December 2016 - Banach - mathdept.okstate.edu

Abstract of a paper by Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Daugavet property and separability in Banach spaces” by Abraham Rueda Zoca<https://arxiv.org/find/math/1/au:+Zoca_A/0/1/0/all/0/1>. Abstract: We give a characterisation of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$-embedded space with another non-zero Banach space. The former characterisation also motivates the introduction of two indices related to the Daugavet property and a short study of them.. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.09698

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Abstract of a paper by Torrey M. Gallagher
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “A weak convergence theorem for mean nonexpansive mappings” by Torrey M. Gallagher<https://arxiv.org/find/math/1/au:+Gallagher_T/0/1/0/all/0/1>. Abstract: In this paper, we prove first that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the analogous result for closed, convex (not necessarily bounded) subsets of uniformly convex Opial spaces. These results generalize the classical theorems for nonexpansive maps of Browder and Petryshyn in Hilbert space and Opial in reflexive spaces satisfying Opial's condition. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.09390

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Abstract of a paper by Miek Messerschmidt
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “A Pointwise Lipschitz Selection Theorem $” by Miek Messerschmidt<https://arxiv.org/find/math/1/au:+Messerschmidt_M/0/1/0/all/0/1>. Abstract: We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.08435

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Abstract of a paper by J.M. Calabuig, J. Rodríguez, P. Rueda, E.A. Sánchez-Pérez
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “On p-Dunford integrable functions with values in Banach spaces” by J.M. Calabuig<https://arxiv.org/find/math/1/au:+Calabuig_J/0/1/0/all/0/1>, J. Rodríguez<https://arxiv.org/find/math/1/au:+Rodriguez_J/0/1/0/all/0/1>, P. Rueda<https://arxiv.org/find/math/1/au:+Rueda_P/0/1/0/all/0/1>, E.A. Sánchez-Pérez<https://arxiv.org/find/math/1/au:+Sanchez_Perez_E/0/1/0/all/0/1>. Abstract: Let $(\Omega, \Sigma, \mu)$ be a complete probability space, $X$ a Banach space and $1\leq p<\infty$. In this paper we discuss several aspects of $p$-Dunford integrable functions $f: \Omega\rightarrow X$. Special attention is paid to the compactness of the Dunford operator of $f$. We also study the $p$-Bochner integrability of the composition $u\circ f: \Omega\rightarrow Y$, where $u$ is a $p$-summing operator from $X$ to another Banach space $Y$. Finally, we also provide some tests of $p$-Dunford integrability by using $w^*$-thick subsets of $X^*$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.08087

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Abstract of a paper by W. Cuellar-Carrera
by Bentuo Zheng (bzheng) 01 Dec '16

01 Dec '16

This is an announcement for the paper “Non-ergodic Banach spaces are near Hilbert” by W. Cuellar-Carrera<https://arxiv.org/find/math/1/au:+Cuellar_Carrera_W/0/1/0/all/0/1>. Abstract: We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ $(2<p<\infty)$ is ergodic. This reinforces the conjecture that $\el_2$ is the only non ergodic Banach space. As an application of our criterion for ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $\ell_p$, for $1\leq p<2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.05500

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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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