- Banach - mathdept.okstate.edu

Abstract of a paper by Emanuel Milman and Liran Rotem
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "Complemented Brunn--Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures" by Emanuel Milman and Liran Rotem. Abstract: Elementary proofs of sharp isoperimetric inequalities on a normed space $(\Real^n,\norm{\cdot})$ equipped with a measure $\mu = w(x) dx$ so that $w^p$ is homogeneous are provided, along with a characterization of the corresponding equality cases. When $p \in (0,\infty]$ and in addition $w^p$ is assumed concave, the result is an immediate corollary of the Borell--Brascamp--Lieb extension of the classical Brunn--Minkowski inequality, providing an elementary proof of a recent result of Cabr\'e--Ros Oton--Serra. When $p \in (-1/n,0)$, the relevant property turns out to be a novel ``complemented Brunn--Minkowski" inequality, which we show is always satisfied by $\mu$ when $w^p$ is homogeneous. This gives rise to a new class of measures, which are ``complemented" analogues of the class of convex measures introduced by Borell, but which have vastly different properties. The resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of Ca\~{n}ete--Rosales and Howe. The isoperimetric and Brunn-Minkowski type inequalities extend to the non-homogeneous setting, under a certain log-convexity assumption on the density. Finally, we obtain functional, Sobolev and Nash-type versions of the studied inequalities. Archive classification: math.FA math.MG Remarks: 37 pages Submitted from: emanuel.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.5695 or http://arXiv.org/abs/1308.5695

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Abstract of a paper by Ryan Causey
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "Estimation of the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces" by Ryan Causey. Abstract: For each ordinal $\alpha< \omega_1$, we prove the existence of a separable, reflexive Banach space with a basis and Szlenk index $\omega^{\alpha+1}$ which is universal for the class of separable, reflexive Banach spaces $X$ such that the Szlenk indices $Sz(X), Sz(X^*)$ do not exceed $\omega^\alpha$. Archive classification: math.FA Submitted from: rcausey(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.5416 or http://arXiv.org/abs/1308.5416

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Abstract of a paper by Volker W. Thurey
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "The complex angle in normed spaces" by Volker W. Thurey. Abstract: We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This `angle' has some simple properties, which are known from the usual angle in real inner product spaces. But to do ordinary Euclidean geometry real angles are necessary. We show that even in a complex normed space there are many pure real valued `angles'. The situation improves yet in inner product spaces. There we can use the known theory of orthogonal systems to find many pairs of vectors with real angles, and to do geometry which is based on the Greeks 2000 years ago. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46C05, 30E99 Remarks: 21 pages Submitted from: volker(a)thuerey.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.5412 or http://arXiv.org/abs/1308.5412

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Abstract of a paper by Miguel Lacruz and Luis Rodriguez-Piazza
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "Localizing algebras and invariant subspaces" by Miguel Lacruz and Luis Rodriguez-Piazza. Abstract: It is shown that the algebra $L^\infty(\mu)$ of all bounded measurable functions with respect to a finite measure $\mu$ is localizing on the Hilbert space $L^2(\mu)$ if and only if the measure $\mu$ has an atom. Next, it is shown that the algebra $H^\infty({\mathbb D})$ of all bounded analytic multipliers on the unit disc fails to be localizing, both on the Bergman space $A^2({\mathbb D})$ and on the Hardy space $H^2({\mathbb D}).$ Then, several conditions are provided for the algebra generated by a diagonal operator on a Hilbert space to be localizing. Finally, a theorem is provided about the existence of hyperinvariant subspaces for operators with a localizing subspace of extended eigenoperators. This theorem extends and unifies some previously known results of Scott Brown and Kim, Moore and Pearcy, and Lomonosov, Radjavi and Troitsky. Archive classification: math.OA Mathematics Subject Classification: 47L10, 47A15 Remarks: 15 pages, submitted to J. Operator Theory Submitted from: lacruz(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.4995 or http://arXiv.org/abs/1308.4995

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Abstract of a paper by Markus Passenbrunner
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "Unconditionality of orthogonal spline systems in $L^p$" by Markus Passenbrunner. Abstract: Given any natural number $k$ and any dense point sequence $(t_n)$, we prove that the corresponding orthonormal spline system is an unconditional basis in reflexive $L^p$. Archive classification: math.FA Remarks: 33 pages Submitted from: markus.passenbrunner(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.5055 or http://arXiv.org/abs/1308.5055

