Banach December 2018

banach@mathdept.okstate.edu

1 participants
19 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 Hans-Olav Tylli, Henrik Wirzenius
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property” by Hans-Olav Tylli<https://arxiv.org/search/math?searchtype=author&query=Tylli%2C+H>, Henrik Wirzenius<https://arxiv.org/search/math?searchtype=author&query=Wirzenius%2C+H>. Abstract: We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_0$ into $\mathcal K(Z)/\mathcal A(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a non-separable space $c_0(Γ)$ into $\mathcal K(Z_{FJ})/\mathcal A(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.09402

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Abstract of a paper by Gustavo Araújo, Joedson Santos
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “On the Maurey--Pisier and Dvoretzky--Rogers theorems” by Gustavo Araújo<https://arxiv.org/search/math?searchtype=author&query=Ara%C3%BAjo%2C+G>, Joedson Santos<https://arxiv.org/search/math?searchtype=author&query=Santos%2C+J>. Abstract: A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space $E$, the infumum of the $q$ such that the identity map $id_{E}$ is absolutely $\left( q,1\right) $-summing is precisely $\cot E$. In the same direction, the Dvoretzky--Rogers Theorem asserts $id_{E}$ fails to be absolutely $\left( p,p\right) $-summing, for all $p\geq1$. In this note, among other results, we unify both theorems by charactering the parameters $q$ and $p$ for which the identity map is absolutely $\left( q,p\right)$-summing. We also provide a result that we call \textit{strings of coincidences} that characterize a family of coincidences between classes of summing operators. We illustrate the usefulness of this result by extending classical result of Diestel, Jarchow and Tonge and the coincidence result of Kwapień. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.09183

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Abstract of a paper by Ersin Kızgut
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “The Cesàro operator on smooth sequence spaces of finite type” by Ersin Kızgut<https://arxiv.org/search/math?searchtype=author&query=K%C4%B1zgut%2C+E>. Abstract: The discrete Cesàro operator $\mathsf{C}$ is investigated in the class of smooth sequence spaces $λ_0(A)$ of finite type. This class contains properly the power series spaces of finite type. Of main interest is its spectrum, which is distinctly different in the cases when $λ_0(A)$ is nuclear and when it is not. The nuclearity of $λ_0(A)$ is characterized via certain properties of the spectrum of $\mathsf{C}$. Moreover, $\mathsf{C}$ is always power bounded and uniformly mean ergodic on $λ_0(A)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.08493

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Abstract of a paper by M.D. Acosta, J.L. Dávila
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “A basis of $\R ^n$ with good isometric properties and some applications to denseness of norm attaining operators” by M.D. Acosta<https://arxiv.org/search/math?searchtype=author&query=Acosta%2C+M+D>, J.L. Dávila<https://arxiv.org/search/math?searchtype=author&query=D%C3%A1vila%2C+J+L>. Abstract: We characterize real Banach spaces $Y$ such that the pair $(\ell_\infty ^n, Y)$ has the Bishop-Phelps-Bollobás property for operators. To this purpose it is essential the use of an appropriate basis of the domain space $\R^n$. As a consequence of the mentioned characterization, we provide examples of spaces $Y$ satisfying such property. For instance, finite-dimensional spaces, uniformly convex spaces, uniform algebras and $L_1(μ)$ ($μ$ a positive measure) satisfy the previous property. https://arxiv.org/abs/1811.08387

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Abstract of a paper by William B.Johnson, Gilles Pisier, Gideon Schechtman
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Ideals in $L(L_1)$” by William B.Johnson<https://arxiv.org/search/math?searchtype=author&query=Johnson%2C+W+B>, Gilles Pisier<https://arxiv.org/search/math?searchtype=author&query=Pisier%2C+G>, Gideon Schechtman<https://arxiv.org/search/math?searchtype=author&query=Schechtman%2C+G>. Abstract: The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a question from A. Pietsch's 1978 book "Operator Ideals". The proof also shows that $L(C[0,1])$ contains a continuum of closed ideals. Finally, a duality argument yields that $L(\ell_\infty)$ has a continuum of closed ideals. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.06571

