Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Hana Bendova
by alspach＠math.okstate.edu 30 Sep '13

30 Sep '13

This is an announcement for the paper "Quantitative Grothendieck property" by Hana Bendova. Abstract: A Banach space $X$ is Grothendieck if the weak and the weak$^*$ convergence of sequences in the dual space $X^*$ coincide. The space $\ell^\infty$ is a classical example of a Grothendieck space due to Grothendieck. We introduce a quantitative version of the Grothendieck property, we prove a quantitative version of the above-mentioned Grothendieck's result and we construct a Grothendieck space which is not quantitatively Grothendieck. We also establish the quantitative Grothendieck property of $L^\infty(\mu)$ for a $\sigma$-finite measure $\mu$. Archive classification: math.FA Mathematics Subject Classification: 46B26, 46B04, 46A20 Remarks: 9 pages, 0 figures, submitted to the Journal of Mathematical The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.4684 or http://arXiv.org/abs/1309.4684

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Abstract of a paper by Simon Lucking
by alspach＠math.okstate.edu 30 Sep '13

30 Sep '13

This is an announcement for the paper "The Daugavet property and translation-invariant subspaces" by Simon Lucking. Abstract: Let $G$ be an infinite, compact abelian group and let $\varLambda$ be a subset of its dual group $\varGamma$. We study the question which spaces of the form $C_\varLambda(G)$ or $L^1_\varLambda(G)$ and which quotients of the form $C(G)/C_\varLambda(G)$ or $L^1(G)/L^1_\varLambda(G)$ have the Daugavet property. We show that $C_\varLambda(G)$ is a rich subspace of $C(G)$ if and only if $\varGamma \setminus \varLambda^{-1}$ is a semi-Riesz set. If $L^1_\varLambda(G)$ is a rich subspace of $L^1(G)$, then $C_\varLambda(G)$ is a rich subspace of $C(G)$ as well. Concerning quotients, we prove that $C(G)/C_\varLambda(G)$ has the Daugavet property, if $\varLambda$ is a Rosenthal set, and that $L^1_\varLambda(G)$ is a poor subspace of $L^1(G)$, if $\varLambda$ is a nicely placed Riesz set. Archive classification: math.FA Mathematics Subject Classification: 46B04, 43A46 Remarks: 20 pages Submitted from: simon.luecking(a)fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.4567 or http://arXiv.org/abs/1309.4567

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Abstract of a paper by Spiros Argyros, Kevin Beanland and Pavlos Motakis
by alspach＠math.okstate.edu 30 Sep '13

30 Sep '13

This is an announcement for the paper "Strictly singular operators in Tsirelson like spaces" by Spiros Argyros, Kevin Beanland and Pavlos Motakis. Abstract: For each $n \in \mathbb{N}$ a Banach space $\mathfrak{X}_{0,1}^n$ is constructed is having the property that every normalized weakly null sequence generates either a $c_0$ or $\ell_1$ spreading models and every infinite dimensional subspace has weakly null sequences generating both $c_0$ and $\ell_1$ spreading models. The space $\mathfrak{X}_{0,1}^n$ is also quasiminimal and for every infinite dimensional closed subspace $Y$ of $\mathfrak{X}_{0,1}^n$, for every $S_1,S_2,\ldots,S_{n+1}$ strictly singular operators on $Y$, the operator $S_1S_2\cdots S_{n+1}$ is compact. Moreover, for every subspace $Y$ as above, there exist $S_1,S_2,\ldots,S_n$ strictly singular operators on $Y$, such that the operator $S_1S_2\cdots S_n$ is non-compact. Archive classification: math.FA Remarks: 45 pages Submitted from: kbeanland(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.4358 or http://arXiv.org/abs/1309.4358

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Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 30 Sep '13

30 Sep '13

This is an announcement for the paper "Some remarks on generalised lush spaces" by Jan-David Hardtke. Abstract: X. Huang et al. recently introduced the notion of generalised lush (GL) spaces, which, at least for separable spaces, is a generalisation of the concept of lushness introduced by K. Boyko et al. in 2007. The main result of Huang et al. is that every GL-space has the so called Mazur-Ulam property (MUP). In this note, we will prove some properties of GL-spaces (further than those already established by Huang et al.), for example, every $M$-ideal in a GL-space is again a GL-space, ultraproducts of GL-spaces are again GL-spaces, and if the bidual $X^{**}$ of a Banach space $X$ is GL, then $X$ itself still has the MUP. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 15 pages Submitted from: hardtke(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.4358 or http://arXiv.org/abs/1309.4358

