: 28673 bytes, is(are) stored in gzipped form as 0610336.gz with size 10kb. The corresponding postcript file has gzipped size 47kb. Submitted from: rmahadevan(a)udec.cl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610336 or http://arXiv.org/abs/math.FA/0610336 or by email in unzipped form by transmitting an empty message with subject line uget 0610336 or in gzipped form by using subject line get 0610336 to: math(a)arXiv.org.

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Abstract of a paper by Hun Hee Lee
by Dale Alspach 10 Oct '06

10 Oct '06

This is an announcement for the paper "Unconditionality with respect to orthonormal systems in noncommutative $L_2$ spaces" by Hun Hee Lee. Abstract: Orthonormal systems in commutative $L_2$ spaces can be used to classify Banach spaces. When the system is complete and satisfies certain norm condition the unconditionality with respect to the system characterizes Hilbert spaces. As a noncommutative analogue we introduce the notion of unconditionality of operator spaces with respect to orthonormal systems in noncommutative $L_2$ spaces and show that the unconditionality characterizes operator Hilbert spaces when the system is complete and satisfy certain norm condition. The proof of the main result heavily depends on free probabilistic tools such as contraction principle for $*$-free Haar unitaries, comparision of averages with respect to $*$-free Haar unitaries and $*$-free circular elements and $K$-covexity, type 2 and cotype 2 with respect to $*$-free circular elements. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: 47L25; 46L53 Remarks: 18 pages The source file(s), Unc-NoncomONS.tex: 56149 bytes, is(are) stored in gzipped form as 0610245.gz with size 15kb. The corresponding postcript file has gzipped size 92kb. Submitted from: lee.hunhee(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610245 or http://arXiv.org/abs/math.FA/0610245 or by email in unzipped form by transmitting an empty message with subject line uget 0610245 or in gzipped form by using subject line get 0610245 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi, Alain Louveau, and Christian Rosendal
by Dale Alspach 10 Oct '06

10 Oct '06

This is an announcement for the paper "The complexity of classifying separable Banach spaces up to isomorphism" by Valentin Ferenczi, Alain Louveau, and Christian Rosendal. Abstract: It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 46B03; 03E15 The source file(s), ComplexityIsomorphism14.tex: 82408 bytes, is(are) stored in gzipped form as 0610289.gz with size 25kb. The corresponding postcript file has gzipped size 101kb. Submitted from: rosendal(a)math.uiuc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610289 or http://arXiv.org/abs/math.FA/0610289 or by email in unzipped form by transmitting an empty message with subject line uget 0610289 or in gzipped form by using subject line get 0610289 to: math(a)arXiv.org.

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Abstract of a paper by Fran\c coise Lust-Piquard and Quanhua Xu
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators" by Fran\c coise Lust-Piquard and Quanhua Xu. Abstract: We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0,\;\8)$. Let $E(\M)$ be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$ corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that $$\forall\; x\in E(\M)\quad \|T(x)\|^2\le K^2\,\|T\|^2 f(x^*x+xx^*),$$ where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for $E(\M)$ when $E$ is either 2-concave or 2-convex and $q$-concave for some $q<\8$. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: Primary 46L52; Secondary 46L50; 47A63 Remarks: 14 pages. To appear in J. Funct. Anal The source file(s), petitgro.tex: 50432 bytes, is(are) stored in gzipped form as 0609356.gz with size 16kb. The corresponding postcript file has gzipped size 74kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609356 or http://arXiv.org/abs/math.FA/0609356 or by email in unzipped form by transmitting an empty message with subject line uget 0609356 or in gzipped form by using subject line get 0609356 to: math(a)arXiv.org.

