Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Andrew Swift
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “On coarse Lipschitz embeddability into $c_0(\kappa)$” by Andrew Swift<https://arxiv.org/find/math/1/au:+Swift_A/0/1/0/all/0/1>. Abstract: In 1994, Jan Pelant proved that a metric property related to the notion of paracompactness called the uniform Stone property characterizes a metric space's uniform embeddability into $c_0(\kappa)$ for some cardinality $\kappa$. In this paper it is shown that coarse Lipschitz embeddability of a metric space into $c_0(\kappa)$ can be characterized in a similar manner. It is also shown that coarse, uniform, and bi-Lipschitz embeddability into $c_0(\kappa)$ are equivalent notions for normed linear spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.04623

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Abstract of a paper by Spiros A. Argyros, Pavlos Motakis, Bünyamin Sari
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “A study of conditional spreading sequences ” by Spiros A. Argyros<https://arxiv.org/find/math/1/au:+Argyros_S/0/1/0/all/0/1>, Pavlos Motakis<https://arxiv.org/find/math/1/au:+Motakis_P/0/1/0/all/0/1>, Bünyamin Sari<https://arxiv.org/find/math/1/au:+Sari_B/0/1/0/all/0/1>. Abstract: It is shown that every conditional spreading sequence can be decomposed into two well behaved parts, one being unconditional and the other being convex block homogeneous, i.e. equivalent to its convex block sequences. This decomposition is then used to prove several results concerning the structure of spaces with conditional spreading bases as well as results in the theory of conditional spreading models. Among other things, it is shown that the space $C(\omega^{omega})$ is universal for all spreading models, i.e., it admits all spreading sequences, both conditional and unconditional, as spreading models. Moreover, every conditional spreading sequence is generated as a spreading model by a sequence in a space that is quasi-reflexive of order one. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.04443

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Abstract of a paper by S. Rajesh
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Fixed points of left reversible semigroup of isometry mappings in Banach spaces” by S. Rajesh<https://arxiv.org/find/math/1/au:+Rajesh_S/0/1/0/all/0/1>. Abstract: In this paper, we prove the existence of a common fixed point in $C(K)$, the Chebyshev center of $K$, for a left reversible semigroup of isometry mappings. This existence result improves the results obtained by Lim et al. and Brodskii and Milman. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.04087

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Abstract of a paper by Debmalya Sain, Puja Ghosh, Kallol Paul
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces” by Debmalya Sain<https://arxiv.org/find/math/1/au:+Sain_D/0/1/0/all/0/1>, Puja Ghosh<https://arxiv.org/find/math/1/au:+Ghosh_P/0/1/0/all/0/1>, Kallol Paul<https://arxiv.org/find/math/1/au:+Paul_K/0/1/0/all/0/1>. Abstract: We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space $X$, which answers a question raised recently by one of the authors in \cite{S} [D. Sain, \textit{Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, Journal of Mathematical Analysis and Applications, accepted, 2016}]. We prove that $T\in B(X)$ is left symmetric if and only if $T$ is the zero operator. If $X$ is two-dimensional then the same characterization can be obtained without the smoothness assumption. We also explore the properties of right symmetric linear operators defined on a finite dimensional real Banach space. In particular, we prove that smooth linear operators on a finite-dimensional strictly convex and smooth real Banach space can not be right symmetric. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.03663

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Abstract of a paper by S. Rajesh, P. Veeramani
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings” by S. Rajesh<https://arxiv.org/find/math/1/au:+Rajesh_S/0/1/0/all/0/1>, P. Veeramani<https://arxiv.org/find/math/1/au:+Veeramani_P/0/1/0/all/0/1>. Abstract: Let $(A,B)$ be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and $T: A\cup B\rightarrow A\cup B$ be a continuous and asymptotically relatively nonexpansive map. We prove that there exists $x\in A\cup B$ such that $\|x - Tx\| = \emph{dist}(A, B)$ whenever $T(A)\subset B, T(B)\subset A$. Also, we establish that if $T(A)\subset A, T(B)\subset B$, then there exist $x\in A$ and $y\in B$ such that $Tx=x, Ty=y$ and $\|x - y\| = \emph{dist}(A, B)$. We prove the aforesaid results when the pair $(A,B)$ has the rectangle property and property $UC$. In case of $A=B$, we obtain, as a particular case of our results, the basic fixed point theorem for asymptotically nonexpansive maps by Goebel and Kirk. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.02484