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Abstract of a paper by Vincent Lafforgue and Assaf Naor
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "A doubling subset of $L_p$ for $p>2$ that is inherently infinite dimensional" by Vincent Lafforgue and Assaf Naor. Abstract: It is shown that for every $p\in (2,\infty)$ there exists a doubling subset of $L_p$ that does not admit a bi-Lipschitz embedding into $\R^k$ for any $k\in \N$. Archive classification: math.MG 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/1308.4554 or http://arXiv.org/abs/1308.4554

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Abstract of a paper by Robert Bogucki
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "On explicit constructions of auerbach bases in separable Banach spaces" by Robert Bogucki. Abstract: This paper considers explicit constructions of Auerbach bases in separable Banach spaces. Answering the question of A. Pe{\l}czy{\'n}ski, we prove by construction the existence of Auerbach basis in arbitrary subspace of $c_0$ of finite codimension and in the space $C(K)$ for $K$ compact countable metric space. Archive classification: math.FA Mathematics Subject Classification: 46B15, 46B20 Submitted from: r.bogucki(a)students.mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.4429 or http://arXiv.org/abs/1308.4429

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Abstract of a paper by Deping Ye
by alspach＠math.okstate.edu 30 Aug '13

30 Aug '13

This is an announcement for the paper "On the $L_p$ geominimal surface area and related inequalities" by Deping Ye. Abstract: In this paper, we introduce the $L_p$ Geominimal surface area for all $-n\neq p<1$, which extends the classical Geominimal surface area ($p=1$) by Petty and the $L_p$ Geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ Geominimal surface area is motivated by recent work on the extension of the $L_p$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the $L_p$ Geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and the Santal\'{o} style inequality. Some cyclic inequalities are established to obtain the monotonicity of the $L_p$ Geominimal surface area. Comparison between the $L_p$ Geominimal surface area and the (formal) $p$-surface area is also provided. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 53A15 Submitted from: deping.ye(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.4196 or http://arXiv.org/abs/1308.4196

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Conference Announcement - 7th Conference on Function Spaces: May 2014
by Krzysztof Jarosz 27 Aug '13

27 Aug '13

7th Conference on Function Spaces will take place at the SIUE campus between May 20 and May 24, 2014. The Conference will follow the same format as the previous one: http://www.siue.edu/MATH/conference2010/ If you consider attending the Conference it would help our preparation if you could email us at kjarosz(a)siue.edu checking one of the following: I will participate, It is too early to decide, but I will likely come, Keep me on the mailing list but chances of me coming are rather low Comments: Could you also pass this information to your colleagues and graduate students? We received a small grant to cover some of the local expenses but at this point we are unable to offer any meaningful travel support. We are however applying for an NSF grant to defer travel and local cost for "graduate students, postdocs, young nontenured faculty, women and members of underrepresented groups" (NSF priority) as well as for invited speakers. Since the NSF founded the previous conferences in this series we are quite hopeful that they will provide participants' support again. Knowing well in advance the potential participants will increase chances for an adequate support. Sincerely yours, Krzysztof Jarosz Department of Mathematics and Statistics Southern Illinois University Edwardsville Edwardsville, IL 62026-1653, USA tel.: (618) 650-2354 fax: (618) 650-3771 e-mail: kjarosz(a)siue.edu http://www.siue.edu/~kjarosz/

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Abstract of a paper by Mirna Dzamonja
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Isomorphic universality and the number of pairwise non-isomorphic in the class of Banach spaces" by Mirna Dzamonja. Abstract: We study isomorphic universality of Banach spaces of a given density and a number of pairwise non-isomorphic models in the same class. We show that in the Cohen model the isomorphic universality number for Banach spaces of density $\aleph_1$ is $\aleph_2$, and analogous results are true for other cardinals (Theorem 1.2(1)) and that adding just one Cohen real to any model destroys the universality of Banach spaces of density $\aleph_1$ (Theorem 1.5). We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces and use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embeddings (very positive embeddings), is high (Theorem 4.8(1)), and similarly for the number of pairwise non-isomorphic models (Theorem 4.8(2)). Archive classification: math.LO math.FA Mathematics Subject Classification: 03E75, 46B26, 46B03, 03C45, 06E15 Submitted from: h020(a)uea.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.3640 or http://arXiv.org/abs/1308.3640

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