Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Volume inequalities and additive maps of convex bodies" by Franz E. Schuster. Abstract: Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A40, 52A39 Citation: Mathematika 53 (2006), 211–234 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7290 or http://arXiv.org/abs/1207.7290

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Abstract of a paper by Rolf Schneider and Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Rotation invariant Minkowski classes of convex bodies" by Rolf Schneider and Franz E. Schuster. Abstract: A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1; T2 such that M + T1 = T2, and T1; T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 33C55 Citation: Mathematika 54 (2007), 1–13 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7286 or http://arXiv.org/abs/1207.7286

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Abstract of a paper by Cleon S. Barroso, Michel P. Reboucas and Marcus A. M. Marrocos
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces" by Cleon S. Barroso, Michel P. Reboucas and Marcus A. M. Marrocos. Abstract: In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a fundamental biorthogonal system, then there exists a continuous vector field $f\colon E\to E$ such that the autonomous differential equation $u'=f(u)$ has no solutions at any time. The proof relies on a key result asserting that every infinite-dimensional Fr\'echet space with a fundamental biorthogonal system possesses a nontrivial separable quotient. The later, is the byproduct of a mixture of known results on barrelledness and two fundamental results of Banach space theory (namely, a result of Pe{\l}czy\'nski on Banach spaces containing $L_1(\mu)$ and the $\ell_1$-theorem of Rosenthal). Next, we introduce a natural notion of weak-approximate solutions for the non-autonomous Cauchy-Peano problem in Banach spaces, and prove that a necessary and sufficient condition for the existence of such an approximation is the absence of $\ell_1$-isomorphs inside the underline space. We also study a kind of algebraic genericity for the Cauchy-Peano problem in spaces $E$ having complemented subspaces with unconditional Schauder basis. It is proved that if $\mathscr{K}(E)$ denotes the family of all continuous vector fields $f\colon E\to E$ for which $u'=f(u)$ has no solutions at any time, then $\mathscr{K}(E)\bigcup \{0\}$ is spaceable in sense that it contains a closed infinite dimensional subspace of $C(E)$, the locally convex space of all continuous vector fields on $E$ with the linear topology of uniform convergence on bounded sets. Archive classification: math.FA Remarks: 13 pages Submitted from: cleonbar(a)mat.ufc.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.6777 or http://arXiv.org/abs/1207.6777

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Abstract of a paper by Jamilson Ramos Campos
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Multiple Cohen strongly $p$-summing operators, ideals, coherence and compatibility" by Jamilson Ramos Campos. Abstract: Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly $p$-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals is discussed in detail. Archive classification: math.FA Submitted from: jamilson(a)dce.ufpb.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.6664 or http://arXiv.org/abs/1207.6664

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Abstract of a paper by Bjorn Kjos-Hanssen
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Effective Banach spaces" by Bjorn Kjos-Hanssen. Abstract: This thesis addresses Pour-El and Richards' fourth question from their book "Computability in analysis and physics", concerning the relation between higher order recursion theory and computability in analysis. Among other things it is shown that there is a computability structure that is uncountable. The example given is a structure on the Banach space of bounded linear operators on the set of almost periodic functions. Archive classification: math.LO math.FA Mathematics Subject Classification: 03D Remarks: Master's thesis, University of Oslo, 1997. Adviser: Dag Normann. The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.6622 or http://arXiv.org/abs/1207.6622

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nsc
by Dale Alspach 13 Aug '12

13 Aug '12

Dear Colleagues, The Institute of Mathematics of the Hebrew University is planning to organize a memorial conference for Joram Lindenstrauss. The conference, titled "Banach spaces: geometry and analysis" will be held at the Institute of Advanced Studies of the Hebrew University in Jerusalem , May 26-31, 2013. More details will follow. Organizers Gideon Schechtman, Tomek Szankowski, Benjamin Weiss.

