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Abstract of a paper by Mohammad Imdad, Rqeeb Gubran and Md Ahmadullah
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Using an implicit function to prove common fixed point theorems” by Mohammad Imdad, Rqeeb Gubran and Md Ahmadullah. Abstract: We observe that, Theorem 2 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105) is not correct in its present form. In this paper, by correcting and enriching aforementioned theorem, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike many other cases, our main results deduce a nonlinear order-theoretic version of a well-known fixed point theorem (proved for quasi-contraction) due to $\'{C}iri\'{c}$ (Proc. Amer. Math. Soc. (54) 267-273, 1974). Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105). The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.05743

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Abstract of a paper by Chafiq Benhida, Kamila Klis-Garlicka and Marek Ptak
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Skew-symmetric operators and reflexivity” by Chafiq Benhida, Kamila Klis-Garlicka and Marek Ptak. Abstract: In contrast to the subspaces of all C-symmetric operators, we show that the subspaces of all skew-C symmetric operators are reflexive and even hyperreflexive with the constant $\kappa(\C^s)\leqslant 3$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.05724

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Abstract of a paper by Luis Garcia-Lirola and Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Unconditional almost squareness and applications to spaces of Lipschitz functions” by Luis Garcia-Lirola and Abraham Rueda Zoca. Abstract: We introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of any dual almost square Banach space. We also take advantage of this notion to provide some criterion of non-duality of some subspaces of scalar as well as vector valued Lipschitz functions. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.04699

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Abstract of a paper by Asuman G. Aksoy and Qidi Peng
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “On a Theorem of S. N. Bernstein for Banach Spaces” by Asuman G. Aksoy and Qidi Peng. Abstract: This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We prove that if $X$ is an infinite-dimensional Banach space and $(Y_n)$ is a nested sequence of subspaces of $X$ such that $Y_n\subset Y_{n+1}$ and $\bar{Y_n}\subset Y_{n+1}$ for any $n\in\mathbb{N}$ and if $(d_n)$ be a decreasing sequence of positive numbers tending to 0, then for any $0<c\leq 1$ there exists $x_c\in X$ such that the distance $\rho (x_c, Y_n)$ from $x_c$ to $Y_n$ satisfies $$cd_n\leq\rho(x_c, Y_n)\leq 4c d_n$$. We prove the above by first improving Borodin's result \cite{Borodin} for Banach spaces by weakening the condition on the sequence $(d_n)$. Lastly, we compare subsequences $(d(\phi_n))$ under different choices of $(\phi_n)$ and examine their effects on approximation. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.04592

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Abstract of a paper by Y. Deng, M. O’Brien, V. G. Troitsky
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Unbounded Norm Convergence in Banach Lattices” by Y. Deng, M. O’Brien, V. G. Troitsky. Abstract: A net $(x_\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x\in X$ if $|x_\alpha-x|\hat u$ converges to 0 in order for all $u\in X_+$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_\alpha)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\||x_\alpha-x|\hat u\|\rightarrow 0$ for all $u\in X_+$. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.03538

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Abstract of a paper by Ryan M. Causey
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Szlenk and $w^*$-dentability indices of $C(K)$” by Ryan M. Causey. Abstract: Given any compact, Hausdorff space $K$ and $1<p<\infty$, we compute the Szlenk and $w^*$-dentability indices of the spaces $C(K)$ and $L_p(C(K))$ We show that if $K$ is compact, Hausdorff, scattered, $CB(K)$ is the Cantor-Bendixson index of $K$, and $\eta$ is the minimum ordinal such that $CB(K)\leq \omega\eta$, then $S_z(C(K))=\omega\eta$ and $D_z(C(K))=S_z(L_p(C(K)))=\omega_1+eta$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.01969

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Abstract of a paper by Sheldon Dantas, Sun Kwang Kim, Han Ju Lee
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “The Bishop-Phelps-Bollobás point property” by Sheldon Dantas, Sun Kwang Kim, Han Ju Lee. Abstract: In this article, we study a version of the Bishop-Phelps-Bollob\'as property. We investigate a pair of Banach spaces $(X, Y)$ such that every operator from $X$ into $Y$ is approximated by operators which attains its norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop-Phelps-Bollob\'as point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs $(X, Y)$ which have and fail this property. Some stability results are obtained about $\ell_1$ and $\ell_\infty$ sums of Banach spaces and we also study this property for bilinear mappings. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.00245

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Abstract of a paper by Matthieu Fradelizi, Mathieu Meyer and Vlad Yaskin
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “On the volume of sections of a convex body by cones” by Matthieu Fradelizi, Mathieu Meyer and Vlad Yaskin. Abstract: Let $K$ be a convex body in $R_n$. We prove that in small codimensions, the sections of a convex body through the centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding convex intersection bodies. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.05351

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Abstract of a paper by Lei Li, Edurado Nieto and Antonio M. Peralta.
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Geometric implications of the $M(r,s)$-properties and the uniform Kadec-Klee property in JB$^*$-triples” by Lei Li, Edurado Nieto and Antonio M. Peralta. Abstract: We explore new implications of the $M(r,s)$and $M^*(r,s)$properties for Banach spaces. We show that a Banach space $X$ satisfying property $M(1,s)$ for some $0<s\leq 1$ admitting a point $x_0$ in its unit sphere at which the relative weak and norm topologies agree, satisfies the generalized Gossez-Lami Dozo property. We establish sufficient conditions, in terms of the $(r,s)$-Lipschitz weak$^*$ Kadec-Klee property on a Banach space $X$ to guarantee that its dual space satisfies the UKK$^*$ property. We determine appropriate conditions to assure that a Banach space X satisfies the $(r,s)$-Lipschitz weak$^*$ Kadec-Klee property. These results are applied to prove that every spin factor satisfies the UKK property, and consequently, the KKP and the UKK properties are equivalent for real and complex JB$^*$-triples. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.04119

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Abstract of a paper by Mohammed Bachir
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “On the Krein-Milman theorem for convex compact metrizable sets” by Mohammed Bachir. Abstract: The Krein-Milman theorem (1940) states that a convex compact subset of a Hausdorff locally convex topological space, is the closed convex hull of its extreme points. We prove in this paper that in the metrizable case, the situation is better: every convex compact metrizable subset of a Hausdorff locally convex topological space, is the closed convex hull of its exposed points. This fails in general for not metrizable compact convex subsets. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.08473

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