Banach April 2017

banach@mathdept.okstate.edu

1 participants
13 discussions

Abstract of a paper by Tomasz Kochanek, Eva Pernecká
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete” by Tomasz Kochanek<https://arxiv.org/find/math/1/au:+Kochanek_T/0/1/0/all/0/1>, Eva Pernecká<https://arxiv.org/find/math/1/au:+Pernecka_E/0/1/0/all/0/1>. Abstract: Let $M$ be a compact subset of a superreflexive Banach space. We prove a certain `weak$^*$-version' of Pe\l czy\'nski's property (V) for the Banach space of Lipschitz functions on $M$. As a consequence, we show that its predual, the Lipschitz-free space $\mathbb{F}(M)$, is weakly sequentially complete. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.07896

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Abstract of a paper by Ivan Yaroslavtsev
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces” by Ivan Yaroslavtsev<https://arxiv.org/find/math/1/au:+Yaroslavtsev_I/0/1/0/all/0/1>. Abstract: In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1, \infty)$ and all purely discontinuous $X$-valued martingales $M$ an $N$ such tha $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb{E}\|N_{\infty}\|_p\leq C_p\mathbb{E}\|M_{\infty}\|_p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.07817

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Abstract of a paper by Fernando Albiac, José L. Ansorena, Ben Wallis
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “On Garling sequence spaces” by Fernando Albiac<https://arxiv.org/find/math/1/au:+Albiac_F/0/1/0/all/0/1>, José L. Ansorena<https://arxiv.org/find/math/1/au:+Ansorena_J/0/1/0/all/0/1>, Ben Wallis<https://arxiv.org/find/math/1/au:+Wallis_B/0/1/0/all/0/1>. Abstract: The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leq p<\infty$ and each nonincreasing weight $w\in c_0-\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(w,p)$. We also show that $g(w,p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.07772

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Abstract of a paper by Daniel Carando, Damián Pinasco, Jorge Tomás Rodríguez
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “On the linear polarization constants of finite dimensional spaces” by Daniel Carando<https://arxiv.org/find/math/1/au:+Carando_D/0/1/0/all/0/1>, Damián Pinasco<https://arxiv.org/find/math/1/au:+Pinasco_D/0/1/0/all/0/1>, Jorge Tomás Rodríguez<https://arxiv.org/find/math/1/au:+Rodriguez_J/0/1/0/all/0/1>. Abstract: We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces $\ell_p^d$: they behave as $d^{1/p}$ if $1\leq p\leq 2$ and as $\sqrt{d}$ if $2\leq p<\infty$. For $p=\infty$ we get the asymptotic behavior up to a logarithmic factor. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.06316

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Abstract of a paper by Gonzalo Flores
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “On an identification of the Lipschitz-free spaces over subsets of $R^n$” by Gonzalo Flores<https://arxiv.org/find/math/1/au:+Flores_G/0/1/0/all/0/1>. Abstract: In this short note, we develop a method for identifying the spaces $Lip_0(U)$ for every nonempty open convex $U$ of $R^n$ and $n\in\mathbb{N}$. Moreover, we show that $\mathbb{F}(U)$ is identified with a quotient of $L_1(U; R^n)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.04405

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Abstract of a paper by Rainis Haller, Paavo Kuuseok, Märt Põldvere
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “On convex combinations of slices of the unit ball in Banach spaces” by Rainis Haller<https://arxiv.org/find/math/1/au:+Haller_R/0/1/0/all/0/1>, Paavo Kuuseok<https://arxiv.org/find/math/1/au:+Kuuseok_P/0/1/0/all/0/1>, Märt Põldvere<https://arxiv.org/find/math/1/au:+Poldvere_M/0/1/0/all/0/1>. Abstract: We prove that the following three properties for a Banach space are all different from each other: every finite convex combination of slices of the unit ball is (1) relatively weakly open, (2) has nonempty interior in relative weak topology of the unit ball, and (3) intersects the unit sphere. In particular, the 1-sum of two Banach spaces does not have property (1), but it has property (2) if both the spaces have property (1); the Banach space $C[0,1]$ does not have property (2), although it has property (3). The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.02919