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Abstract of a paper by Julio Becerra Guerrero, Gines Lopez-Perez, and Abraham Rueda Zoca
by alspach＠math.okstate.edu 30 Sep '13

30 Sep '13

This is an announcement for the paper "Octahedral norms and convex combination of slices in Banach spaces" by Julio Becerra Guerrero, Gines Lopez-Perez, and Abraham Rueda Zoca. Abstract: We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of $w^*$-slices in the dual unit ball has diameter $2$, which answer an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing $\ell_1$ and for every $\varepsilon >0$ there is an equivalent norm so that every convex combination of $w^*$-slices in the dual unit ball has diameter at least $2-\varepsilon$. Archive classification: math.FA Submitted from: glopezp(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.3866 or http://arXiv.org/abs/1309.3866

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Abstract of a paper by Gines Lopez Perez and Jose A. Soler Arias
by alspach＠math.okstate.edu 30 Sep '13

30 Sep '13

This is an announcement for the paper "Weak-star point of continuity property and Schauder bases" by Gines Lopez Perez and Jose A. Soler Arias. Abstract: We characterize the weak-star point of continuity property for subspaces of dual spaces with separable predual and we deduce that the weak-star point of continuity property is determined by subspaces with a Schauder basis in the natural setting of dual spaces of separable Banach spaces. As a consequence of the above characterization we get that a dual space satisfies the Radon-Nikodym property if, and only if, every seminormalized topologically weak-star null tree has a boundedly complete branch, which improves some results in \cite{DF} obtained for the separable case. Also, as a consequence of the above characterization, the following result obtained in \cite{R1} is deduced: {\it every seminormalized basic sequence in a Banach space with the point of continuity property has a boundedly complete subsequence Archive classification: math.FA Submitted from: glopezp(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.3862 or http://arXiv.org/abs/1309.3862

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Abstract of a paper by Luis Bernal-Gonzalez and Manuel Ordonez-Cabrera
by alspach＠math.okstate.edu 23 Sep '13

23 Sep '13

This is an announcement for the paper "Lineability criteria, with applications" by Luis Bernal-Gonzalez and Manuel Ordonez-Cabrera. Abstract: Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability. Archive classification: math.FA Mathematics Subject Classification: 15A03, 26A46, 28A25, 30B40, 46E10, 46E30, 47A16 Remarks: 38 pages Submitted from: lbernal(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.3656 or http://arXiv.org/abs/1309.3656

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Abstract of a paper by Mukhtar Ibragimov and Karimbergen Kudaybergenov
by alspach＠math.okstate.edu 23 Sep '13

23 Sep '13

This is an announcement for the paper "Geometric description of L$_1$-Spaces" by Mukhtar Ibragimov and Karimbergen Kudaybergenov. Abstract: We describe strongly facially symmetric spaces which are isometrically isomorphic to L$_1$-space. Archive classification: math.OA Mathematics Subject Classification: 46B20 Remarks: published in Russian Mathematics, 57, No 5, 2013, 16-21 Submitted from: karim20061(a)yandex.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.3620 or http://arXiv.org/abs/1309.3620

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Abstract of a paper by Marek Balcerzak, Adam Majchrzycki, and Filip Strobin
by alspach＠math.okstate.edu 23 Sep '13

23 Sep '13

This is an announcement for the paper "Uniform openness of multiplication in Banach spaces $L _p$" by Marek Balcerzak, Adam Majchrzycki, and Filip Strobin. Abstract: We show that multiplication from $L_p\times L_q$ to $L_1$ (for $p,q\in [1,\infty]$, $1/p+1/q=1$) is a uniformly open mapping. We also prove the uniform openness of the multiplication from $\ell_1\times c_0$ to $\ell_1$. This strengthens the former results obtained by M. Balcerzak, A.~Majchrzycki and A. Wachowicz. Archive classification: math.FA Mathematics Subject Classification: 46B25, 47A06, 54C10 Remarks: 8 pages Submitted from: filip.strobin(a)p.lodz.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.3433 or http://arXiv.org/abs/1309.3433

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Abstract of a paper by Cyril Tintarev
by alspach＠math.okstate.edu 23 Sep '13

23 Sep '13

This is an announcement for the paper "Concentration analysis and cocompactness" by Cyril Tintarev. Abstract: Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions. Archive classification: math.AP math.FA Submitted from: tintarev(a)math.uu.se The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.3431 or http://arXiv.org/abs/1309.3431

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