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Abstract of a paper by Heinrich von Weizsacker
by alspach＠math.okstate.edu 26 Jul '12

26 Jul '12

This is an announcement for the paper "In which spaces every curve is Lebesgue-Pettis-integrable?" by Heinrich von Weizsacker. Abstract: In a real locally convex Hausdorff space the closed convex hull of every metrizable compact set is compact if (and only if) every continuous curve has a Pettis integral with respect to Lebesgue measure. For such spaces there is a natural concept of Bochner integrals. Archive classification: math.FA Mathematics Subject Classification: 46G10 Submitted from: weizsaecker(a)mathematik.uni-kl.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.6034 or http://arXiv.org/abs/1207.6034

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Abstract of a paper by Tuomas Hytonen and Assaf Naor
by alspach＠math.okstate.edu 26 Jul '12

26 Jul '12

This is an announcement for the paper "Pisier's inequality revisited" by Tuomas Hytonen and Assaf Naor. Abstract: Given a Banach space $X$, for $n\in \mathbb N$ and $p\in (1,\infty)$ we investigate the smallest constant $\mathfrak P\in (0,\infty)$ for which every $f_1,\ldots,f_n:\{-1,1\}^n\to X$ satisfy \begin{multline*} \int_{\{-1,1\}^n}\Bigg\|\sum_{j=1}^n \partial_jf_j(\varepsilon)\Bigg\|^pd\mu(\varepsilon)\\\le \mathfrak{P}^p\int_{\{-1,1\}^n}\int_{\{-1,1\}^n}\Bigg\|\sum_{j=1}^n \d_j\Delta f_j(\varepsilon)\Bigg\|^pd\mu(\varepsilon) d\mu(\delta), \end{multline*} where $\mu$ is the uniform probability measure on the discrete hypercube $\{-1,1\}^n$ and $\{\partial_j\}_{j=1}^n$ and $\Delta=\sum_{j=1}^n\partial_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $\mathfrak{P}_p^n(X)$, we show that $\mathfrak{P}_p^n(X)\le \sum_{k=1}^{n}\frac{1}{k}$ for every Banach space $(X,\|\cdot\|)$. This extends the classical Pisier inequality, which corresponds to the special case $f_j=\Delta^{-1}\partial_j f$ for some $f:\{-1,1\}^n\to X$. We show that $\sup_{n\in \N}\mathfrak{P}_p^n(X)<\infty$ if either the dual $X^*$ is a $\mathrm{UMD}^+$ Banach space, or for some $\theta\in (0,1)$ we have $X=[H,Y]_\theta$, where $H$ is a Hilbert space and $Y$ is an arbitrary Banach space. It follows that $\sup_{n\in \N}\mathfrak{P}_p^n(X)<\infty$ if $X$ is a Banach lattice of finite cotype. Archive classification: math.FA Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.5375 or http://arXiv.org/abs/1207.5375

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Abstract of a paper by Boris Rubin
by alspach＠math.okstate.edu 26 Jul '12

26 Jul '12

This is an announcement for the paper "Weighted norm inequalities for k-plane transforms" by Boris Rubin. Abstract: We obtain sharp inequalities for the k-plane transform, the ``j-plane to k-plane'' transform, and the corresponding dual transforms, acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Some generalizations and open problems are discussed. Archive classification: math.FA Mathematics Subject Classification: 44A12 Remarks: 16 pages 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/1207.5180 or http://arXiv.org/abs/1207.5180

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Abstract of a paper by Boris Rubin
by alspach＠math.okstate.edu 26 Jul '12

26 Jul '12

This is an announcement for the paper "On the Funk-Radon-Helgason inversion method in integral geometry" by Boris Rubin. Abstract: The paper deals with totally geodesic Radon transforms on constant curvature spaces. We study applicability of the historically the first Funk-Radon-Helgason method of mean value operators to reconstruction of continuous and $L^p$ functions from their Radon transforms. New inversion formulas involving Erd\'elyi-Kober type fractional integrals are obtained. Particular emphasis is placed on the choice of the differentiation operator in the spirit of the recent Helgason's formula. Archive classification: math.FA Mathematics Subject Classification: 44A12 Remarks: 29 pages 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/1207.5178 or http://arXiv.org/abs/1207.5178

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