July 2015 - Banach - mathdept.okstate.edu

Abstract of a paper by P. Rueda and E.A. Sanchez-Perez
by alspach＠math.okstate.edu 16 Jul '15

16 Jul '15

This is an announcement for the paper "The Dvoretsky-Rogers Theorem for vector valued integrals on function spaces" by P. Rueda and E.A. Sanchez-Perez. Abstract: We show a Dvoretsky-Rogers type Theorem for the adapted version of the $q$-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guaranty that the space has to be finite dimensional, contrarily to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals ---our vector valued version of convergence in the weak topology--- is equivalent to the convergence with respect to the norm. Examples and applications are also given. Archive classification: math.FA Mathematics Subject Classification: 46B15, 46B50, 46E30, 46G10 Submitted from: easancpe(a)mat.upv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.03033 or http://arXiv.org/abs/1507.03033

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Abstract of a paper by Denny H. Leung and Wee-Kee Tang
by alspach＠math.okstate.edu 16 Jul '15

16 Jul '15

This is an announcement for the paper "Persistence of Banach lattices under nonlinear order isomorphisms" by Denny H. Leung and Wee-Kee Tang. Abstract: Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection between them that preserves order. We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to c_0, and for Banach lattices that contain a closed sublattice order isomorphic to c_0. Archive classification: math.FA Mathematics Subject Classification: 46B42 Submitted from: weekeetang(a)ntu.edu.sg The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.02759 or http://arXiv.org/abs/1507.02759

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Abstract of a paper by Petr Hajek, Gilles Lancien and Eva Pernecka
by alspach＠math.okstate.edu 16 Jul '15

16 Jul '15

This is an announcement for the paper "Lipschitz-free spaces over metric spaces homeomorphic to the Cantor space" by Petr Hajek, Gilles Lancien and Eva Pernecka. Abstract: In this note we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property. This answers a question by G. Godefroy. We also prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. Archive classification: math.FA Submitted from: gilles.lancien(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.02701 or http://arXiv.org/abs/1507.02701

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Abstract of a paper by Daniel Carando, Damian Pinasco and Jorge Tomas Rodriguez
by alspach＠math.okstate.edu 16 Jul '15

16 Jul '15

This is an announcement for the paper "Non-linear Plank Problems and polynomial inequalities" by Daniel Carando, Damian Pinasco and Jorge Tomas Rodriguez. Abstract: In this article we study plank type problems for polynomials on a Banach space $X$. Our aim is to find sufficient conditions on the positive real numbers $a_1, \ldots, a_n,$ such that for continuous polynomials $P_1,\ldots,P_n:X\rightarrow \mathbb C$ of degrees $k_1,\ldots,k_n$, there exists a norm one element $\textbf{z}\in X$ for which $|P_i(\textbf{z})| \ge a_i^{k_i}$ for $i=1,\ldots,n.$ In order to do this, we prove some new inequalities for the norm of the product of polynomials, which are of an independent interest. Archive classification: math.FA Remarks: 18 pages Submitted from: jtrodrig(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.02316 or http://arXiv.org/abs/1507.02316

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Abstract of a paper by Mohammed Bachir and Joel Blot
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "A useful lemma for Lagrange multiplier rules in infinite dimension" by Mohammed Bachir and Joel Blot. Abstract: We give some reasonable and usable conditions on a sequence of norm one in a dual banach space under which the sequence does not converges to the origin in the $w^*$-topology. These requirements help to ensure that the Lagrange multipliers are nontrivial, when we are interested for example on the infinite dimensional infinite-horizon Pontryagin Principles for discrete-time problems. Archive classification: math.FA Submitted from: mohammed.bachir(a)univ-paris1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.01919 or http://arXiv.org/abs/1507.01919

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Abstract of a paper by Ryan M Causey
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "An alternate description of the Szlenk index with applications" by Ryan M Causey. Abstract: We discuss an alternate method for computing the Szlenk index of an arbitrary $w^*$ compact subsets of the dual of a Banach space. We discuss consequences of this method as well as offer simple, alternative proofs of a number of results already found in the literature. Archive classification: math.FA Submitted from: CAUSEYRM(a)mailbox.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.01993 or http://arXiv.org/abs/1507.01993

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Abstract of a paper by Andreas Seeger and Tino Ullrich
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "Haar projection numbers and failure of unconditional convergence in Sobolev spaces" by Andreas Seeger and Tino Ullrich. Abstract: For $1<p<\infty$ we determine the precise range of $L_p$ Sobolev spaces for which the Haar system is an unconditional basis. We also consider the natural extensions to Triebel-Lizorkin spaces and prove upper and lower bounds for norms of projection operators depending on properties of the Haar frequency set. Archive classification: math.CA math.FA Mathematics Subject Classification: 46E35, 46B15, 42C40 Submitted from: seeger(a)math.wisc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.01211 or http://arXiv.org/abs/1507.01211

