Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Ersin Kizgut, Elif Uyanik, and Murat Yurdakul
by alspach＠math.okstate.edu 30 Dec '14

30 Dec '14

This is an announcement for the paper "On operator relations between locally convex spaces" by Ersin Kizgut, Elif Uyanik, and Murat Yurdakul. Abstract: A linear operator $T:X \to Y$ between vector spaces is called strictly singular if for any infinite dimensional closed vector subspace $M$ of $X$, the restriction of $T$ on $M$ is not a topological isomorphism. In this note we introduced some sufficient conditions on domain and range spaces such that any bounded linear operator in between is strictly singular, and give some examples of spaces satisfying these conditions. Archive classification: math.FA Remarks: 15 pages, presented in the context of 8th Australian and New The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.5761 or http://arXiv.org/abs/1412.5761

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Abstract of a paper by Karl-Mikael Perfekt
by alspach＠math.okstate.edu 30 Dec '14

30 Dec '14

This is an announcement for the paper "On M-ideals and o-O type spaces" by Karl-Mikael Perfekt. Abstract: We consider pairs of Banach spaces (M_0, M) such that M_0 is defined in terms of a little-o condition, and M is defined by the corresponding big-O condition. The construction is general and pairs include function spaces of vanishing and bounded mean oscillation, vanishing weighted and weighted spaces of functions or their derivatives, M\"obius invariant spaces of analytic functions, Lipschitz-H\"older spaces, etc. It has previously been shown that the bidual M_0** of M_0 is isometrically isomorphic with M. The main result of this paper is that M_0 is an M-ideal in M. This has several useful consequences: M_0 has Pelczynskis properties (u) and (V), M_0 is proximinal in M, and M_0* is a strongly unique predual of M, while M_0 itself never is a strongly unique predual. Archive classification: math.FA Remarks: 9 pages Submitted from: karlmikp(a)math.ntnu.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.5486 or http://arXiv.org/abs/1412.5486

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Abstract of a paper by Liran Rotem
by alspach＠math.okstate.edu 30 Dec '14

30 Dec '14

This is an announcement for the paper "A letter: The log-Brunn-Minkowski inequality for complex bodies" by Liran Rotem. Abstract: In this short note we explain why the log-Brunn-Minkowski conjecture is correct for complex convex bodies. We do this by relating the conjecture to the notion of complex interpolation, and appealing to a general theorem by Cordero-Erausquin. Archive classification: math.MG math.FA Submitted from: liranro1(a)post.tau.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.5321 or http://arXiv.org/abs/1412.5321

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Abstract of a paper by Daniel Carando, Andreas Defant, and Pablo Sevilla-Peris
by alspach＠math.okstate.edu 30 Dec '14

30 Dec '14

This is an announcement for the paper "Almost sure-sign convergence of Hardy-type Dirichlet series" by Daniel Carando, Andreas Defant, and Pablo Sevilla-Peris. Abstract: Hartman proved in 1939 that the width of the largest possible strip in the complex plane, on which a Dirichlet series $\sum_n a_n n^{-s}$ is uniformly a.s.-sign convergent (i.e., $\sum_n \varepsilon_n a_n n^{-s}$ converges uniformly for almost all sequences of signs $\varepsilon_n =\pm 1$) but does not convergent absolutely, equals $1/2$. We study this result from a more modern point of view within the framework of so called Hardy-type Dirichlet series with values in a Banach space. Archive classification: math.FA Mathematics Subject Classification: 30B50, 30H10, 46G20 Submitted from: dcarando(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.5030 or http://arXiv.org/abs/1412.5030

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Abstract of a paper by Peide Liu and Maofa Wang
by alspach＠math.okstate.edu 30 Dec '14

30 Dec '14

This is an announcement for the paper "Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent" by Peide Liu and Maofa Wang. Abstract: In this paper, the classical Dellacherie's theorem about stochastic process is extended to variable exponent Lebesgue spaces. As its applications, we obtain variable exponent analogues of several famous inequalities in classical martingale theory, including convexity lemma, Burkholder-Gundy-Davis' inequality and Chevalier's inequality. Moreover, we investigate some other equivalent relations between variable exponent martingale Hardy spaces. Archive classification: math.FA Submitted from: pdliu(a)whu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.8146 or http://arXiv.org/abs/1412.8146

