Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Jean-Matthieu Auge
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Linear operators with wild dynamics" by Jean-Matthieu Auge. Abstract: If $X$ is a separable infinite dimensional Banach space, we construct a bounded and linear operator $R$ on $X$ such that $$ A_R=\{x \in X, \|R^tx\| \rightarrow \infty\} $$ is not dense and has non empty interior with the additional property that $R$ can be written $I+K$, where $I$ is the identity and $K$ is a compact operator. This answers two recent questions of H\'ajek and Smith. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, Secondary 47A15, 47A16 Remarks: 14 pages Submitted from: jean-matthieu.auge(a)math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2044 or http://arXiv.org/abs/1204.2044

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Joram lindenstrauss
by Dale Alspach 29 Apr '12

29 Apr '12

Joram Lindenstrauss died today after a long illness. His influence on Banach space theory has been enormous. Personally, I benefited from his visits to Ohio State while I was a graduate student and early on learned much from his books written with Lior Tzafriri. Dale Alspach

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Abstract of a paper by Geng Tian, Youqing Ji, and Yang Cao
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Schauder bases and operator theory II: (SI) Schauder operators" by Geng Tian, Youqing Ji, and Yang Cao. Abstract: In this paper, we will show that for an operator $T$ which is injective and has dense range, there exists an invertible operator $X$ (in fact we can find $U+K$, where $U$ is an unitary operator and $K$ is a compact operator with norm less than a given positive real number) such that $XT$ is strongly irreducible. As its application, strongly irreducible operators always exist in the orbit of Schauder matrices. Archive classification: math.FA Mathematics Subject Classification: 47A55, 47A53, 47A16, Secondary 54H20 Remarks: It is the 3rd version of our paper Submitted from: caoyang(a)jlu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.1587 or http://arXiv.org/abs/1204.1587

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Abstract of a paper by Afonso S. Bandeira, Edgar Dobriban, Dustin G. Mixon, and William F. Sawin
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Certifying the restricted isometry property is hard" by Afonso S. Bandeira, Edgar Dobriban, Dustin G. Mixon, and William F. Sawin. Abstract: This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is hard for NP under randomized polynomial-time reductions. Our reduction is from the NP-complete clique decision problem, and it uses ideas from matroid theory. As a consequence of our result, it is impossible to efficiently test for RIP provided NP \not\subseteq BPP, an assumption which is slightly stronger than P \neq NP. Archive classification: math.FA cs.IT math.IT Remarks: 7 pages Submitted from: dmixon(a)princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.1580 or http://arXiv.org/abs/1204.1580

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Abstract of a paper by Soeren Christensen, Joscha Prochno, and Stiene Riemer
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "An inversion formula for Orlicz norms and sequences of random variables" by Soeren Christensen, Joscha Prochno, and Stiene Riemer. Abstract: Given an Orlicz function $M$, we show which random variables $\xi_i$, $i=1,...,n$ generate the associated Orlicz norm, i.e., which random variables yield $\mathbb{E} \max\limits_{1\leq i \leq n}|x_i\xi_i| \sim \norm{(x_i)_{i=1}^n}_M$. As a corollary we obtain a representation for the distribution function in terms of $M$ and $M'$ which can be easily applied to many examples of interest. Archive classification: math.FA math.PR Mathematics Subject Classification: 46B09, 60E15 Remarks: 11 pages Submitted from: prochno(a)math.uni-kiel.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.1242 or http://arXiv.org/abs/1204.1242

