Banach June 2005

banach@mathdept.okstate.edu

1 participants
8 discussions

Abstract of a paper by U. Bader, A. Furman, T. Gelander, and N. Monod
by Dale Alspach 21 Jun '05

21 Jun '05

This is an announcement for the paper "Property (T) and rigidity for actions on Banach spaces" by U. Bader, A. Furman, T. Gelander, and N. Monod. Abstract: We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when $1<p< 2+\e$. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces. Archive classification: Group Theory; Functional Analysis The source file(s), ftlp14.tex: 137939 bytes, is(are) stored in gzipped form as 0506361.gz with size 43kb. The corresponding postcript file has gzipped size 152kb. Submitted from: monod(a)math.uchicago.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.GR/0506361 or http://arXiv.org/abs/math.GR/0506361 or by email in unzipped form by transmitting an empty message with subject line uget 0506361 or in gzipped form by using subject line get 0506361 to: math(a)arXiv.org.

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Abstract of a paper by George Androulakis
by Dale Alspach 16 Jun '05

16 Jun '05

This is an announcement for the paper "A new method of constructing invariant subspaces" by George Androulakis. Abstract: The method of compatible sequences is introduced in order to produce non-trivial (closed) invariant subspaces of (bounded linear) operators. Also a topological tool is used which is new in the search of invariant subspaces: the extraction of continuous selections of lower semicontinuous set valued functions. The advantage of this method over previously known methods is that if an operator acts on a reflexive Banach space then it has a non-trivial invariant subspace if and only if there exist compatible sequences (their definition refers to a fixed operator). Using compatible sequences a result of Aronszajn-Smith is proved for reflexive Banach spaces. Also it is shown that if $X$ be a separable reflexive Banach space, $T \in {\mathcal L} (X)$, and $A$ is any closed ball of $X$, then there exists $v \in A$ such that either $Tv=0$, or $\overline{\text{Span}}\, \text{Orb}_T (Tv)$ is a non-trivial invariant subspace of $T$, or there exists a continuous function $f:A \to A$ where $A$ is endowed with the weak topology, such that $f(x) \in \overline{\text{Span}}\, \{ T^k x : k \in {\mathbb N} \} $ for all $x \in A$ and $f(v)=v$. Archive classification: Functional Analysis Mathematics Subject Classification: 47A15 The source file(s), ISP.tex: 60926 bytes, is(are) stored in gzipped form as 0506284.gz with size 17kb. The corresponding postcript file has gzipped size 78kb. Submitted from: giorgis(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0506284 or http://arXiv.org/abs/math.FA/0506284 or by email in unzipped form by transmitting an empty message with subject line uget 0506284 or in gzipped form by using subject line get 0506284 to: math(a)arXiv.org.

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Abstract of a paper by Manor Mendel, Assaf Naor
by Dale Alspach 16 Jun '05

16 Jun '05

This is an announcement for the paper "Scaled Enflo type is equivalent to Rademacher type" by Manor Mendel and Assaf Naor. Abstract: We introduce the notion of scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B20; 51F99 Remarks: 5 pages The source file(s), enflo-rademacher.tex: 16927 bytes, is(are) stored in gzipped form as 0506215.gz with size 5kb. The corresponding postcript file has gzipped size 47kb. Submitted from: mendelma(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0506215 or http://arXiv.org/abs/math.FA/0506215 or by email in unzipped form by transmitting an empty message with subject line uget 0506215 or in gzipped form by using subject line get 0506215 to: math(a)arXiv.org.

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Abstract of a paper by Shahar Mendelson, Alain Pajor and Nicole Tomczak-Jaegermann
by Dale Alspach 15 Jun '05

15 Jun '05

This is an announcement for the paper "Reconstruction and subgaussian operators" by Shahar Mendelson, Alain Pajor and Nicole Tomczak-Jaegermann. Abstract: We present a randomized method to approximate any vector $v$ from some set $T \subset \R^n$. The data one is given is the set $T$, and $k$ scalar products $(\inr{X_i,v})_{i=1}^k$, where $(X_i)_{i=1}^k$ are i.i.d. isotropic subgaussian random vectors in $\R^n$, and $k \ll n$. We show that with high probability, any $y \in T$ for which $(\inr{X_i,y})_{i=1}^k$ is close to the data vector $(\inr{X_i,v})_{i=1}^k$ will be a good approximation of $v$, and that the degree of approximation is determined by a natural geometric parameter associated with the set $T$. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to $\{-1,1\}$-valued vectors with i.i.d, symmetric entries, yields new information on the geometry of faces of random $\{-1,1\}$-polytope; we show that a $k$-dimensional random $\{-1,1\}$-polytope with $n$ vertices is $m$-neighborly for very large $m\le {ck/\log (c' n/k)}$. The proofs are based on new estimates on the behavior of the empirical process $\sup_{f \in F} \left|k^{-1}\sum_{i=1}^k f^2(X_i) -\E f^2 \right|$ when $F$ is a subset of the $L_2$ sphere. The estimates are given in terms of the $\gamma_2$ functional with respect to the $\psi_2$ metric on $F$, and hold both in exponential probability and in expectation. Archive classification: Functional Analysis; Probability Mathematics Subject Classification: 46B07, 47B06, 41A45; 94B75, 52B05 Remarks: 31 pages; no figures; submitted The source file(s), MPT_subgaussian.tex: 75209 bytes, is(are) stored in gzipped form as 0506239.gz with size 24kb. The corresponding postcript file has gzipped size 106kb. Submitted from: alain.pajor(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0506239 or http://arXiv.org/abs/math.FA/0506239 or by email in unzipped form by transmitting an empty message with subject line uget 0506239 or in gzipped form by using subject line get 0506239 to: math(a)arXiv.org.