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Abstract of a paper by O. Delgado, M. Mastylo, E.A. Sanchez-Perez
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “Strong factorizations of operators with applications to Fourier and Cesáro transforms” by O. Delgado<https://arxiv.org/find/math/1/au:+Delgado_O/0/1/0/all/0/1>, M. Mastylo<https://arxiv.org/find/math/1/au:+Mastylo_M/0/1/0/all/0/1>, E.A. Sanchez-Perez<https://arxiv.org/find/math/1/au:+Sanchez_Perez_E/0/1/0/all/0/1>. Abstract: Consider two continuous linear operators $T: X_1(\mu)\rightarrow Y_1(\nu)$ and $S: X_2(\mu)\rightarrow Y_2(\nu)$ between Banach function spaces related to different $\sigma$-finite measures $\mu$ and $\nu$. We characterize by means of weighted norm inequalities when $T$ can be strongly factored through $S$, that is, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all $f\in X_1(\mu)$. For the case of spaces with Schauder basis our characterization can be improved, as we show when $S$ is for instance the Fourier operator, or the Ces\`aro operator. Our aim is to study the case when the map $T$ is besides injective. Then we say that it is a representing operator ---in the sense that it allows to represent each elements of the Banach function space $X(\mu)$ by a sequence of generalized Fourier coefficients, providing a complete characterization of these maps in terms of weighted norm inequalities. Some examples and applications involving recent results on the Hausdorff-Young and the Hardy-Littlewood inequalities for operators on weighted Banach function spaces are also provided. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.02260

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Abstract of a paper by Witold Marciszewski, Grzegorz Plebanek
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “Extension operators and twisted sums of $c_0$ and $C(K)$ spaces” by Witold Marciszewski<https://arxiv.org/find/math/1/au:+Marciszewski_W/0/1/0/all/0/1>, Grzegorz Plebanek<https://arxiv.org/find/math/1/au:+Plebanek_G/0/1/0/all/0/1>. Abstract: We investigate the following problem posed by Cabello Sanch\'ez, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give the first examples of compact spaces $K$ providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either $K$ is the Cantor cube $2^{\omega_1}$ or $K$ is a separable scattered compact space of height 3 and weight $\omega_1$, then every twisted sum of $c_0$ and $C(K)$ is trivial. We also construct nontrivial twisted sums of $c_0$ and $C(K)$ for $K$ belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces $K\sebset L$which do not admit an extension operator $C(K)\rightarrow C(L)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.02139

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Abstract of a paper by Petr Hajek, Thomas Schlumprecht
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “Coarse embeddings into $c_0(\Gamma)$” by Petr Hajek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>. Abstract: Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices to let $\lambda=\mathbb{N}_{\omega}$. If $X$ is a Banach space with $dens(X)\geq\lambda$, which admits a coarse (or uniform) embedding into any $c_0(\Gamma)$, then $X$ fails to have nontrivial cotype, i.e. $X$ contains $\ell_{\infty}^n$ $C$-uniformly for every $C>1$. In the special case when $X$ has a symmetric basis, we may even conclude that it is linearly isomorphic with $c_0(dens (X))$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.01891

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Abstract of a paper by Hannes Thiel
by Bentuo Zheng (bzheng) 01 Apr '17

01 Apr '17

This is an announcement for the paper “Preduals for spaces of operators involving Hilbert spaces and trace-class operators” by Hannes Thiel<https://arxiv.org/find/math/1/au:+Thiel_H/0/1/0/all/0/1>. Abstract: Continuing the study of preduals of spaces $L(H, Y)$ of bounded, linear maps, we consider the situation that $H$ is a Hilbert space. We establish a natural correspondence between isometric preduals of $L(H, Y)$ and isometric preduals of $Y$. The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements $L(H, Y)$ in its bidual is automatically a right $L(H)$-module map. As an application, we show that isometric preduals of $L(S_1)$, the algebra of operators on the space of trace-class operators, correspond to isometric preduals of $S_1$ itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of $S_1$ making its multiplication separately weak* continuous. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.01169

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Banach April 2017