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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Abstract of a paper by Leandro CandidoApoorva Khare
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “The non-compact normed space of norms on a finite-dimensional Banach space” by Leandro Candido<https://arxiv.org/search/math?searchtype=author&query=Candido%2C+L>Apoorva Khare<https://arxiv.org/search/math?searchtype=author&query=Khare%2C+A>. Abstract: We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $\mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k \geqslant 2$ the metric subspace $\{ \| \cdot \|_p : p \in [1,\infty] \}$ maps isometrically and monotonically to $[0, \log k]$ (or $[0,1]$ by scaling the norm), again unlike in the Banach-Mazur compactum. Our analysis goes through embedding the above quotient space into a normed space, and reveals an implicit functorial construction of function spaces with diameter norms (as well as a variant of the distortion). In particular, we realize the above quotient space of norms as a normed space. We next study the parallel setting of the - also hitherto unexplored - metric space $\mathcal{S}([n])$ of all metrics on a finite set of $n$ elements, revealing the connection between log-distortion and diameter norms. In particular, we show that $\mathcal{S}([n])$ is also a normed space. We demonstrate embeddings of equivalence classes of finite metric spaces (parallel to the Gromov-Hausdorff setting), as well as of $\mathcal{S}([n-1])$, into $\mathcal{S}([n])$. We conclude by discussing extensions to norms on an arbitrary Banach space and to discrete metrics on any set, as well as some questions in both settings above. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.06188

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Abstract of a paper by Emiel Lorist
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “The $\ell^s$-boundedness of a family of integral operators on UMD Banach function spaces” by Emiel Lorist<https://arxiv.org/search/math?searchtype=author&query=Lorist%2C+E>. Abstract: We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.12833

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Abstract of a paper by Colin Petitjean, Antonín Procházka
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “On exposed points of Lipschitz free spaces” by Colin Petitjean<https://arxiv.org/search/math?searchtype=author&query=Petitjean%2C+C>, Antonín Procházka<https://arxiv.org/search/math?searchtype=author&query=Proch%C3%A1zka%2C+A>. Abstract: In this note we prove that a molecule $d(x,y)^{-1}(δ(x)-δ(y))$ is an exposed point of the unit ball of a Lispchitz free space $\mathcal F(M)$ if and only if the metric segment $[x,y]=\{z \in M \; : \; d(x,y)=d(z,x)+d(z,y) \}$ is reduced to $\{x,y\}$. This is based on a recent result due to Aliaga and Pernecká which states that the class of Lipschitz free spaces over closed subsets of M is closed under arbitrary intersections when M has finite diameter. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.12031

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Abstract of a paper by Ralph Chill, Alberto Fiorenza, Sebastian Krol
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Interpolation of nonlinear positive or order preserving operators on Banach lattices” by Ralph Chill<https://arxiv.org/search/math?searchtype=author&query=Chill%2C+R>, Alberto Fiorenza<https://arxiv.org/search/math?searchtype=author&query=Fiorenza%2C+A>, Sebastian Krol<https://arxiv.org/search/math?searchtype=author&query=Krol%2C+S>. Abstract: We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo-Peetre operators, and exact partially $K$-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple $(L^1,L^\infty )$ due to Bénilan and Crandall. https://arxiv.org/abs/1810.09684

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Abstract of a paper by Martin Mathieu, Pedro Tradacete
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Strictly singular multiplication operators on $\mathcal L(X)$” by Martin Mathieu<https://arxiv.org/search/math?searchtype=author&query=Mathieu%2C+M>, Pedro Tradacete<https://arxiv.org/search/math?searchtype=author&query=Tradacete%2C+P>. Abstract: Exploiting several $\ell_p$-factorization results for strictly singular operators, we study the strict singularity of the multiplication operator $L_A R_B\colon T\mapsto ATB$ on $\mathcal L(X)$ for various Banach spaces~$X$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.09194

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Abstract of a paper by Jose M. Calabuig, Maite Fernández Unzueta, Fernando Galaz-Fontes, Enrique A. Sánchez Pérez
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Equivalent Norms in a Banach Function Space and the Subsequence Property” by Jose M. Calabuig<https://arxiv.org/search/math?searchtype=author&query=Calabuig%2C+J+M>, Maite Fernández Unzueta<https://arxiv.org/search/math?searchtype=author&query=Unzueta%2C+M+F>, Fernando Galaz-Fontes<https://arxiv.org/search/math?searchtype=author&query=Galaz-Fontes%2C+F>, Enrique A. Sánchez Pérez<https://arxiv.org/search/math?searchtype=author&query=P%C3%A9rez%2C+E+A+S>. Abstract: Given a finite measure space $(Ω,Σ,μ)$, we show that any Banach space $X(μ)$ consisting of (equivalence classes of) real measurable functions defined on $Ω$ such that $f χ_A \in X(μ) $ and $ \|f χ_A \| \leq \|f\|, \, f \in X(μ), \ A \in Σ$, and having the subsequence property, is in fact an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05714

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Abstract of a paper by R.M. Causey
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “Lipschitz subtype” by R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M> Abstract: We give necessary and sufficient conditions for a Lipschitz map, or more generally a uniformly Lipschitz family of maps, to factor the Hamming cubes. This is an extension to Lipschitz maps of a particular spatial result of Bourgain, Milman, and Wolfson. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05707

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Abstract of a paper by M. Alikhani
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “On pseudo weakly compact operators of order P” by M. Alikhani<https://arxiv.org/search/math?searchtype=author&query=Alikhani%2C+M>. Abstract: In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spaces $ X$ and $ Y $ such that projective tensor and injective tensor products between $ X $ and $ Y $ have the $ p $-sequentially Right property.\ Finally, we introduce two properties for the Banach spaces, namely $ p $-sequentially Right$ ^{\ast} $ and weak $ p $-sequentially Right$ ^{\ast} $ properties and obtain some characterizations of these properties. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05638

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Abstract of a paper by R.M. Causey
by Bentuo Zheng (bzheng) 02 Dec '18

02 Dec '18

This is an announcement for the paper “The ξ,ζ-Dunford Pettis property” by R.M. Causey<https://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M>. Abstract: Using the hierarchy of weakly null sequences introduced by Argyros, Merkourakis, and Tsarpalias, we introduce two new families of operator classes. The first family simultaneously generalizes the completely continuous operators and the weak Banach-Saks operators. The second family generalizes the class $\mathfrak{DP}$. We study the distinctness of these classes, and prove that each class is an operator ideal. We also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. We produce a number of examples, including the higher ordinal Schreier and Baernstein spaces. We prove ordinal analogues of several known results for Banach spaces with the Dunford-Pettis, hereditary Dunford-Pettis property, and hereditary by quotients Dunford-Pettis property. For example, we prove that for any $0\leqslant ξ, ζ<ω_1$, a Banach space $X$ has the hereditary $ω^ξ, ω^ζ$-Dunford Pettis property if and only if every seminormalized, weakly null sequence either has a subsequence which is an $\ell_1^{ω^ξ}$-spreading model or a $c_0^{ω^ζ}$-spreading model. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1810.05196

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Banach December 2018