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Abstract of a paper by Ngai-Ching Wong
by Dale Alspach 15 Jun '05

15 Jun '05

This is an announcement for the paper "The triangle of operators, topologies, bornologies" by Ngai-Ching Wong. Abstract: This paper discusses two common techniques in functional analysis: the topological method and the bornological method. In terms of Pietsch's operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Grothendieck (via Banach space operators), Randtke (via continuous seminorms) and Hogbe-Nlend (via convex bounded sets) are compared. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: 47L20, 46A03, 46A11, 46A17 Remarks: 33 pages The source file(s), triangle05_ArXiV.tex: 98877 bytes, is(are) stored in gzipped form as 0506183.gz with size 27kb. The corresponding postcript file has gzipped size 124kb. Submitted from: wong(a)math.nsysu.edu.tw The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0506183 or http://arXiv.org/abs/math.FA/0506183 or by email in unzipped form by transmitting an empty message with subject line uget 0506183 or in gzipped form by using subject line get 0506183 to: math(a)arXiv.org.

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Abstract of a paper by Piotr W. Nowak
by Dale Alspach 10 Jun '05

10 Jun '05

This is an announcement for the paper "A metric space not quasi-isometrically embeddable into any uniformly convex Banach space" by Piotr W. Nowak. Abstract: We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed. Archive classification: Metric Geometry; Functional Analysis Remarks: 6 pages, 2 figures The source file(s), Quasi-isometricnon-embeddabilityintouniformlyconvexBanachspaces.tex: 18532 byt, figuramain.eps: 7608 bytes, figure1.eps: 3766 bytes, is(are) stored in gzipped form as 0506178.tar.gz with size 9kb. The corresponding postcript file has gzipped size 44kb. Submitted from: pnowak(a)math.vanderbilt.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0506178 or http://arXiv.org/abs/math.MG/0506178 or by email in unzipped form by transmitting an empty message with subject line uget 0506178 or in gzipped form by using subject line get 0506178 to: math(a)arXiv.org.

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Abstract of a paper by Koenraad M.R. Audenaert
by Dale Alspach 05 Jun '05

05 Jun '05

This is an announcement for the paper "A norm compression inequality for block partitioned positive semidefinite matrices" by Koenraad M.R. Audenaert. Abstract: Let $A$ be a positive semidefinite matrix, block partitioned as $$ A=\twomat{B}{C}{C^*}{D}, $$ where $B$ and $D$ are square blocks. We prove the following inequalities for the Schatten $q$-norm $||.||_q$, which are sharp when the blocks are of size at least $2\times2$: $$ ||A||_q^q \le (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 1\le q\le 2, $$ and $$ ||A||_q^q \ge (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 2\le q. $$ These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer being sharp: $$ ||A||_q^q \le \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 1\le q\le 2, $$ and $$ ||A||_q^q \ge \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 2\le q. $$ Archive classification: Functional Analysis Mathematics Subject Classification: 15A60 Remarks: 24 pages The source file(s), normcompr_v3.tex: 50189 bytes, is(are) stored in gzipped form as 0505680.gz with size 16kb. The corresponding postcript file has gzipped size 79kb. Submitted from: kauden(a)imperial.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505680 or http://arXiv.org/abs/math.FA/0505680 or by email in unzipped form by transmitting an empty message with subject line uget 0505680 or in gzipped form by using subject line get 0505680 to: math(a)arXiv.org.

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Abstract of a paper by Rubin Boris
by Dale Alspach 05 Jun '05

05 Jun '05

This is an announcement for the paper "The generalized Busemann-Petty problem with weights" by Rubin Boris. Abstract: The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies with lower-dimensional smaller sections necessarily have smaller volume. We study the weighted version of this problem corresponding to the physical situation when bodies are endowed with mass distribution and the relevant sections are measured with attenuation. Archive classification: Functional Analysis Mathematics Subject Classification: 52A38; 44A12 Remarks: 12 pages The source file(s), sol1.tex: 32080 bytes, is(are) stored in gzipped form as 0505666.gz with size 11kb. The corresponding postcript file has gzipped size 57kb. Submitted from: borisr(a)math.lsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505666 or http://arXiv.org/abs/math.FA/0505666 or by email in unzipped form by transmitting an empty message with subject line uget 0505666 or in gzipped form by using subject line get 0505666 to: math(a)arXiv.org.

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Banach June 2005