- Banach - mathdept.okstate.edu

Abstract of a paper by Ryan M. Causey
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “The Szlenk index of $L_p(X)$ and $A_p$” by Ryan M. Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: Given a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1<p<\infty$, we provide an optimal relationship between the Szlenk index of $K$ and the Szlenk index of an associated subset of $L_p(X)^*$. As an application, given a Banach space X, we prove an optimal estimate of the Szlenk index of $L_p(X)$ in terms of the Szlenk index of $X$. This extends a result of H\'ajek and Schlumprecht to uncountable ordinals. More generally, given an operator $A: X\rightarrow Y$, we provide an estimate of the Szlenk index of the "pointwise $A$" operator $A_p: L_p(X)\rightarrow L_p(Y)$ in terms of the Szlenk index of $A$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.06226

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Abstract of a paper by Trond A. Abrahamsen, Vegard Lima
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Relatively weakly open convex combinations of slices” by Trond A. Abrahamsen<https://arxiv.org/find/math/1/au:+Abrahamsen_T/0/1/0/all/0/1>, Vegard Lima<https://arxiv.org/find/math/1/au:+Lima_V/0/1/0/all/0/1>. Abstract: We show that $c_0$, and in fact $C(K)$ for any scattered compact Hausdorff space $K$, have the property that finite convex combinations of slices of the unit ball are relatively weakly open. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.06169

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Abstract of a paper by Loic Gaillard, Pascal Lefèvre
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Lacunary Müntz spaces: isomorphisms and Carleson embeddings” by Loic Gaillard<https://arxiv.org/find/math/1/au:+Gaillard_L/0/1/0/all/0/1>, Pascal Lefèvre<https://arxiv.org/find/math/1/au:+Lefevre_P/0/1/0/all/0/1>. Abstract: In this paper we prove that $M_{\Lambda}^p$ is almost isometric to $\ell_p$ in the canonical way when $\Lambda$ is lacunary with a large ratio. On the other hand, our approach can be used to study also the Carleson measures for M\"untz spaces $M_{\Lambda}^p$ when $\Lambda$ is lacunary. We give some necessary and some sufficient conditions to ensure that a Carleson embedding is bounded or compact. In the hilbertian case, the membership to Schatten classes is also studied. When $\Lambda$ behaves like a geometric sequence the results are sharp, and we get some characterizations. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.05807

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Abstract of a paper by Artsrun Sargsyan, Martin Grigoryan
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Universal function for a weighted space $L^1_u[0,1]$” by Artsrun Sargsyan<https://arxiv.org/find/math/1/au:+Sargsyan_A/0/1/0/all/0/1>, Martin Grigoryan<https://arxiv.org/find/math/1/au:+Grigoryan_M/0/1/0/all/0/1>. Abstract: It is shown that there exist such a function g from $L^1[0,1]$ and a weight function $0<u(x)<=1$ that g is universal for the weighted space $L^1_u[0,1]$ with respect to signs of its Fourier-Walsh coefficients. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.05776

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Abstract of a paper by Razvan Anisca, Valentin Ferenczi, Yolanda Moreno
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “On the classification of positions and of complex structures in Banach spaces” by Razvan Anisca<https://arxiv.org/find/math/1/au:+Anisca_R/0/1/0/all/0/1>, Valentin Ferenczi<https://arxiv.org/find/math/1/au:+Ferenczi_V/0/1/0/all/0/1>, Yolanda Moreno<https://arxiv.org/find/math/1/au:+Moreno_Y/0/1/0/all/0/1>. Abstract: A topological setting is defined to study the complexities of the relation of equivalence of embeddings (or "position") of a Banach space into another and of the relation of isomorphism of complex structures on a real Banach space. The following results are obtained: a) if $X$ is not uniformly finitely extensible, then there exists a space $Y$ for which the relation of position of $Y$ inside $X$ reduces the relation $E_0$ and therefore is not smooth; b) the relation of position of $\ell_p$ inside $\ell_p$, or inside $L_p$, $p\neq 2$, reduces the relation $E_1$ and therefore is not reducible to an orbit relation induced by the action of a Polish group; c) the relation of position of a space inside another can attain the maximum complexity $E_{max}$; d) there exists a subspace of $L_p$, $1\leq p<2$, on which isomorphism between complex structures reduces $E_1$ and therefore is not reducible to an orbit relation induced by the action of a Polish group. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.04263

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Abstract of a paper by Jens Flemming, Daniel Gerth
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Injectivity and weak$^*$-to-weak continuity suffice for convergence rates in $\ell_1$-regularization” by Jens Flemming<https://arxiv.org/find/math/1/au:+Flemming_J/0/1/0/all/0/1>, Daniel Gerth<https://arxiv.org/find/math/1/au:+Gerth_D/0/1/0/all/0/1>. Abstract: We show that the convergence rate of $\ell_1$-regularization for linear ill-posed equations is always $O(\delta)$ if the exact solution is sparse and if the considered operator is injective and weak$^*$-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.03460

