Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by J F Toland
by Bentuo Zheng (bzheng) 04 Mar '18

04 Mar '18

This is an announcement for the paper “Localizing Weak Convergence in $L_{\infty}$” by J F Toland<https://arxiv.org/find/math/1/au:+Toland_J/0/1/0/all/0/1>. Abstract: For a general measure space $(X, \sL, \l)$ the pointwise nature of weak convergence in $\Li$ is investigated using singular functionals analogous to $\d$-functions in the theory of continuous functions on topological spaces. The implications for pointwise behaviour in $X$ of weakly convergent sequences in $\Li$ are inferred and the composition mapping $u\mapsto F(u)$ is shown to be sequentially weakly continuous on $\Li$ when $F:\RR \to \RR$ is continuous. When $\sB$ is the Borel $\sigma$-algebra of a locally compact Hausdorff topological space $(X, \rho)$ and $f \in L_\infty(X, \sB, \l)^*$ is arbitrary, let $\nu$ be the finitely additive measure in the integral representation of $f$ on $L_\infty(X, \sB, \l)$, and let $\hat{\nu}$ be the Borel measure in the integral representation of $f$ restricted to $C_0(X, \rho)$. From a minimax formula for $\hat{\nu}$ in terms $\nu$ it emerges that when $(X, \rho)$ is not compact, $\hat{\nu}$ may be zero when $\nu$ is not, and the set of $\nu$ for which $\hat{\nu}$ has a singularity with respect to $\l$ can be characterised. Throughout, the relation between $\d$-functions and the analogous singular functionals on $\Li$ is explored and weak convergence in $L_\infty(X,\sB,\l)$ is localized about points of $(X_{\infty}, \rho_{\infty})$, the one-point compactification of $(X, \rho)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1802.01878

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Abstract of a paper by Anna Kamińska, Yves Raynaud
by Bentuo Zheng (bzheng) 04 Mar '18

04 Mar '18

This is an announcement for the paper “Abstract Lorentz spaces and Köthe duality” by Anna Kamińska<https://arxiv.org/find/math/1/au:+Kaminska_A/0/1/0/all/0/1>, Yves Raynaud<https://arxiv.org/find/math/1/au:+Raynaud_Y/0/1/0/all/0/1>. Abstract: Given a fully symmetric Banach function space $E$ and a decreasing positive weight $w$ on $I=(0, a), 0<a\leq\infty$, the generalized Lorentz space $\Lambda_{E, w}$ is defined as the symmetrization of the canonical copy $E_w$ of $E$ on the measure space associated with the weight. If $E$ is an Orlicz space then $\Lambda_{E, w}$ is an Orlicz-Lorentz space. An investigation of the K\"othe duality of these classes is developed that is parallel to preceding works on Orlicz-Lorentz spaces. First a class of functions $M_{E, w}$, which does not need to be even a linear space, is similarly defined as the symmetrization of the space $w.E_w$. Let also $Q_{E, w}$ be the smallest fully symmetric Banach function space containing $M_{E, w}$. Then the K\"othe dual of the class $M_{E, w}$ is identified as the Lorentz space $\Lambda_{E’, w}$, while the K\"othe dual of $\Lambda_{E, w}$ is Q_{E’, w}$. The space $Q_{E, w}$ is also characterized in terms of Halperin's level functions. These results are applied to concrete Banach function spaces. In particular the K\"othe duality of Orlicz-Lorentz spaces is revisited at the light of the new results. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1802.01728

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Abstract of a paper by Leandro Candido
by Bentuo Zheng (bzheng) 04 Mar '18

04 Mar '18

This is an announcement for the paper “On the geometry of the Banach spaces $C([0, \alpha]\times K)$ for some scattered ♣-compacta” by Leandro Candido<https://arxiv.org/find/math/1/au:+Candido_L/0/1/0/all/0/1>. Abstract: For some non-metrizable scattered $K$ compacta, constructed under the assumption of the Ostaszewski's ♣-principle, we study the geometry of the Banach spaces of the form $C(M\times K)$ where $M$ is a countable compact metric space. In particular, we classify up to isomorphism all the complemented subspaces of $C([0, \omega]\times K)$ and $C([0, \omega^{\omega}]\times K)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1802.01164

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Geometrical Aspects of Banach Spaces 2018, June 25-29, 2018, University of Birmingham, UK
by Olga Maleva 14 Feb '18