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Abstract of a paper by Tomasz Kania and Niels Jakob Laustsen
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "Ideal structure of the algebra of bounded operators acting on a Banach space" by Tomasz Kania and Niels Jakob Laustsen. Abstract: We construct a Banach space $Z$ such that the lattice of closed two-sided ideals of the Banach algebra $\mathscr{B}(Z)$ of bounded operators on $Z$ is as follows: $$ \{0\}\subset \mathscr{K}(Z)\subset\mathscr{E}(Z) \raisebox{-.5ex}% {\ensuremath{\overset{\begin{turn}{30}$\subset$\end{turn}}% {\begin{turn}{-30}$\subset$\end{turn}}}}\!\!% \begin{array}{c}\mathscr{M}_1\\[1mm]\mathscr{M}_2\end{array}\!\!\!% \raisebox{-1.25ex}% {\ensuremath{\overset{\raisebox{1.25ex}{\ensuremath{\begin{turn}{-30}$\subset$\end{turn}}}}% {\raisebox{-.25ex}{\ensuremath{\begin{turn}{30}$\subset$\end{turn}}}}}}\,\mathscr{B}(Z). $$ We then determine which kinds of approximate identities (bounded/left/right), if any, each of the four non-trivial closed ideals of $\mathscr{B}(Z)$ contain, and we show that the maximal ideal $\mathscr{M}_1$ is generated as a left ideal by two operators, but not by a single operator, thus answering a question left open in our collaboration with Dales, Kochanek and Koszmider (\emph{Studia Math.} 2013). In contrast, the other maximal ideal $\mathscr{M}_2$ is not finitely generated as a left ideal. The Banach space $Z$ is the direct sum of Argyros and Haydon's Banach space $X_{\text{AH}}$ which has very few operators and a certain subspace $Y$ of $X_{\text{AH}}$. The key property of~$Y$ is that every bounded operator from $Y$ into $X_{\text{AH}}$ is the sum of a scalar multiple of the inclusion mapping and a compact operator. Archive classification: math.FA math.RA Remarks: 21 pp Submitted from: tomasz.marcin.kania(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.01213 or http://arXiv.org/abs/1507.01213

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Abstract of a paper by Manaf Adnan Saleh Saleh
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "Nonlinear Operator Ideals Between Metric Spaces and Banach Spaces, Part I" by Manaf Adnan Saleh Saleh. Abstract: In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear versions of operator ideals. We introduce several examples of nonlinear ideals and the relationships between them. For every space ideal $\mathsf{A}$ can be generated by a special nonlinear ideal which consists of those Lipschitz operators admitting a factorization through a Banach space $\mathbf{M}\in\mathsf{A}$. We investigate products and quotients of nonlinear ideals. We devote to constructions three types of new nonlinear ideals from given ones. A ``new'' is a rule defining nonlinear ideals $\mathfrak{A}^{L}_{new}$, $\textswab{A}^{L}_{new}$, and $\textfrak{A}^{L}_{new}$ for every $\mathfrak{A}$, $\textswab{A}^{L}$, and $\textfrak{A}^{L}$ respectively, are called a Lipschitz procedure. Considering the class of all stable objects for a given Lipschitz procedure we obtain nonlinear ideals having special properties. We present the concept of a (strongly) $p-$Banach nonlinear ideal ($0<p<1$) and prove that the nonlinear ideals of Lipschitz nuclear operators, Lipschitz Hilbert operators, products and quotient are strongly $r-$Banach nonlinear ideals ($0<r<1$). Archive classification: math.FA Mathematics Subject Classification: 47Bxx, 46B28 Submitted from: manaf-adnan.saleh(a)uni-jena.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.00861 or http://arXiv.org/abs/1507.00861

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Abstract of a paper by Ivan Soprunov and Artem Zvavitch
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "Bezout Inequality for Mixed volumes" by Ivan Soprunov and Artem Zvavitch. Abstract: In this paper we consider the following analog of Bezout inequality for mixed volumes: $$V(P_1,\dots,P_r,\Delta^{n-r})V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(P_i,\Delta^{n-1})\ \text{ for }2\leq r\leq n.$$ We show that the above inequality is true when $\Delta$ is an $n$-dimensional simplex and $P_1, \dots, P_r$ are convex bodies in $\mathbb{R}^n$. We conjecture that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be an $n$-dimensional simplex. We prove that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to $\Delta$), which confirms the conjecture when $\Delta$ is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality. Archive classification: math.MG math.FA Mathematics Subject Classification: Primary 52A39, 52B11, 52A20, Secondary 52A23 Remarks: 18 pages, 2 figures Submitted from: i.soprunov(a)csuohio.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1507.00765 or http://arXiv.org/abs/1507.00765

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