Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Ramon van Handel
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "Chaining, Interpolation, and Convexity" by Ramon van Handel. Abstract: We show that classical chaining bounds on the suprema of random processes in terms of entropy numbers can be systematically improved when the underlying set is convex: the entropy numbers need not be computed for the entire set, but only for certain "thin" subsets. This phenomenon arises from the observation that real interpolation can be used as a natural chaining mechanism. Unlike the general form of Talagrand's generic chaining method, which is sharp but often difficult to use, the resulting bounds involve only entropy numbers but are nonetheless sharp in many situations in which classical entropy bounds are suboptimal. Such bounds are readily amenable to explicit computations in specific examples, and we discover some old and new geometric principles for the control of chaining functionals as special cases. Archive classification: math.PR math.FA math.MG Mathematics Subject Classification: 60B11, 60G15, 41A46, 46B20, 46B70 Remarks: 20 pages Submitted from: rvan(a)princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.05906 or http://arXiv.org/abs/1508.05906

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Abstract of a paper by Andreas Defant and Mieczyslaw Mastylo
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "$L^p$-norms and Mahler's measure of polynomials on the $n$-dimensional torus" by Andreas Defant and Mieczyslaw Mastylo. Abstract: We prove Nikol'skii type inequalities which for polynomials on the $n$-dimensional torus $\mathbb{T}^n$ relate the $L^p$-with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0 <p <q < \infty$). Among other things we show that $C=\sqrt{q/p}$ is the best constant such that $\|P\|_{L^q}\leq C^{\text{deg}(P)} \|P\|_{L^p}$ for all homogeneous polynomials $P$ on $\mathbb{T}^n$. We also prove an exact inequality between the $L^p$-norm of a polynomial $P$ on $\mathbb{T}^n$ and its Mahler measure $M(P)$, which is the geometric mean of $|P|$ with respect to the normalized Lebesgue measure on $\mathbb{T}^n$. Using extrapolation we transfer this estimate into a Khintchine-Kahane type inequality, which, for polynomials on $\mathbb{T}^n$, relates a certain exponential Orlicz norm and Mahler's measure. Applications are given, including some interpolation estimates. Archive classification: math.FA Mathematics Subject Classification: 11R06, 11C08 Submitted from: mastylo(a)amu.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.05556 or http://arXiv.org/abs/1508.05556

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Abstract of a paper by Andreas Defant and Mieczyslaw Mastylo
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "Bohnenblust-Hille inequalities for Lorentz spaces via interpolation" by Andreas Defant and Mieczyslaw Mastylo. Abstract: We prove that the Lorentz sequence space $\ell_{\frac{2m}{m+1},1}$ is, in a~precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust-Hille type inequality for $m$-linear forms or $m$-homogeneous polynomials on $\mathbb{C}^n$. Motivated by this result we develop methods for dealing with subtle Bohnenblust-Hille type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei-Fournier inequalities involving mixed type spaces, we prove multilinear and polynomial Bohnenblust-Hille type inequalities in Lorentz spaces with subpolynomial and subexponential constants. Improving a remarkable result of Balasubramanian-Calado-Queff\'{e}lec, we show an application to the theory of Dirichlet series. Archive classification: math.FA Mathematics Subject Classification: 46B70, 47A53 Submitted from: mastylo(a)amu.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.05554 or http://arXiv.org/abs/1508.05554

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Abstract of a paper by Jose Miguel Zapata
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "On conditional weak topologies under a simplified approach derived the framework of conditional sets" by Jose Miguel Zapata. Abstract: The purpose of this manuscript is to introduce a simplified approach derived from the framework of conditional sets, which is a novel approach to study dynamic and conditional settings, as those that arise in mathematical finance. Under this approach, and with the aim of providing an analytic basis for the study of dynamic and conditional risk measures, we carry out a study of the conditional weak topologies and conditional weak compactness, extending some well-known results to this framework and culminating with the proof of conditional versions of Eberlein-\v{S}mulian and Amir-Lindenstrauss Theorems. In pursuing this aim we study the algebraic structure of conditional spaces conditionally finitely generated and state conditional versions of Baire Category Theorem and Uniform Boundedness Principle. Archive classification: math.FA Submitted from: jmzg1(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.05112 or http://arXiv.org/abs/1508.05112