14 Feb '18

Dear Colleagues, Please find below a first announcement of the workshop which we hope will be of interest to you. Please feel free to forward this information further. *Geometrical Aspects of Banach Spaces 2018** **June 25-29, 2018** **University of Birmingham** **Birmingham, United Kingdom** * Main speakers: Javier Alejandro Chavez Dominguez, University of Oklahoma Gilles Godefroy, University Paris VI Sophie Grivaux, University of Lille William Johnson*, Texas A&M University David Preiss, University of Warwick Gideon Schechtman, Weizmann Institute of *TBC The homepage of the workshop is: http://tinyurl.com/banach-workshop OR http://web.mat.bham.ac.uk/O.Maleva/banach/ We invite you to register an interest in attending the workshop and contributing a talk by _*30 April 2018*_ via http://tinyurl.com/banach-workshop/reg.php or http://web.mat.bham.ac.uk/O.Maleva/banach/reg.php We expect to be able to cover some expenses (such as accommodation and local travel) for participants at an early stage of their career (PhD students, postdocs). Please indicate if you are requesting financial support during registration (use the "free comments" field). Sincerely, Olga Maleva and Jan Kolar (organisers)

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Neal Carothers
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

Dear All, I am forwarding the following sad news to you about Neal Carothers. Bentuo > Dear All, > > > > It is with a heavy heart I am sharing with you the sad news that Dr. Neal > Carothers passed away on January 29, 2018. > > > > There are not enough words to properly describe the impact Neal made on > this > department and the math community during his tenure at BGSU. It is touching > to see all of the kind words and memories shared about Neal from the math > community. Dr. Steven Seubert who was a close friend and colleague of Neal > did a good job portraying Neal’s impactful professional career. I am > honored > to have Steve’s permission to share his message about Neal with you: > > > > “It is with the greatest sadness that I have learned of the passing of Neal > Carothers. I am indebted to Neal not only on a professional, but a > personal > level as well. He was a gifted mathematician, exceptional educator, and > expositor of the highest degree. I not only learned a lot of mathematics > from > him > > (and from his book 'Real Analysis"), but learned a lot about how to > share it > with others; his love of and passion for mathematics was evident and > infectious and he had a profound effect on my career. No doubt this was > also > true for the generations of students he trained at BGSU. > > > > He amassed the usual research credentials publishing more than a dozen > refereed research articles in some of the most prestigious journals in his > field and he supervised three students who earned their PhD under his > tutelage. He was a popular speaker delivering over a dozen invited > addresses > and wrote two books. > > > > On the administrative end, Neal served the department in all of the most > important capacities, as a member of its Advisory, Personnel, and Promotion > and Tenure Committees, and from 1999-2007, as its Chair and with impressive > results. A hallmark of his leadership was to have nurtured strong graduate > programs and fostered strong relationships with departments and programs > all > across campus by his presence at the college and university levels. > > > > I am certain that one of Neal's most cherished accomplishments would be his > reputed text 'Real analysis’. I am including below a copy of one review for > this exceptional text which compares it to several other well-known > books by > some of the most revered of all mathematicians of our era. When the review > states that Neal's book 'will sit happily alongside classics such as > Apostol's Mathematical analysis, Royden's Real analysis, Rudin's Real and > complex analysis and Hewitt and Stromberg's Real and abstract analysis,' it > is acknowledging Neal as a supreme expositor and master of his trade.' > > > > You will be sorely missed, my friend.” > > > > > > > > [IMAGE] > > > > At this time, we do not have specific arrangements for the funeral, but you > might want to check any update with Dunn Funeral Home at > http://www.dunnfuneralhome.com/obituaries/Neal-L-Carothers?obId=2936577#/cel > > ebrationWall > > > RIP, Dear Neal. > > Hanfeng > > Hanfeng Chen > > Professor and Chair

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Abstract of a paper by Joanna Garbulińska-Węgrzyn
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “A note on the universal separable Banach space with an unconditional Schauder basis with constant K” by Joanna Garbulińska-Węgrzyn<https://arxiv.org/find/math/1/au:+Garbulinska_Wegrzyn_J/0/1/0/all/0/1>. Abstract: Using the technique of Fraisse theory we construct a universal object in the class of separable Banach spaces with an unconditional Schauder basis with constant $K$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.10064

