Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Casey Kelleher, Daniel Miller, Trenton Osborn and Anthony Weston
by alspach＠math.okstate.edu 25 Apr '12

25 Apr '12

This is an announcement for the paper "Polygonal equalities and virtual degeneracy in $L$-spaces" by Casey Kelleher, Daniel Miller, Trenton Osborn and Anthony Weston. Abstract: Cases of equality in the classical $p$-negative type inequalities for $L_{p}(\mu)$-spaces are characterized for each $p \in (0,2)$ according to a new property called virtual degeneracy. For each $p \in (0,2)$, this leads to a complete classification of the subsets of $L_{p}$-spaces that have strict $p$-negative type. It follows that if $0 < p < q \leq 2$ and if $(\Omega_{1}, \mu_{1})$ and $(\Omega_{2}, \mu_{2})$ are measure spaces, then no subset of $L_{q}(\Omega_{2}, \mu_{2})$ is isometric to any linear subspace $W$ of $L_{p}(\Omega_{1}, \mu_{1})$ that contains a pair of disjointly supported unit vectors. Under these circumstances it is also the case that no subset of $L_{q}(\Omega_{2}, \mu_{2})$ is isometric to any subset of $L_{p}(\Omega_{1}, \mu_{1})$ that has non-empty interior. We conclude the paper by examining virtually degenerate subspaces of $L_{p}(\mu)$-spaces. Archive classification: math.FA Mathematics Subject Classification: 46B04 Remarks: 9 pages Submitted from: westona(a)canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.5837 or http://arXiv.org/abs/1203.5837

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Abstract of a paper by Fabio Jose Bertoloto
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "Duality of certain Banach spaces of vector-valued holomorphic functions" by Fabio Jose Bertoloto. Abstract: In this work we study the vector-valued Hardy spaces H p (D; F ) (1 ≤ p ≤ ∞) and their relationship with RNP, ARNP and the UMDP properties. By following the approach of Taylor in the scalar-valued case, we prove that, when F and F have the ARNP property, then H p (D; F ) and H q (D; F ) are canonically topologically isomorphic (for p, q ∈ (1, ∞) conjugate indices) if and only if F has the UMDP. Archive classification: math.FA Mathematics Subject Classification: 46G20, 46G10, 30H10 Submitted from: bertoloto(a)famat.ufu.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.5322 or http://arXiv.org/abs/1203.5322

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Abstract of a paper by Jeremy Avigad and Jason Rute
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "Oscillation and the mean ergodic theorem" by Jeremy Avigad and Jason Rute. Abstract: Let B be a uniformly convex Banach space, let T be a nonexpansive linear operator, and let A_n x denote the ergodic average (1/n) sum_{i<n} T^n x. A generalization of the mean ergodic theorem due to Garrett Birkhoff asserts that the sequence (A_n x) converges, which is equivalent to saying that for every epsilon > 0, the sequence has only finitely many fluctuations greater than epsilon. Drawing on calculations by Kohlenbach and Leustean, we provide a uniform bound on the number of fluctuations that depends only on rho := || x || / epsilon and a modulus, eta, of uniform convexity for B. Specifically, we show that the sequence of averages (A_n x) has O(rho^2 log rho * eta(1/(8 rho))^{-1})-many epsilon-fluctuations, and if B is a Hilbert space, the sequence has O(rho^3 log rho)-many epsilon-fluctuations. The proof is fully explicit, providing a remarkably uniform, quantitative, and constructive formulation of the mean ergodic theorem. Archive classification: math.DS math.FA math.LO Mathematics Subject Classification: 37A30, 03F60 Submitted from: avigad(a)cmu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.4124 or http://arXiv.org/abs/1203.4124

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Abstract of a paper by Joscha Prochno and Stiene Riemer
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "On the maximum of random variables on product spaces" by Joscha Prochno and Stiene Riemer. Abstract: Let $\xi_i$, $i=1,...,n$, and $\eta_j$, $j=1,...,m$ be iid p-stable respectively q-stable random variables, $1<p<q<2$. We prove estimates for $\Ex_{\Omega_1} \Ex_{\Omega_2}\max_{i,j}\abs{a_{ij}\xi_i(\omega_1)\eta_j(\omega_2)}$ in terms of the $\ell_p^m(\ell_q^n)$-norm of $(a_{ij})_{i,j}$. Additionally, for p-stable and standard gaussian random variables we prove estimates in terms of the $\ell_p^m(\ell_{M_{\xi}}^n)$-norm, $M_{\xi}$ depending on the Gaussians. Furthermore, we show that a sequence $\xi_i$, $i=1,\ldots,n$ of iid $\log-\gamma(1,p)$ distributed random variables ($p\geq 2$) generates a truncated $\ell_p$-norm, especially $\Ex \max_{i}\abs{a_i\xi_i}\sim \norm{(a_i)_i}_2$ for $p=2$. As far as we know, the generating distribution for $\ell_p$-norms with $p\geq 2$ has not been known up to now. Archive classification: math.FA math.PR Remarks: 17 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/1203.3788 or http://arXiv.org/abs/1203.3788

