Banach December 2012

banach@mathdept.okstate.edu

1 participants
19 discussions

Abstract of a paper by Daniel Carando and Pablo Sevilla-Peris
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "Extendibility of bilinear forms on Banach sequence spaces" by Daniel Carando and Pablo Sevilla-Peris. Abstract: We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize $c_0$ in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace. Archive classification: math.FA 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/1212.0777 or http://arXiv.org/abs/1212.0777

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Abstract of a paper by Sergey V. Astashkin and Lech Maligranda
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "A short proof of some recent results related to Ces{\`a}ro function spaces" by Sergey V. Astashkin and Lech Maligranda. Abstract: We give a short proof of the recent results that, for every $1\leq p< \infty,$ the Ces{\`a}ro function space $Ces_p(I)$ is not a dual space, has the weak Banach-Saks property and does not have the Radon-Nikodym property. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46B42 Remarks: 4 pages Submitted from: lech.maligranda(a)ltu.se The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1212.0346 or http://arXiv.org/abs/1212.0346

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Abstract of a paper by Anil Kumar Karn
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "Orthogonality in $\ell _p$-spaces and its bearing on ordered Banach spaces" by Anil Kumar Karn. Abstract: We introduce a notion of $p$-orthogonality in a general Banach space $1 \le p \le \infty$. We use this concept to characterize $\ell _p$-spaces among Banach spaces and also among complete order smooth $p$-normed spaces. We further introduce a notion of $p$-orthogonal decomposition in order smooth $p$-normed spaces. We prove that if the $\infty$-orthogonal decomposition holds in an order smooth $\infty$-normed space, then the $1$-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique. Archive classification: math.FA Mathematics Subject Classification: Primary: 46B40, Secondary: 46B45, 47B60 Submitted from: anilkarn(a)niser.ac.in The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1212.0054 or http://arXiv.org/abs/1212.0054

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Abstract of a paper by Silvia Lassalle and Pablo Turco
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "Operators ideals and approximation properties" by Silvia Lassalle and Pablo Turco. Abstract: We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this framework. We say that a Banach space enjoys the $\A$-approximation property if the identity map is uniformly approximable on $\A$-compact sets by finite rank operators. The Grothendieck's classic approximation property is the $\K$-approximation property for $\K$ the ideal of compact operators and the $p$-approximation property is obtained as the $\mathcal N^p$-approximation property for $\mathcal N^p$ the ideal of right $p$-nuclear operators. We introduce a way to measure the size of $\A$-compact sets and use it to give a norm on $\K_\A$, the ideal of $\A$-compact operators. Most of our results concerning the operator Banach ideal $\K_\A$ are obtained for right-accessible ideals $\A$. For instance, we prove that $\K_\A$ is a dual ideal, it is regular and we characterize its maximal hull. A strong concept of approximation property, which makes use of the norm defined on $\K_\A$, is also addressed. Finally, we obtain a generalization of Schwartz theorem with a revisited $\epsilon$-product. Archive classification: math.FA Mathematics Subject Classification: 46G20, 46B28, 47B07 Remarks: 22 Pages Submitted from: paturco(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.7366 or http://arXiv.org/abs/1211.7366

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Abstract of a paper by Kallol Paul, Debmalya Sain and Kanhaiya Jha
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "On strong orthogonality and strictly convex normed linear spaces" by Kallol Paul, Debmalya Sain and Kanhaiya Jha. Abstract: We introduce the notion of strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element $ x $ of the unit sphere $ S_{X}$ to be an exposed point of the unit ball $ B_X .$ We then prove that a normed linear space is strictly convex iff for each element x of the unit sphere there exists a bounded linear operator A on X which attains its norm only at the points of the form $ \lambda x $ with $ \lambda \in S_{K} $. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 47A30 Submitted from: kalloldada(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.6489 or http://arXiv.org/abs/1211.6489

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Abstract of a paper by Ruidong Wang, Xujian Huang, and Dongni Tan
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "On the numerical radius of Lipschitz operators in Banach spaces" by Ruidong Wang, Xujian Huang, and Dongni Tan. Abstract: We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index $1$. As an application, we show that real lush spaces and $C$-rich subspaces have Lipschitz numerical index $1$. Moreover, using the G$\hat{a}$teaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the $c_0$-, $l_1$-, and $l_\infty$-sums of spaces and vector-valued function spaces. From this, we show that the $C(K)$ spaces, $L_1(\mu)$-spaces and $L_\infty(\nu)$ spaces have Lipschitz numerical index $1$. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, secondary 47A12 Remarks: 23 pages Submitted from: huangxujian86(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.5753 or http://arXiv.org/abs/1211.5753

