- Banach - mathdept.okstate.edu

Abstract of a paper by Ryan M. Causey
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The Szlenk index of $C(K,X)$” by Ryan M. Causey. Abstract: Given any Banach space X and any w∗-compact subset $K$ of $X^*$, we compute the Szlenk index of the $w^*$ closed, convex hull of $K$ as a function of the Szlenk index of $K$. As a consequence, for any compact, Hausdorff topological space $K$ and any Banach space $X$, we compute the the Szlenk index of $C(K, X)$ as a function of the Szlenk index of $X$ and the Cantor-Bendixson index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product in terms of $S_z(X)$ and $S_z(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\ell_1$ or $c_0$ index of a Banach space. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.07875

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Abstract of a paper by Emanuele Casini, Enrico Miglierina and Lukasz Piasecki
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Weak$^*$ Fixed Point Property and Polyhedrality in Lindenstrauss Spaces” by Emanuele Casini, Enrico Miglierina and Lukasz Piasecki. Abstract: The main aim of this paper is to study the $w^*$-fixed point property for nonexpansive mappings in the duals of separable Lindestrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of $w^*$-closed subsets of the dual sphere is equivalent to the $w^*$-fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable $w^*$-fixed point property. The last geometrical notion was introduced by Fonf and Vesel$\'{y}$ in 2004 as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be seen as a polyhedral concept for the predual space. Finally, we prove several relationships between various polyhedrality notions in the framework of Lindenstrauss spaces and we provide some examples showing that none of the considered implications can be reverted. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.07587

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Abstract of a paper by Pavlos Motakis and Thomas Schlumprecht
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “A metric interpretation of reflexivity for Banach spaces” by Pavlos Motakis and Thomas Schlumprecht. Abstract: We define two metrics $d_{1, \alpha}$ and $d_{\infty, \alpha}$ on each Schreier family $S_{\alpha}, \alpha<\omega_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $\alpha<\omega_1$, so that there is no mapping $\Phi: S_{\alpha}\rightarrow X$ for which $$cd_{\infty, \alpha}(A, B)\leq \|\Phi(A)-\Phi(B)\|\leq Cd_{1, \alpha}(A, B)$$ for all $A, B\in S_{\alpha}$. Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $\alpha$ that max$(S_z(X), Sz(X^*))\leq\alpha$ if and only if $S_{\alpha}d_{1,\alpha}$ does not bi-Lipschitzly embed into $X$. Here $S_z(Y)$ denotes the Szlenk index of a Banach space $Y$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.07271

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Abstract of a paper by Rainis Haller, Katrlin Pirk and Mart Poldvere
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Diametral strong diameter two property of Banach spaces is stable under direct sums with $1$-norm” by Rainis Haller, Katrlin Pirk and Mart Poldvere. Abstract: We prove that the diametral strong diameter $2$ property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under $1$-sums, i.e., the direct sum of two spaces with the diametral strong diameter $2$ property equipped with the $1$-norm has again this property. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1603.08082

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Abstract of a paper by Pablo M. Berna and Oscar Blasco
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Characterization of greedy bases in Banach spaces” by Pablo M. Berna and Oscar Blasco. Abstract: We shall present a new characterization of greedy bases and $1$-greedy bases in terms of certain functionals defined using distances to one dimensional subspaces generated by the basis. We also introduce a new property that unifies the notions of unconditionality and democracy and allows us to recover a better dependence on the constants. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.07260

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Abstract of a paper by Tomasz kiwerski and Pawel Kilwicz
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Isomorphic copies of $l^\infty$ in Cesàro-Orlicz function spaces” by Tomasz kiwerski and Pawel Kilwicz. Abstract: We characterize Ces$\`$aro-Orlicz function spaces $C_{es_{\phi}}$ containing isomorphic copy of $\ell_{\infty}$. We also describe the subspaces $C_{es_{\phi}}$ of all order continuous elements of $C_{es_{\phi}}$. Finally, we study the monotonicity structure of the spaces $C_{es_{\phi}}$ and $(C_{es_{\phi}})_a$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.07257

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Abstract of a paper by Sun Kwang Kim, Han Ju Lee, Miguel Martin and Javier Meri
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “On a second numerical index for Banach spaces” by Sun Kwang Kim, Han Ju Lee, Miguel Martin and Javier Meri. Abstract: We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces$\ldots$ which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollob$\'$as property for numerical radius is given. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.06198

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Abstract of a paper by Ersin Kizgut, Elif Uyanik and Murat Yurdakul
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Remarks on bounded operators in $\ell$-Köthe spaces” by Ersin Kizgut, Elif Uyanik and Murat Yurdakul. Abstract: For locally convex spaces $X$ and $Y$, the continuous linear map $T: X\rightarrow Y$ is said to be bounded if it maps zero neighborhoods of $X$ into bounded sets of $Y$. We denote $(X, Y)\in B$ when every operator between $X$ and $Y$ is bounded. For a Banach space $\ell$ with a monotone norm $\|\cdot\|$ in which the canonical system $(e_n)$ forms an unconditional basis, we consider $\ell$ -K$\"$othe spaces as a generalization of usual K$\"$othe spaces. In this note, we characterize $\ell$ -K$\"$othe spaces $\ell(a_{pn})$ and $\ell(b_{sm})$ such that $(\ell(a_{pn}),\ell(b_{sm}))\in B$. A pair $(X, Y)$ is said to have the bounded factorization property, and denoted $(X, Y)\in BF$ , if each linear continuous operator $T: X\rightarrow X$ that factors over $Y$ is bounded. We also prove that injective tensor products of some classical K$\"$othe spaces have bounded factorization property. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.05298

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Abstract of a paper by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Lipschitz slices versus linear slices in Banach spaces” by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda Zoca. Abstract: The aim of this note is study the topology generated by Lipschitz slices in the unit sphere of a Banach space. We prove that the above topology agrees with the weak topology in the unit sphere and, as a consequence, we obtain Lipschitz characterizations of classical linear topics in Banach spaces, as Radon-Nikodym property, convex point of continuity property and strong regularity, which shows that the above classical linear properties only depend on the natural uniformity in the Banach space given by the metric and the linear structure. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.04430

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Abstract of a paper by Jarno Talponen
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Unconditional and bimonotone structures in high density Banach spaces” by Jarno Talponen. Abstract: It is shown that every normalized weakly null sequence of length $k_{\lambda}$ in a Banach space has a subsequence of length $\lambda$ which is an unconditional basic sequence; here $k_{\lambda}$ is a large cardinal depending on a given infinite cardinal $\lambda$. Transfinite topological games on Banach spaces are analyzed which determine the existence of a long unconditional basic sequence. Then 'asymptotic disentanglement' condition in a transfinite setting is studied which ensures a winning strategy for the unconditional basic sequence builder in the above game. The following problem is investigated: When does a Markushevich basic sequence with length uncountable regular cardinal $k$ admit a subsequence of the same length which is a bimonotone basic sequence? Stabilizations of projectional resolutions of the identity (PRI) are performed under a density contravariance principle to gain some additional strong regularity properties, such as bimonotonicity. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.04408

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