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Abstract of a paper by Ondrej Kurka
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "Non-universal families of separable Banach spaces" by Ondrej Kurka. Abstract: We prove that if $ C $ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $ C $ is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for $ C $ but still not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces. Archive classification: math.FA Mathematics Subject Classification: 46B04, 54H05 (Primary) 46B15, 46B20, 46B25 (Secondary) Submitted from: kurka.ondrej(a)seznam.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.05059 or http://arXiv.org/abs/1508.05059

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Abstract of a paper by Timur Oikhberg and Pedro Tradacete
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "Almost disjointness preservers" by Timur Oikhberg and Pedro Tradacete. Abstract: We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an ``almost disjointness preserving'' operator is well approximable by a disjointess preserving one. However, this approximation is not always possible, as our examples show. Archive classification: math.FA Mathematics Subject Classification: 47B38, 46B42 Remarks: 43 pages Submitted from: ptradace(a)math.uc3m.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.04074 or http://arXiv.org/abs/1508.04074

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Abstract of a paper by D.T. Dzadzaeva and M.A. Pliev
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "Narrow operators on lattice-normed spaces and vector measures" by D.T. Dzadzaeva and M.A. Pliev. Abstract: We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic lattice-normed space is order narrow. Archive classification: math.FA Mathematics Subject Classification: Primary 46B99, Secondary 46G12 Submitted from: martin.weber(a)tu-dresden.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.03995 or http://arXiv.org/abs/1508.03995

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Abstract of a paper by Tomasz Kobos
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "An uniform estimate of the relative projection constant" by Tomasz Kobos. Abstract: The main goal of the paper is to provide a quantitative lower bound greater than $1$ for the relative projection constant $\lambda(Y, X)$, where $X$ is a subspace of $\ell_{2p}^m$ space and $Y \subset X$ is an arbitrary hyperplane. As a consequence, we establish that for every integer $n \geq 4$ there exists an $n$-dimensional normed space $X$ such that for an every hyperplane $Y$ and every projection $P:X \to Y$ the inequality $||P|| > 1 + \left (2 \left ( n + 3 \right )^{2} \right )^{-100(n+3)^2}$ holds. This gives a non-trivial lower bound in a variation of problem proposed by Bosznay and Garay in $1986$. Archive classification: math.FA Mathematics Subject Classification: 47A58, 41A65, 47A30, 52A21 Remarks: 15 pages Submitted from: tkobos(a)wp.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.03518 or http://arXiv.org/abs/1508.03518

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Abstract of a paper by Abasov, N., Megaled, A., and Pliev, M
by alspach＠math.okstate.edu 28 Aug '15

28 Aug '15

This is an announcement for the paper "Dominated oprerators from a lattice-normed space to a sequence Banach lattice" by Abasov,N., Megaled,A., and Pliev,M. Abstract: Abstract. We show that every dominated linear operator from an Banach-Kantorovich space over atomless Dedekind complete vector lattice to a sequence Banach lattice $l_p({\Gamma})$ or $c_0({\Gamma})$ is narrow. As a conse- quence, we obtain that an atomless Banach lattice cannot have a finite dimensional decomposition of a certain kind. Finally we show that if a linear dominated operator T from lattice-normed space V to Banach- Kantorovich space W is order narrow then the same is its exact dominant $\ls T\rs$. Archive classification: math.FA Mathematics Subject Classification: 47H30, 46B42 Submitted from: martin.weber(a)tu-dresden.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.03275 or http://arXiv.org/abs/1508.03275

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Abstract of a paper by Bruno de Mendonca Braga
by alspach＠math.okstate.edu 11 Aug '15

11 Aug '15

This is an announcement for the paper "Duality on Banach spaces and a Borel parametrized version of Zippin's theorem" by Bruno de Mendonca Braga. Abstract: Let SB be the standard coding for separable Banach spaces as subspaces of $C(\Delta)$. In these notes, we show that if $\mathbb{B} \subset \text{SB}$ is a Borel subset of spaces with separable dual, then the assignment $X \mapsto X^*$ can be realized by a Borel function $\mathbb{B}\to \text{SB}$. Moreover, this assignment can be done in such a way that the functional evaluation is still well defined (Theorem $1$). Also, we prove a Borel parametrized version of Zippin's theorem, i.e., we prove that there exists $Z \in \text{SB}$ and a Borel function that assigns for each $X \in \mathbb{B}$ an isomorphic copy of $X$ inside of $Z$ (Theorem $5$). Archive classification: math.FA Submitted from: demendoncabraga(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1508.02066 or http://arXiv.org/abs/1508.02066

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Banach