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Abstract of a paper by V. Mykhaylyuk, M. Popov, B. Randrianantoanina, and G. Schechtman
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "Narrow and $\ell_2$-strictly singular operators from $L_p$" by V. Mykhaylyuk, M. Popov, B. Randrianantoanina, and G. Schechtman. Abstract: In the first part of the paper we prove that for $2 < p, r < \infty$ every operator $T: L_p \to \ell_r$ is narrow. This completes the list of sequence and function Lebesgue spaces $X$ with the property that every operator $T:L_p \to X$ is narrow. Next, using similar methods we prove that every $\ell_2$-strictly singular operator from $L_p$, $1<p<\infty$, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H.~P.~Rosenthal asserts that if an operator $T$ on $L_1[0,1]$ satisfies the assumption that for each measurable set $A \subseteq [0,1]$ the restriction $T \bigl|_{L_1(A)}$ is not an isomorphic embedding, then $T$ is narrow. (Here $L_1(A) = \{x \in L_1: {\rm supp} \, x \subseteq A\}$.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being $\ell_2$-strictly singular, for operators on $L_p[0,1]$, $1<p<2$, to be narrow. We define a notion of a ``gentle'' growth of a function and we prove that for $1 < p < 2$ every operator $T$ on $L_p$ which, for every $A\subseteq[0,1]$, sends a function of ``gentle" growth supported on $A$ to a function of arbitrarily small norm is narrow. Archive classification: math.FA Mathematics Subject Classification: Primary 47B07, secondary 47B38, 46B03, 46E30 Remarks: Dedicated to the memory of Joram Lindenstrauss Submitted from: randrib(a)muohio.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.4854 or http://arXiv.org/abs/1211.4854

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Abstract of a paper by Claudia Correa and Daniel V. Tausk
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "On extensions of $c_0$-valued operators" by Claudia Correa and Daniel V. Tausk. Abstract: We study pairs of Banach spaces $(X,Y)$, with $Y\subset X$, for which the thesis of Sobczyk's theorem holds, namely, such that every bounded $c_0$-valued operator defined in $Y$ extends to $X$. In this case, we say that $Y$ has the $c_0$-extension property in $X$. We are mainly concerned with the case when $X$ is a $C(K)$ space and $Y\equiv C(L)$ is a Banach subalgebra of $C(K)$. The main result of the article states that, if $K$ is a compact line and $L$ is countable, then $C(L)$ has the $c_0$-extension property in $C(K)$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46E15, 54F05 Remarks: 16 pages Submitted from: tausk(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.4830 or http://arXiv.org/abs/1211.4830

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Abstract of a paper by Itai Ben Yaacov and C. Ward Henson
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "Generic orbits and type isolation in the Gurarij space" by Itai Ben Yaacov and C. Ward Henson. Abstract: We study model-theoretic aspects of the separable Gurarij space $\bG$, in particular type isolation and the existence of prime models, without use of formal logic. \begin{enumerate} \item If $E$ is a finite-dimensional Banach space, then the set of isolated types over $E$ is dense, and there exists a prime Gurarij over $E$. This is the unique separable Gurarij space $\bG$ extending $E$ with the unique Hahn-Banach extension property (\emph{property $U$}), and the orbit of $\id\colon E \hookrightarrow \bG$ under the action of $\Aut(\bG)$ is a dense $G_\delta$ in the space of all linear isometric embeddings $E \hookrightarrow \bG$. \item If $E$ is infinite-dimensional then there are no non realised isolated types, and therefore no prime model over $E$ (unless $\bG \cong E$), and all orbits of embeddings $E \hookrightarrow \bG$ are meagre. On the other hand, there are Gurarij spaces extending $E$ with property $U$. \end{enumerate} We also point out that the class of Gurarij space is the class of models of an $\aleph_0$-categorical theory with quantifier elimination, and calculate the density character of the space of types over $E$, answering a question of Avil\'es et al. Archive classification: math.FA math.LO Submitted from: begnac.arxiv(a)free.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.4814 or http://arXiv.org/abs/1211.4814

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Banach December 2012