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Abstract of a paper by Debmalya Sain, Kallol Paul, Arpita Mal
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “On extreme contractions between real Banach spaces” by Debmalya Sain<https://arxiv.org/find/math/1/au:+Sain_D/0/1/0/all/0/1>, Kallol Paul<https://arxiv.org/find/math/1/au:+Paul_K/0/1/0/all/0/1>, Arpita Mal<https://arxiv.org/find/math/1/au:+Mal_A/0/1/0/all/0/1>. Abstract: We completely characterize extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, perhaps for the very first time. In order to obtain the desired characterization, we introduce the notions of (weakly) compatible point pair (CPP) and $\mu$−compatible point pair ($\mu$−CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all rank one extreme contractions in $L(\ell_4^2, \ell_4^2)$ and $L(\ell_4^2, H)$, where ℍ is any Hilbert space. We also prove that there does not exist any rank one extreme contractions in $L(H, \ell_p^2)$, whenever $p$ is even and $H$ is any Hilbert space. We further study extreme contractions between infinite-dimensional Banach spaces and obtain some analogous results. Finally, we characterize real Hilbert spaces among real Banach spaces in terms of CPP, that substantiates our motivation behind introducing these new geometric notions. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.09980

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Abstract of a paper by Tiexin Guo, Erxin Zhang, Yachao Wang, George Yuan
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Kirk's Fixed Point Theorem in Complete Random Normed modules” by Tiexin Guo<https://arxiv.org/find/math/1/au:+Guo_T/0/1/0/all/0/1>, Erxin Zhang<https://arxiv.org/find/math/1/au:+Zhang_E/0/1/0/all/0/1>, Yachao Wang<https://arxiv.org/find/math/1/au:+Wang_Y/0/1/0/all/0/1>, George Yuan<https://arxiv.org/find/math/1/au:+Yuan_G/0/1/0/all/0/1>. Abstract: Recently, stimulated by financial applications and $L_0$--convex optimization, Guo, et.al introduced the notion of $L_0$--convex compactness for an $L_0$--convex subset of a Hausdorff topological module over the topological algebra $L_0(P, K)$, where K is the scalar field of real or complex numbers and $L_0(P, K)$ the algebra of equivalence classes of $K$--valued measurable functions defined on a $\sigma$--finite measure space $(\omega, F, \mu)$, endowed with the topology of convergence locally in measure. A complete random normed module (briefly, RN module), as a random generalization of a Banach space, is just such a kind of topological module, this paper further introduces the notion of random normal structure and gives various kinds of determination theorems for a closed $L_0$--convex subset to have random normal structure or $L_0$--convex compactness, in particular we prove a characterization theorem for a closed $L_0$--convex subset to have $L_0$--convex compactness, which can be regarded as a generalization of the famous James characterization theorem for a closed convex subset of a Banach space to be weakly compact. Based on these preparations, we generalize the classical Kirk's fixed point theorem from a Banach space to a complete RN module as follows: Let $(E, \|\cdot\|)$ be a complete RN module and $V\subset E$ a nonempty $L_0$--convexly compact closed $L_0$--convex subset with random normal structure, then every nonexpansive self—mapping $f$ from $V$ to $V$ has a fixed point in $V$. The generalized Kirk's fixed point theorem is also of fundamental importance in random functional analysis, for example, we can derive from it a very general random fixed point theorem, which unifies and improves several known random fixed point theorems for random nonexpansive mappings. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.09341

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Abstract of a paper by C.S. Kubrusly, B.P. Duggal
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “On Weak Supercyclicity I” by C.S. Kubrusly<https://arxiv.org/find/math/1/au:+Kubrusly_C/0/1/0/all/0/1>, B.P. Duggal<https://arxiv.org/find/math/1/au:+Duggal_B/0/1/0/all/0/1>. Abstract: This paper provides conditions (i) to distinguish weak supercyclicity form supercyclicity for operators acting on normed and Banach spaces, and also (ii) to ensure when weak supercyclicity implies weak stability. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.08091

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Abstract of a paper by Elia Bruè, Simone Di Marino, Federico Stra
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Linear Lipschitz and $C^1$ extension operators through random projection” by Elia Bruè<https://arxiv.org/find/math/1/au:+Brue_E/0/1/0/all/0/1>, Simone Di Marino<https://arxiv.org/find/math/1/au:+Marino_S/0/1/0/all/0/1>, Federico Stra<https://arxiv.org/find/math/1/au:+Stra_F/0/1/0/all/0/1>. Abstract: We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and $C^1$ functions. This way we prove more directly a result by Lee and Naor and we generalize the $C^1$ extension theorem by Whitney to Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.07533

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