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Abstract of a paper by M.D. Voisei
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "The sum and chain rules for maximal monotone operators" by M.D. Voisei. Abstract: This paper is primarily concerned with the problem of maximality for the sum $A+B$ and composition $L^{*}ML$ in non-reflexive Banach space settings under qualifications constraints involving the domains of $A,B,M$. Here $X$, $Y$ are Banach spaces with duals $X^{*}$, $Y^{*}$, $A,B:X\rightrightarrows X^{*}$, $M:Y\rightrightarrows Y^{*}$ are multi-valued maximal monotone operators, and $L:X\rightarrow Y$ is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented. Archive classification: Functional Analysis Mathematics Subject Classification: 47H05, 46N10 Remarks: 17 pages, submitted to Set-Valued Analysis The source file(s), tscr.tex: 42800 bytes, is(are) stored in gzipped form as 0609296.gz with size 12kb. The corresponding postcript file has gzipped size 60kb. Submitted from: mvoisei(a)utpa.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609296 or http://arXiv.org/abs/math.FA/0609296 or by email in unzipped form by transmitting an empty message with subject line uget 0609296 or in gzipped form by using subject line get 0609296 to: math(a)arXiv.org.

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Abstract of a paper by Frank Oertel and Mark Owen
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "On utility-based super-replication prices of contingent claims with unbounded payoffs" by Frank Oertel and Mark Owen. Abstract: Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at $-\infty$ we prove that the utility-based super-replication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite {\it loss-entropy}. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is the closure of the latter under a suitable weak topology. Central to our proof is the representation of a cone $C_U$ of utility-based super-replicable contingent claims as the polar cone to the set of finite loss-entropy pricing measures. The cone $C_U$ is defined as the closure, under a relevant weak topology, of the cone of all (sufficiently integrable) contingent claims that can be dominated by a zero-financed terminal wealth. We investigate also the natural dual of this result and show that the polar cone to $C_U$ is generated by those separating measures with finite loss-entropy. The full two-sided polarity we achieve between measures and contingent claims yields an economic justification for the use of the cone $C_U$, and an open question. Archive classification: Probability; Functional Analysis; Optimization and Control Mathematics Subject Classification: 1B16, 46N10, 60G44 The source file(s), 051102reversed.tex: 29375 bytes, is(are) stored in gzipped form as 0609403.gz with size 10kb. The corresponding postcript file has gzipped size 53kb. Submitted from: f.oertel(a)ucc.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.PR/0609403 or http://arXiv.org/abs/math.PR/0609403 or by email in unzipped form by transmitting an empty message with subject line uget 0609403 or in gzipped form by using subject line get 0609403 to: math(a)arXiv.org.

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Abstract of a paper by Frank Oertel and Mark Owen
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "Utility-based super-replication prices of unbounded contingent claims and duality of cones" by Frank Oertel and Mark Owen. Abstract: Consider a financial market in which an agent trades with utility-induced restrictions on wealth. We prove that the utility-based super-replication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite entropy. Central to our proof is the representation of a cone $C_\V$ of utility-based super-replicable contingent claims as the polar cone of the set of finite entropy separating measures. $C_\V$ is shown to be the closure, under a relevant weak topology, of the cone of all (sufficiently integrable) contingent claims that can be dominated by a zero-financed terminal wealth. As our approach shows, those terminal wealths need {\it not} necessarily stem from {\it admissible} trading strategies only. We investigate also the natural dual of this result, and show that the polar cone of $C_\V$ is the cone generated by separating measures with {\it finite loss-entropy}. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, we prove that the former set is the closure of the latter under a suitable weak topology. Finally, we show how our framework can be applied to another field of mathematical economics and how it sheds a different light on Farkas' Lemma and its infinite dimensional version there. Archive classification: Probability; Functional Analysis; Optimization and Control Mathematics Subject Classification: 1B16, 46N10, 60G44 The source file(s), Final_Version_17_01_06_submitted.tex: 56116 bytes, is(are) stored in gzipped form as 0609402.gz with size 17kb. The corresponding postcript file has gzipped size 82kb. Submitted from: f.oertel(a)ucc.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.PR/0609402 or http://arXiv.org/abs/math.PR/0609402 or by email in unzipped form by transmitting an empty message with subject line uget 0609402 or in gzipped form by using subject line get 0609402 to: math(a)arXiv.org.

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