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Abstract of a paper by Rogerio Fajardo, Pedro Kaufmann and Leonardo Pellegrini
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Spaceability in sets of operators on $C(K)$" by Rogerio Fajardo, Pedro Kaufmann and Leonardo Pellegrini. Abstract: We prove that if $K$ is a compact Hausdorff space satisfying either condition \item $K$ contains a nontrivial convergent sequence, or \item $C(K)$ is isomorphic to its square, then there exists an infinite-dimensional closed subspace of the space of operators on $C(K)$, each nonzero element of which does \emph{not} have the form $gI+S$, where $g\in C(K)$, $S$ is weakly compact and $I$ is the identity operator. This comes in contrast with what happens in $C(K)$ spaces with \emph{few operators} in the sense of Koszmider [P. Koszmider, P., Banach spaces of continuous functions with few operators. Math. Ann. 300 (2004), no. 1, 151 - 183.], which are precisely $C(K)$ spaces where \emph{every} operator is of the form $gI+S$. In addition we show that, in case $C(K)$ has few operators, there is an opertator $J$ on $C(K\times\{0,1\})=C(K)^2$ such that each operator on $C(K\times\{0,1\})$ is of the form $gI+hJ+S$, where $g,h\in C(K\times\{0,1\})$ and $S$ is strictly singular, in connection to a result by Ferenczi [V. Ferenczi,Uniqueness of complex structure and real hereditarily indecomposable Banach spaces. Adv. Math. 213 (2007), no. 1, 462 - 488.]. Archive classification: math.FA Submitted from: leoime(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.6855 or http://arXiv.org/abs/1203.6855

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Abstract of a paper by Alexander Barvinok
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Approximations of convex bodies by polytopes and by projections of spectrahedra" by Alexander Barvinok. Abstract: We prove that for any compact set B in R^d and for any epsilon >0 there is a finite subset X of B of |X|=d^{O(1/epsilon^2)} points such that the maximum absolute value of any linear function ell: R^d --> R on X approximates the maximum absolute value of ell on B within a factor of epsilon sqrt{d}. We also prove that for any finite set B in Z^d and for any positive integer k there is a convex set C in R^d containing B such that C is an affine image of a section of the cone of rxr positive semidefinite matrices for r=d^{O(k)} and such that for any linear function ell: R^d --> R with integer coefficients the maximum absolute value of ell on B and the maximum absolute value of ell on C coincide provided the former does not exceed k. Archive classification: math.MG math.FA math.OC Mathematics Subject Classification: 52A20, 52A27, 52A21, 52B55, 90C22 Remarks: 11 pages Submitted from: barvinok(a)umich.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.0471 or http://arXiv.org/abs/1204.0471

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Abstract of a paper by Eli Glasner and Michael Megrelishvili
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Banach representations and affine compactifications of dynamical systems" by Eli Glasner and Michael Megrelishvili. Abstract: To every Banach space V we associate a compact right topological affine semigroup E(V). We show that a separable Banach space V is Asplund if and only if E(V) is metrizable, and it is Rosenthal (i.e. it does not contain an isomorphic copy of $l_1$) if and only if E(V) is a Rosenthal compactum. We study representations of compact right topological semigroups in E(V). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily non-sensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility. Archive classification: math.DS math.FA math.GN Remarks: 43 pages Submitted from: megereli(a)math.biu.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.0432 or http://arXiv.org/abs/1204.0432

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Abstract of a paper by Mohammad Sadegh Asgari
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "New characterizations of fusion bases and Riesz fusion bases in hilbert spaces" by Mohammad Sadegh Asgari. Abstract: In this paper we investigate a new notion of bases in Hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in Hilbert spaces. We also introduce a new definition of fusion dual sequence associated with a fusion basis and show that the operators of a fusion dual sequence are continuous projections. Next we define the fusion biorthogonal sequence, Bessel fusion basis, Hilbert fusion basis and obtain some characterizations of them. we study orthonormal fusion systems and Riesz fusion bases for Hilbert spaces. we consider the stability of fusion bases under small perturbations. We also generalized a result of Paley-Wiener [13] to the situation of fusion basis. Archive classification: math.FA Mathematics Subject Classification: Primary 42C15, Secondary 46C99 Remarks: 14 pages Submitted from: moh.asgari(a)iauctb.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.6279 or http://arXiv.org/abs/1203.6279

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Abstract of a paper by Christian Rosendal
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Global and local boundedness of Polish groups" by Christian Rosendal. Abstract: We present a comprehensive theory of boundedness properties for Polish groups developed with a main focus on Roelcke precompactness (precompactness of the lower uniformity) and Property (OB) (boundedness of all isometric actions on separable metric spaces). In particular, these properties are characterised by the orbit structure of isometric actions on metric spaces and isometric or continuous affine representations on separable Banach spaces. Archive classification: math.FA math.GR Mathematics Subject Classification: Primary: 22A25, Secondary: 03E15, 46B04 Submitted from: rosendal(a)math.uic.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.6047 or http://arXiv.org/abs/1203.6047

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