April 2016 - Banach - mathdept.okstate.edu

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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Abstract of a paper by Marek Cuth and Michal Johanis
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals” by Marek Cuth and Michal Johanis. Abstract: We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_{\infty}$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of $\ell_{\infty}$ into the dual" and "containing as good copy of $\ell_1$ as possible" in a general Banach space. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.04131

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Abstract of a paper by Tepper L. Gill
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The S-basis and M-basis Problems for Separable Banach Spaces” by Tepper L. Gill. Abstract: This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense embedding in $\mcB$ and $\mcB$ is a continuous dense embedding in $\mcH_2$. This is the best possible improvement of a theorem due to Mazur (see \cite{BA} and also \cite{PE1}). The second objective is show how $\mcH_2$ allows us to provide a positive answer to the Marcinkiewicz-basis (M-basis) problem. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.03547

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Abstract of a paper by Szymon Draga and Tomasz Kochanek
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The Szlenk power type and tensor products of Banach spaces” by Szymon Draga and Tomasz Kochanek. Abstract: We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $\omega$. We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.03461

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Abstract of a paper by Beata Randrianantoanina
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “On sign embeddings and narrow operators on $L_2$” by Beata Randrianantoanina. Abstract: The goal of this note is two-fold. First we present a brief overview of "weak" embeddings, with a special emphasis on sign embeddings which were introduced by H. P. Rosenthal in the early 1980s. We also discuss the related notion of narrow operators, which was introduced by A. Plichko and M. Popov in 1990. We give examples of applications of these notions in the geometry of Banach spaces and in other areas of analysis. We also present some open problems. In the second part we prove that Rosenthal's celebrated characterization of narrow operators on $L_1$ is also true for operators on $L_2$. This answers, for $p=2$, a question posed by Plichko and Popov in 1990. For $1<p<2$ the problem remains open. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.02710

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

30 Apr '16

This is an announcement for the paper “Metrical Characterizations of super weakly compact operators” by Ryan M. Causey and Stephen J. Dilworth. Abstract: We characterize super weakly compact operators as those through which binary tree and diamond and Laakso graphs may not be factored with uniform distortion. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.01810

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Abstract of a paper by Sheldon Dantas
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The strong Bishop-Phelps-Bollobás property” by Sheldon Dantas. Abstract: In this paper we introduce the strong Bishop-Phelps-Bollob$\'$as property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a Banach space $X$ is uniformly convex if and only if the pair $(X, N)$ satisfies the sBPBp. Positive results of pairs of Banach spaces $(X, Y)$ satisfying this property are given and concrete pairs of Banach spaces $(X, Y)$ failing it are exhibited. A complete characterization of the sBPBp for the pairs $(\ell_p, \ell_q)$ is also provided. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.01461

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