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July 2015

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Abstract of a paper by O. Delgado and E. A. Sanchez Perez
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "Strong extensions for $q$-summing operators acting in $p$-convex function spaces for $1 \le p \le q$" by O. Delgado and E. A. Sanchez Perez. Abstract: Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $\sigma$-finite measure $\mu$. We combine the structure of the spaces $L^p(\mu)$ and $L^q(\xi)$ for constructing the new space $S_{X_p}^{\,q}(\xi)$, where $\xi$ is a probability Radon measure on a certain compact set associated to $X$. We show some of its properties, and the relevant fact that every $q$-summing operator $T$ defined on $X$ can be continuously (strongly) extended to $S_{X_p}^{\,q}(\xi)$. This result turns out to be a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provide (strong) factorizations for $q$-summing operators through $L^q$-spaces when $1 \le q \le p$. Thus, our result completes the picture, showing what happens in the complementary case $1\le p\le q$, opening the door to the study of the multilinear versions of $q$-summing operators also in these cases. Archive classification: math.FA Mathematics Subject Classification: 46E30, 47B38 Submitted from: easancpe(a)mat.upv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.09010 or http://arXiv.org/abs/1506.09010

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Abstract of a paper by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki
by alspach＠math.okstate.edu 09 Jul '15

09 Jul '15

This is an announcement for the paper "Rethinking Polyhedrality for Lindenstrauss Spaces" by Emanuele Casini, Enrico Miglierina, and Lukasz Piasecki. Abstract: A recent example by the authors (see arXiv:1503.09088 [math.FA]) shows that an old result of Zippin about the existence of an isometric copy of $c$ in a separable Lindenstrauss space is incorrect. The same example proves that some characterizations of polyhedral Lindenstrauss spaces, based on the result of Zippin, are false. The main result of the present paper provides a new characterization of polyhedrality for the preduals of $\ell_{1}$ and gives a correct proof for one of the older. Indeed, we prove that for a space $X$ such that $X^{*}=\ell_{1}$ the following properties are equivalent: (1) $X$ is a polyhedral space; (2) $X$ does not contain an isometric copy of $c$; (3) $\sup\left\{ x^{*}(x)\,:\, x^{*}\in\mathrm{ext}\left(B_{X^{*}}\right)\setminus D(x)\right\} <1$ for each $x\in S_{X}$, where $D(x)=\left\{ x^{*}\in S_{X^{*}}:x^{*}(x)=1\right\}$. By known theory, from our result follows that a generic Lindenstrauss space is polyhedral if and only if it does not contain an isometric copy of $c$. Moreover, a correct version of the result of Zippin is derived as a corollary of the main result. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 46B25 Submitted from: enrico.miglierina(a)unicatt.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.08559 or http://arXiv.org/abs/1506.08559

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