Banach March 2019

banach@mathdept.okstate.edu

2 participants
16 discussions

Informal Analysis Seminar February 24-25, 2018
by Artem Zvavitch 05 Sep '19

05 Sep '19

Dear Colleagues, The Analysis group at Kent State University is happy to announce a meeting of the Informal Analysis Seminar, which will be held at the Department of Mathematical Sciences at Kent State University, February 24-25. The seminar will feature plenary speakers Robert Connelly (Cornell University), and Peter Sternberg (Indiana University) Each speaker will deliver a four hour lecture series designed to be accessible for graduate students. Funding is available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Posters can be submitted electronically in PDF format. Further information, and an online registration form, can be found online http://www.math.kent.edu/informal We encourage you to register as soon as possible, but to receive support and/or help with hotel reservation, please register before January 29, 2018. Finally, please feel free to forward this email to any colleagues or students who you think may be interested in attending. Best regards, The Kent State Analysis Group

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Abstract of a paper by Florent P. Baudier, Pavlos Motakis, Thomas Schlumprecht, András Zsák
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “On the bi-Lipschitz geometry of lamplighter graphs” by Florent P. Baudier<https://arxiv.org/search/math?searchtype=author&query=Baudier%2C+F+P>, Pavlos Motakis<https://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P>, Thomas Schlumprecht<https://arxiv.org/search/math?searchtype=author&query=Schlumprecht%2C+T>, András Zsák<https://arxiv.org/search/math?searchtype=author&query=Zs%C3%A1k%2C+A>. Abstract: In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most $6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.07098

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Abstract of a paper by Sofiya Ostrovska, Mikhail Ostrovskii
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Generalized transportation cost spaces” by Sofiya Ostrovska<https://arxiv.org/search/math?searchtype=author&query=Ostrovska%2C+S>, Mikhail Ostrovskii<https://arxiv.org/search/math?searchtype=author&query=Ostrovskii%2C+M>. Abstract: The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete infinite metric spaces transportation cost spaces on which do not contain isometric copies of $\ell_1$, this result answers a question raised by Cuth and Johanis (2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of $\ell_1$; (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to $\ell_\infty^d$ of the corresponding dimension, and that for all finite metric spaces $M$, except a very special class, the infimum of all seminorms for which the embedding of $M$ into the corresponding seminormed space is isometric, is not a seminorm. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.10334

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Abstract of a paper by Denka Kutzarova, Pavlos Motakis
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Asymptotically symmetric spaces with hereditarily non-unique spreading models” by Denka Kutzarova<https://arxiv.org/search/math?searchtype=author&query=Kutzarova%2C+D>, Pavlos Motakis<https://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P>. Abstract: We examine a variant of a Banach space $\mathfrak{X}_{0,1}$ defined by Argyros, Beanland, and the second named author that has the property that it admits precisely two spreading models in every infinite dimensional subspace. We prove that this space is asymptotically symmetric and thus it provides a negative answer to a problem of Junge, the first. named author, and Odell. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.10098

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Abstract of a paper by Spiros A. Argyros, Pavlos Motakis
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “On the complete separation of asymptotic structures in Banach spaces” by Spiros A. Argyros<https://arxiv.org/search/math?searchtype=author&query=Argyros%2C+S+A>, Pavlos Motakis<https://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P>. Abstract: Let $(e_i)_i$ denote the unit vector basis of $\ell_p$, $1\leq p< \infty$, or $c_0$. We construct a reflexive Banach space with an unconditional basis that admits $(e_i)_i$ as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-$\ell_p$ or $c_0$ subspace. This solves a problem of E. Odell. We also construct a space with a unique $\ell_1$ spreading model and no subspace with a uniformly unique $\ell_1$ spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights. https://arxiv.org/abs/1902.10092