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Abstract of a paper by Stiene Riemer and Carsten Schuett
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "On the expectation of the norm of random matrices with non-identically distributed" by Stiene Riemer and Carsten Schuett. Abstract: We give estimates for the expectation of the norm of random matrices with independent but not necessarily identically distributed entries. Archive classification: math.FA Submitted from: riemer(a)math.uni-kiel.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.3713 or http://arXiv.org/abs/1203.3713

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Abstract of a paper by Denny H. Leung and Ya-Shu Wang
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "Compact and weakly compact disjointness preserving operators on spaces of differentiable functions" by Denny H. Leung and Ya-Shu Wang. Abstract: A pair of functions defined on a set X with values in a vector space E is said to be disjoint if at least one of the functions takes the value $0$ at every point in X. An operator acting between vector-valued function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. We characterize compact and weakly compact disjointness preserving operators between spaces of Banach space-valued differentiable functions. Archive classification: math.FA Mathematics Subject Classification: 46E40, 46E50, 47B33, 47B38 Submitted from: matlhh(a)nus.edu.sg The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.3607 or http://arXiv.org/abs/1203.3607

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Abstract of a paper by Yang Cao, Geng Tian, and Bingzhe Hou
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "Schauder bases and operator theory" by Yang Cao, Geng Tian, and Bingzhe Hou. Abstract: In this paper, we firstly give a matrix approach to the bases of a separable Hilbert space and then correct a mistake appearing in both review and the English translation of the Olevskii's paper. After this, we show that even a diagonal compact operator may map an orthonormal basis into a conditional basis. Archive classification: math.FA Mathematics Subject Classification: Primary 47B37, 47B99, Secondary 54H20, 37B99 Remarks: 17 pages 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/1203.3603 or http://arXiv.org/abs/1203.3603

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Abstract of a paper by A. Ibort, P. Linares, and J.G. Llavona
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "On the representation of orthogonally additive polynomials in $\ell_p$" by A. Ibort, P. Linares, and J.G. Llavona. Abstract: We present a new proof of a Sundaresan's result which shows that the space of orthogonally additive polynomials $\mathcal{P}_o(^k\ell_p)$ is isometrically isomorphic to $\ell_{p/p-k}$ if $k<p<\infty$ and to $\ell_\infty$ if $1\leq p\leq k$. Archive classification: math.FA Mathematics Subject Classification: 46G25, 46B42, 46M05 Citation: Publ. Res. Inst. Math. Sci., 45 (2) 519-24 (2009) Submitted from: albertoi(a)math.uc3m.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.2968 or http://arXiv.org/abs/1203.2968

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Abstract of a paper by A. Ibort, P. Linares, and J.G. Llavona
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "A representation theorem for orthogonally additive polynomials in Riesz spaces" by A. Ibort, P. Linares, and J.G. Llavona. Abstract: The aim of this article is to prove a representation theorem for orthogonally additive polynomials in the spirit of the recent theorem on representation of orthogonally additive polynomials on Banach lattices but for the setting of Riesz spaces. To this purpose the notion of $p$--orthosymmetric multilinear form is introduced and it is shown to be equivalent to the or\-tho\-go\-na\-lly additive property of the corresponding polynomial. Then the space of positive orthogonally additive polynomials on an Archimedean Riesz space taking values on an uniformly complete Archimedean Riesz space is shown to be isomorphic to the space of positive linear forms on the $n$-power in the sense of Boulabiar and Buskes of the original Riesz space. Archive classification: math.FA Mathematics Subject Classification: 46A40, 46G25, 47B65 Citation: Rev. Mat. Complutense, 25 (1) 21-30 (2012) Submitted from: albertoi(a)math.uc3m.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.2379 or http://arXiv.org/abs/1203.2379

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Abstract of a paper by Sjoerd Dirksen
by alspach＠math.okstate.edu 27 Mar '12

27 Mar '12

This is an announcement for the paper "Noncommutative and vector-valued Boyd interpolation theorems" by Sjoerd Dirksen. Abstract: We present a new, elementary proof of Boyd's interpolation theorem. Our approach naturally yields a vector-valued as well as a noncommutative version of this result and even allows for the interpolation of certain operators on $l^1$-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob's maximal inequality and the dual Doob inequality for noncommutative symmetric spaces. We apply our results to prove the Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales in these spaces. Archive classification: math.FA math.OA math.PR Submitted from: sjoerd.dirksen(a)hcm.uni-bonn.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1203.1653 or http://arXiv.org/abs/1203.1653

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