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Abstract of a paper by Emanuele Casini, Enrico Miglierina, Łukasz Piasecki, Roxana Popescu
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Stability constants of the weak$^*$ fixed point property for the space $\ell_1$” by Emanuele Casini<https://arxiv.org/find/math/1/au:+Casini_E/0/1/0/all/0/1>, Enrico Miglierina<https://arxiv.org/find/math/1/au:+Miglierina_E/0/1/0/all/0/1>, Łukasz Piasecki<https://arxiv.org/find/math/1/au:+Piasecki_L/0/1/0/all/0/1>, Roxana Popescu<https://arxiv.org/find/math/1/au:+Popescu_R/0/1/0/all/0/1>. Abstract: The main aim of the paper is to study some quantitative aspects of the stability of the weak∗ fixed point property for nonexpansive maps in $\ell_1$ (shortly, $w^*$-fpp). We focus on two complementary approaches to this topic. First, given a predual $X$ of $\ell_1$ such that the $\sigma(\ell_1, X)$-fpp holds, we precisely establish how far, with respect to the Banach-Mazur distance, we can move from X without losing the $w^*$-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in $\ell_1$ containing all $\sigma(\ell_1, X)$-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the $w^*$-fpp in the restricted framework of preduals of $\ell_1$. Namely, we show that every predual $X$ of $\ell_1$ with a distance from $c_0$ strictly less than 3, induces a weak∗ topology on $\ell_1$ such that the $\sigma(\ell_1, X)$-fpp holds. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.02133

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Abstract of a paper by Nik Weaver
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “On the unique predual problem for Lipschitz spaces” by Nik Weaver<https://arxiv.org/find/math/1/au:+Weaver_N/0/1/0/all/0/1>. Abstract: For any metric space $X$, the predual of Lip$(X)$ is unique. If $X$ has finite diameter or is complete and convex --- in particular, if it is a Banach space --- then the predual of Lip$_0(X)$ is unique. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.01812

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Abstract of a paper by Richard Lechner
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Factorization in $SL_{\infty}$” by Richard Lechner<https://arxiv.org/find/math/1/au:+Lechner_R/0/1/0/all/0/1>. Abstract: We show that the non-separable Banach space $SL_{\infty}$ is primary. This is achieved by directly solving the infinite dimensional factorization problem in $SL_{\infty}$. In particular, we bypass Bourgain's localization method. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.00622

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Abstract of a paper by Cormac Walsh
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces” by Cormac Walsh<https://arxiv.org/find/math/1/au:+Walsh_C/0/1/0/all/0/1>. Abstract: We study the horofunction boundaries of Hilbert and Thompson geometries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.07508

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Abstract of a paper by Benjamin Miesch, Maël Pavón
by Bentuo Zheng (bzheng) 01 Nov '16

01 Nov '16

This is an announcement for the paper “Ball Intersection Properties in Metric Spaces ” by Benjamin Miesch<https://arxiv.org/find/math/1/au:+Miesch_B/0/1/0/all/0/1>, Maël Pavón<https://arxiv.org/find/math/1/au:+Pavon_M/0/1/0/all/0/1>. Abstract: We show that in complete metric spaces, $4$-hyperconvexity is equivalent to finite hyperconvexity. Moreover, every complete, almost $n$-hyperconvex metric space is $n$-hyperconvex. This generalizes among others results of Lindenstrauss and answers questions of Aronszajn-Panitchpakdi. Furthermore, we prove local-to-global results for externally and weakly externally hyperconvex subsets of hyperconvex metric spaces and find sufficient conditions in order for those classes of subsets to be convex with respect to a geodesic bicombing. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.03307

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