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Abstract of a paper by Thomas Delzant, Vilmos Komornik
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Representations of dual spaces” by Thomas Delzant<https://arxiv.org/search/math?searchtype=author&query=Delzant%2C+T>, Vilmos Komornik<https://arxiv.org/search/math?searchtype=author&query=Komornik%2C+V>. Abstract: We give a nonlinear representation of the duals for a class of Banach spaces. This leads to classroom-friendly proofs of the classical representation theorems $H'=H$ and $(L^p)'=L^q$. Our proofs extend to a family of Orlicz spaces, and yield as an unexpected byproduct a version of the Helly-Hahn-Banach theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.06811

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Abstract of a paper by Joscha Prochno
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Embeddings of Orlicz-Lorentz spaces into $L_1$” by Joscha Prochno<https://arxiv.org/search/math?searchtype=author&query=Prochno%2C+J>. Abstract: In this article, we show that Orlicz-Lorentz spaces $\ell^n_{M,a}$, $n\in\mathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\|\cdot\|_{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $d^n(a,p)$. Our approach is based on combinatorial averaging techniques and we prove a new result of independent interest that relates suitable averages with Orlicz-Lorentz norms. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.05043

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Abstract of a paper by Niels Jakob Laustsen, Jared T. White
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Subspaces that can and cannot be the kernel of a bounded operator on a Banach space” by Niels Jakob Laustsen<https://arxiv.org/search/math?searchtype=author&query=Laustsen%2C+N+J>, Jared T. White<https://arxiv.org/search/math?searchtype=author&query=White%2C+J+T>. Abstract: Given a Banach space $E$, we ask which closed subspaces may be realised as the kernel of a bounded operator $E \rightarrow E$. We prove some positive results which imply in particular that when $E$ is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space $E$ which contains a closed subspace that cannot be realized as the kernel of any bounded operator on $E$. This implies that the Banach algebra $\mathcal{B}(E)$ of bounded operators on $E$ fails to be weak*-topologically left Noetherian. The Banach space $E$ that we use is the dual of Wark's non-separable, reflexive Banach space with few operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.01677

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Abstract of a paper by M.D. Acosta, M. Soleimani-Mourchehkhorti
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Stability results of properties related to the Bishop-Phelps-Bollobás property for operators” by M.D. Acosta<https://arxiv.org/search/math?searchtype=author&query=Acosta%2C+M+D>, M. Soleimani-Mourchehkhorti<https://arxiv.org/search/math?searchtype=author&query=Soleimani-Mourchehkho…>. Abstract: We prove that the class of Banach spaces $Y$ such that the pair $(\ell_1, Y)$ has the Bishop-Phelps-Bollobás property for operators is stable under finite products when the norm of the product is given by an absolute norm. We also provide examples showing that previous stability results obtained for that property are optimal. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.01677

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Abstract of a paper by Aziz Azizov, Vladimir Chilin, Semyon Litvinov
by Bentuo Zheng (bzheng) 01 Mar '19

01 Mar '19

This is an announcement for the paper “Ergodic theorems in Banach ideals of compact operators” by Aziz Azizov<https://arxiv.org/search/math?searchtype=author&query=Azizov%2C+A>, Vladimir Chilin<https://arxiv.org/search/math?searchtype=author&query=Chilin%2C+V>, Semyon Litvinov<https://arxiv.org/search/math?searchtype=author&query=Litvinov%2C+S>. Abstract: Let $\mathcal H$ be a complex infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a fully symmetric sequence space. If $\{s_n(x)\}_{n=1}^\infty$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{s_n(x)\}\subset E\}$ with $\|x\|_{\mathcal C_E}=\|\{s_n(x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. We show that the averages $A_n(T)(x)=\frac1{n+1}\sum\limits_{k = 0}^n T^k(x)$ converge uniformly in $\mathcal C_E$ for any positive Dunford-Schwartz operator $T$ and $x\in\mathcal C_E$. Besides, if $x\in\mathcal B(\mathcal H)\setminus\mathcal K(\mathcal H)$, there exists a Hermitian Dunford-Schwartz operator $T$ such that the sequence $\{A_n(T)(x)\}$ does not converge uniformly. We also show that the averages $A_n(T)$ converge strongly in $(\mathcal C_E,\|\cdot\|_{\mathcal C_E})$ if and only if $E$ is separable and $E\neq l^1$, as sets. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.00759

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Banach March 2019