December 2015 - Banach - mathdept.okstate.edu

Abstract of a paper by Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, and Artem Zvavitch
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Do Minkowski averages get progressively more convex?" by Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, and Artem Zvavitch. Abstract: Let us define, for a compact set $A \subset \mathbb{R}^n$, the Minkowski averages of $A$: $$ A(k) =3D \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldo= ts, a_k\in A\right\}=3D\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{= A + \cdots + A}}\Big). $$ We study the monotonicity of the convergence of $A(= k)$ towards the convex hull of $A$, when considering the Hausdorff distance, the volume deficit and a non-convexity index of Schneider as measures of convergence. For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, we prove that a strong form of monotonic= ity holds, and for the Hausdorff distance, we establish that the sequence is eventually nonincreasing. Archive classification: math.FA math.OC Remarks: 6 pages, including figures. Contains announcement of results th= The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.03718 or http://arXiv.org/abs/1512.03718

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Abstract of a paper by Mathieu Meyer and Shlomo Reisner
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "The isotropy constant and boundary properties of convex bodies" by Mathieu Meyer and Shlomo Reisner. Abstract: Let ${\cal K}^n$ be the set of all convex bodies in $\mathbb R^n$ endowed with the Hausdorff distance. We prove that if $K\in {\cal K}^n$ has positive generalized Gauss curvature at some point of its boundary, then $K$ is not a local maximizer for the isotropy constant $L_K$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 52A20, 53A05 Submitted from: reisner(a)math.haifa.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.02927 or http://arXiv.org/abs/1512.02927

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Abstract of a paper by Julio Becerra Guerrero, Gines Lopez Perez, and Abraham Rueda
by alspach＠math.okstate.edu 16 Dec '15

16 Dec '15

This is an announcement for the paper "Some results on almost square Banach spaces" by Julio Becerra Guerrero, Gines Lopez Perez, and Abraham Rueda. Abstract: We study almost square Banach spaces under a topological point of view. Indeed, we prove that the class of Banach spaces which admits an equivalent norm to be ASQ is that of those Banach spaces which contain an isomorphic copy of $c_0$. We also prove that the symmetric projective tensor products of an almost square Banach space have the strong diameter two property Archive classification: math.FA Remarks: 12 pages Submitted from: arz0001(a)correo.ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.00610 or http://arXiv.org/abs/1512.00610

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Abstract of a paper by Alexandru Aleman and Laurian Suciu
by alspach＠math.okstate.edu 01 Dec '15

01 Dec '15

This is an announcement for the paper "On ergodic operator means in Banach spaces" by Alexandru Aleman and Laurian Suciu. Abstract: We consider a large class of operator means and prove that a number of ergodic theorems, as well as growth estimates known for particular cases, continue to hold in the general context under fairly mild regularity conditions. The methods developed in the paper not only yield a new approach based on a general point of view, but also lead to results that are new, even in the context of the classical Cesaro means. Archive classification: math.FA Submitted from: laurians2002(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.08929 or http://arXiv.org/abs/1511.08929

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Abstract of a paper by Peter G. Casazza, Daniel Freeman, and Richard G. Lynch
by alspach＠math.okstate.edu 01 Dec '15

01 Dec '15

This is an announcement for the paper "Weaving Schauder frames" by Peter G. Casazza, Daniel Freeman, and Richard G. Lynch. Abstract: We extend the concept of weaving Hilbert space frames to the Banach space setting. Similar to frames in a Hilbert space, we show that for any two approximate Schauder frames for a Banach space, every weaving is an approximate Schauder frame if and only if there is a uniform constant $C\geq 1$ such that every weaving is a $C$-approximate Schauder frame. We also study weaving Schauder bases, where it is necessary to introduce two notions of weaving. On one hand, we can ask if two Schauder bases are woven when considered as Schauder frames with their biorthogonal functionals, and alternatively, we can ask if each weaving of two Schauder bases remains a Schauder basis. We will prove that these two notions coincide when all weavings are unconditional, but otherwise they can be different. Lastly, we prove two perturbation theorems for approximate Schauder frames. Archive classification: math.FA Mathematics Subject Classification: 46B20, 42C15 Submitted from: rilynch37(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.06093 or http://arXiv.org/abs/1511.06093

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Abstract of a paper by Jesus M. F. Castillo, Wilson Cuellar, Valentin Ferenczi, and Yolanda Moreno
by alspach＠math.okstate.edu 01 Dec '15

01 Dec '15

This is an announcement for the paper "Complex structures on twisted Hilbert spaces" by Jesus M. F. Castillo, Wilson Cuellar, Valentin Ferenczi, and Yolanda Moreno. Abstract: We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck $Z_2$ space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of K\"othe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant K\"othe function spaces then there are complex structures on it that can be extended to a complex structure of the twisted Hilbert space. Regarding the hyperplane problem we show that no complex structure on $\ell_2$ can be extended to a complex structure on an hyperplane of $Z_2$ containing it. Archive classification: math.FA Submitted from: castillo(a)unex.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.05867 or http://arXiv.org/abs/1511.05867

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Abstract of a paper by Stephane Chretien
by alspach＠math.okstate.edu 01 Dec '15

01 Dec '15

This is an announcement for the paper "On the restricted invertibility problem with an additional constraint for random matrices" by Stephane Chretien. Abstract: The Restricted Invertibility problem is the problem of selecting the largest subset of columns of a given matrix $X$, while keeping the smallest singular value of the extracted submatrix above a certain threshold. In this paper, we address this problem in the simpler case where $X$ is a random matrix but with the additional constraint that the selected columns be almost orthogonal to a given vector $v$. Our main result is a lower bound on the number of columns we can extract from a normalized i.i.d. Gaussian matrix for the worst $v$. Archive classification: math.PR math.FA Submitted from: stephane.chretien(a)npl.co.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.05463 or http://arXiv.org/abs/1511.05463

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Abstract of a paper by Michal Kraus
by alspach＠math.okstate.edu 01 Dec '15

01 Dec '15

This is an announcement for the paper "Quantitative coarse embeddings of quasi-Banach spaces into a Hilbert space" by Michal Kraus. Abstract: We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46A16, 51F99, 46B85 Remarks: 11 pages Submitted from: mkraus(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1511.05214 or http://arXiv.org/abs/1511.05214

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