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Abstract of a paper by Arpita Mal, Kallol Paul, T.S.S.R.K. Rao, Debmalya Sain
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Approximate Birkhoff-James orthogonality and smoothness in the space of bounded linear operators” by Arpita Mal<https://arxiv.org/search/math?searchtype=author&query=Mal%2C+A>, Kallol Paul<https://arxiv.org/search/math?searchtype=author&query=Paul%2C+K>, T.S.S.R.K. Rao<https://arxiv.org/search/math?searchtype=author&query=Rao%2C+T+S+S+R+K>, Debmalya Sain<https://arxiv.org/search/math?searchtype=author&query=Sain%2C+D>. Abstract: We study approximate Birkhoff-James orthogonality of bounded linear operators defined between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}.$ As an application of the results obtained, we characterize smoothness of a bounded linear operator $T$ under the condition that $\mathbb{K}(\mathbb{X},\mathbb{Y}),$ the space of compact linear operators is an $M-$ideal in $\mathbb{L}(\mathbb{X},\mathbb{Y}),$ the space of bounded linear operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.02772

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Abstract of a paper by Niels Jakob Laustsen, Jared T. White
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

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/1811.02399

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Abstract of a paper by Fernando Albiac, Jose L. Ansorena, Marek Cuth, Michal Doucha
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Lipschitz free $p$-spaces for $0<p<1$” by Fernando Albiac<https://arxiv.org/search/math?searchtype=author&query=Albiac%2C+F>, Jose L. Ansorena<https://arxiv.org/search/math?searchtype=author&query=Ansorena%2C+J+L>, Marek Cuth<https://arxiv.org/search/math?searchtype=author&query=Cuth%2C+M>, Michal Doucha<https://arxiv.org/search/math?searchtype=author&query=Doucha%2C+M>. Abstract: This paper initiates the study of the structure of a new class of $p$-Banach spaces, $0<p<1$, namely the Lipschitz free $p$-spaces (alternatively called Arens-Eells $p$-spaces) $\mathcal{F}_{p}(\mathcal{M})$ over $p$-metric spaces. We systematically develop the theory and show that some results hold as in the case of $p=1$, while some new interesting phenomena appear in the case $0<p<1$ which have no analogue in the classical setting. For the former, we, e.g., show that the Lipschitz free $p$-space over a separable ultrametric space is isomorphic to $\ell_{p}$ for all $0<p\le 1$, or that $\ell_p$ isomorphically embeds into $\mathcal{F}_p(\mathcal{M})$ for any $p$-metric space $\mathcal{M}$. On the other hand, solving a problem by the first author and N. Kalton, there are metric spaces $\mathcal{N}\subset \mathcal{M}$ such that the natural embedding from $\mathcal{F}_p(\mathcal{N})$ to $\mathcal{F}_p(\mathcal{M})$ is not an isometry. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.01265

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Abstract of a paper by Earnest Akofor
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “On Lipschitz Retraction of Finite Subsets of Normed Spaces” by Earnest Akofor<https://arxiv.org/search/math?searchtype=author&query=Akofor%2C+E>. Abstract: If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset \cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space and, by Bačák and Kovalev when $X$ is a CAT(0) space, that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ for all $n\geq 2$. We prove that when $X$ is a normed space, the above sequence admits Lipschitz retractions $X(n)\rightarrow X$, $X(n)\rightarrow X(2)$, as well as concrete retractions $X(n)\rightarrow X(n-1)$ that are Lipschitz if $n=2,3$ and Hölder-continuous on bounded sets if $n>3$. We also prove that if $X$ is a geodesic metric space, then each $X(n)$ is a $2$-quasiconvex metric space. These results are relevant to certain questions in the aforementioned previous work which asked whether Lipschitz retractions $X(n)\rightarrow X(n-1)$, $n\geq 2$, exist for $X$ in more general classes of Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.00603

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Banach December 2018