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Abstract of a paper by Eusebio Gardella and Hannes Thiel
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "Quotients of Banach algebras acting on $L^p$-spaces" by Eusebio Gardella and Hannes Thiel. Abstract: We show that the class of Banach algebras that can be isometrically represented on an $L^p$-space, for $p\neq 2$, is not closed under quotients. This answers a question asked by Le Merdy 20 years ago. Our methods are heavily reliant on our earlier study of Banach algebras generated by invertible isometries of $L^p$-spaces. Archive classification: math.OA math.FA Mathematics Subject Classification: Primary: 47L10, 43A15. Secondary: 46J10 Remarks: 7 pages Submitted from: gardella(a)uoregon.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.3985 or http://arXiv.org/abs/1412.3985

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Abstract of a paper by Tomasz Kania
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "On C*-algebras which cannot be decomposed into tensor products with factors infinite-dimensional" by Tomasz Kania. Abstract: We prove that C*-algebras which satisfy a Banach-space property of being a Grothendieck space cannot be decomposed into a tensor product of two infinite-dimensional Banach spaces. By a result of Pfitzner, this class contains all von Neumann algebras and their norm-quotients. We thus strengthen a recent result of Ghasemi who established a similar conclusion for C*-tensor products in the case of SAW*-algebras. In particular, we solve in the negative a problem of Simon Wassermann concerning tensorial decompositions of the Calkin algebra in the category of Banach spaces. Archive classification: math.OA math.FA 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/1412.3621 or http://arXiv.org/abs/1412.3621

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Abstract of a paper by N. Albuquerque, G. Araujo, and D. Pellegrino
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "H\"{o}lder's inequality: some recent and unexpected applications" by N. Albuquerque, G. Araujo, and D. Pellegrino. Abstract: H\"{o}lder's inequality, since its appearance in 1888, has played a fundamental role in Mathematical Analysis and it is, without any doubt, one of the milestones in Mathematics. It may seem strange that, nowadays, it keeps resurfacing and bringing new insights to the mathematical community. In this expository article we show how a variant of H\"{o}lder's inequality (although well-known in PDEs) was essentially overlooked in Functional Analysis and has had a crucial (and in some sense unexpected) influence in very recent and major breakthroughs in Mathematics. Some of these recent advances appeared in 2012-2014 and include the theory of Dirichlet series, the famous Bohr radius problem, certain classical inequalities (such as Bohnenblust--Hille or Hardy--Littlewood), or even Mathematical Physics. Archive classification: math.FA Submitted from: pellegrino(a)pq.cnpq.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.2017 or http://arXiv.org/abs/1412.2017

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Abstract of a paper by S. Gabriyelyan, J. Kcakol, W. Kubis, and W. Marciszewski
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "Networks for the weak topology of Banach and Frechet spaces" by S. Gabriyelyan, J. Kcakol, W. Kubis, and W. Marciszewski. Abstract: We start the systematic study of Fr\'{e}chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $\sigma$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows: If the space $E:=C_{c}(X)$ is a Fr\'echet lcs, then $E$ endowed with its weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fr\'echet lcs to be an $\aleph$-space in the weak topology. We prove that a reflexive Fr\'echet lcs $E$ in the weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $E$ is separable. We show however that the nonseparable Banach space $\ell_{1}(\mathbb{R})$ with the weak topology is an $\aleph$-space. Archive classification: math.FA Mathematics Subject Classification: Primary 46A03, 54H11, Secondary 22A05, 54C35 Remarks: 18 pages Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.1748 or http://arXiv.org/abs/1412.1748

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Abstract of a paper by Michael Megrelishvili
by alspach＠math.okstate.edu 16 Dec '14

16 Dec '14

This is an announcement for the paper "A note on dependence of families having bounded variation" by Michael Megrelishvili. Abstract: We show that for arbitrary linearly ordered set $X$ any bounded family of real valued functions on $X$ with bounded total variation does not contain independent subsequences. As a corollary we generalize Helly's sequential compactness theorem. Archive classification: math.GN math.FA Mathematics Subject Classification: 54F15, 54D30, 06A05 Remarks: 7 pages Submitted from: megereli(a)math.biu.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1412.1515 or http://arXiv.org/abs/1412.1515

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