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Abstract of a paper by A. Aydın, E. Yu. Emelyanov, N. Erkurşun Özcan, M. A. A. Marabeh
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Compact-Like Operators in Lattice-Normed Spaces” by A. Aydın<https://arxiv.org/find/math/1/au:+Aydin_A/0/1/0/all/0/1>, E. Yu. Emelyanov<https://arxiv.org/find/math/1/au:+Emelyanov_E/0/1/0/all/0/1>, N. Erkurşun Özcan<https://arxiv.org/find/math/1/au:+Ozcan_N/0/1/0/all/0/1>, M. A. A. Marabeh<https://arxiv.org/find/math/1/au:+Marabeh_M/0/1/0/all/0/1>. Abstract: A linear operator $T$ between two lattice-normed spaces is said to be $p$-compact if, for any $p$-bounded net $x_{\alpha}$, the net $Tx_{\alpha}$ has a $p$-convergent subnet. $p$-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, AM-compact operators, etc. Similar to M-weakly and L-weakly compact operators, we define $p$-M-weakly and $p$-L-weakly compact operators and study some of their properties. We also study $up$-continuous and $up$-compact operators between lattice-normed vector lattices. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.03073

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Abstract of a paper by Vladimir Kadets, Miguel Martin, Javier Meri, Antonio Perez
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Spear operators between Banach spaces” by Vladimir Kadets<https://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1>, Miguel Martin<https://arxiv.org/find/math/1/au:+Martin_M/0/1/0/all/0/1>, Javier Meri<https://arxiv.org/find/math/1/au:+Meri_J/0/1/0/all/0/1>, Antonio Perez<https://arxiv.org/find/math/1/au:+Perez_A/0/1/0/all/0/1>. Abstract: The aim of this manuscript is to study $\emph{spear operators}$: bounded linear operators $G$ between Banach spaces $X$ and $Y$ satisfying that for every other bounded linear operator $T: X\rightarrow Y$ there exists a modulus-one scalar $\omega$ such that $$\|G+\omega T\|=1+\|T\|$$. To this end, we introduce two related properties, one weaker called the alternative Daugavet property (if rank-one operators $T$ satisfy the requirements), and one stronger called lushness, and we develop a complete theory about the relations between these three properties. To do this, the concepts of spear vector and spear set play an important role. Further, we provide with many examples among classical spaces, being one of them the lushness of the Fourier transform on $L_1$. We also study the relation of these properties with the Radon-Nikod\'ym property, with Asplund spaces, with the duality, and we provide some stability results. Further, we present some isometric and isomorphic consequences of these properties as, for instance, that $\ell_1$ is contained in the dual of the domain of every real operator with infinite rank and the alternative Daugavet property, and that these three concepts behave badly with smoothness and rotundity. Finally, we study Lipschitz spear operators (that is, those Lipschitz operators satisfying the Lipschitz version of the equation above) and prove that (linear) lush operators are Lipschitz spear operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.02977

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New Book
by Fred Dashiell 07 Jan '17

07 Jan '17

The following book has recently been released: "Banach Spaces of Continuous Functions as Dual Spaces" by H.G. Dales, F.K. Dashiell Jr., A.T.-M. Lau, and D. Strauss, CMS Books in Mathematics, Springer, 2016. Springer website: http://www.springer.com/us/book/9783319323473 Amazon: https://www.amazon.com/Banach-Spaces-Continuous-Functions-Mathematics/dp/331 9323474/ Thanks Fred Dashiell

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Abstract of a paper by Florent P. Baudier, Ryan Causey, Stephen DIlworth, Denka Kutzarova, Nirina L. Randrianarivony, Thomas Schlumprecht, Sheng Zhang
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “On the geometry of the countably branching diamond graphs” by Florent P. Baudier<https://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1>, Ryan Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>, Stephen DIlworth<https://arxiv.org/find/math/1/au:+DIlworth_S/0/1/0/all/0/1>, Denka Kutzarova<https://arxiv.org/find/math/1/au:+Kutzarova_D/0/1/0/all/0/1>, Nirina L. Randrianarivony<https://arxiv.org/find/math/1/au:+Randrianarivony_N/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>, Sheng Zhang<https://arxiv.org/find/math/1/au:+Zhang_S/0/1/0/all/0/1>. Abstract: In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_{\omega k})_{k\in\mathbb{N}$ is investigated. In particular it is shown that for every $\epsilon>0$ and $k\in\mathbb{N}, D_{\omega k}$ embeds bi-Lipschiztly with distortion at most $6(1+\epsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_{\omega k})_{k\in\mathbb{N}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of separable reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some problems in renorming theory are also discussed. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01984

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