Banach

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Abstract of a paper by Piotr Koszmider
by alspach＠math.okstate.edu 29 Sep '15

29 Sep '15

This is an announcement for the paper "On the problem of compact totally disconnected reflection of nonmetrizability" by Piotr Koszmider. Abstract: We construct a ZFC example of a nonmetrizable compact space $K$ such that every totally disconnected closed subspace $L\subseteq K$ is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space $K$ must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the the structure of the Banach space $C(K)$, for example, by Holszty\'nski's theorem, if $K$ is a counterexample, then $C(K)$ contains no isometric copy of a nonseparable Banach space $C(L)$ for $L$ totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open. Archive classification: math.GN math.FA math.LO Submitted from: piotr.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.05282 or http://arXiv.org/abs/1509.05282

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Transfinite methods in Banach spaces and algebras of operators
by Dale Alspach 20 Sep '15

20 Sep '15

Dear Colleague, We are pleased to announce that a conference entitled "Transfinite methods in Banach spaces and algebras of operators" will take place at Bedlewo Conference Center, Poland, 18-22 July 2016. The list of speakers will include: Tristan Bice (Salvador), Christina Brech (Sao Paulo), Yemon Choi (Lancaster; tbc), Marek Cuth (Prague), Garth Dales (Lancaster), Alan Dow (North Carolina), Valentin Ferenczi (Sao Paulo), Joanna Garbulinska (Kielce), Gilles Godefroy (Paris VI), Bill Johnson (Texas A&M; tbc), Tomasz Kochanek (IM PAN), Jordi Lopez-Abad (ICMAT Madrid), Pavlos Motakis (Texas A&M), Grzegorz Plebanek (Wroclaw), Jose Rodriguez (Murcia), Thomas Schlumprecht (Texas A&M), Jesus Suarez (Caceres) and Stevo Todorcevic (CRNS, Toronto). For more details, please see the webpage: http://www.impan.pl/~set_theory/Banach2016/ <http://www.impan.pl/%7Eset_theory/Banach2016/> We would be very grateful if you could distribute this email to anybody who might be interested in the conference, including graduate students and early-career researchers. We hope to see you in Bedlewo next summer! Best wishes, Antonio Aviles, Piotr Koszmider, Niels Laustsen (we apologize if you received this e-mail more than once, or if you are not interested)

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Abstract of a paper by Niels Jakob Laustsen, Richard Lechner, and Paul F.X. Mueller
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "Factorization of the identity through operators with large diagonal" by Niels Jakob Laustsen, Richard Lechner, and Paul F.X. Mueller. Abstract: Given a Banach space $X$ with an unconditional basis, we consider the following question: does the identity on $X$ factor through every bounded operator on $X$ with large diagonal relative to the unconditional basis? We show that on Gowers' space with its unconditional basis there exists an operator for which the answer to the question is negative. By contrast, for any operator on the mixed-norm Hardy spaces $H^p(H^q)$, where $1 \leq p,q < \infty$, with the bi-parameter Haar system, this problem always has a positive solution. The one-parameter $H^p$ spaces were treated first by Andrew in 1979. Archive classification: math.FA Mathematics Subject Classification: 46B25, 60G46, 46B07, 46B26, 30H35, 30H10, 46B15, 47B37, 47A53 Remarks: 16 pages, 5 figures Submitted from: Richard.Lechner(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.03141 or http://arXiv.org/abs/1509.03141

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Abstract of a paper by Nick Lindemulder
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "Banach space-valued extensions of linear operators on $L^{\infty}$" by Nick Lindemulder. Abstract: Let $E$ and $G$ be two Banach function spaces, let $T \in \mathcal{L}(E,Y)$, and let ${\langle X,Y \rangle}$ be a Banach dual pair. In this paper we give conditions for which there exists a (necessarily unique) bounded linear operator $T_{Y} \in \mathcal{L}(E(Y),G(Y))$ with the property that \[ {\langle x,T_{Y}e \rangle} = T{\langle x,e \rangle} \quad\quad \forall e \in E(Y), x \in X. \] Our first main result states that, in case ${\langle X,Y \rangle} = {\langle Y^{*}, Y \rangle}$ with $Y$ a reflexive Banach space, for the existence of $T_{Y}$ it sufficient that $T$ is dominated by a positive operator. Our second main result concerns the case that $T$ is an adjoint operator on $L^{\infty}(A)$: we suppose that $E = L^{\infty}(A)$ for a semi-finite measure space $(A,\mathscr{A},\mu)$, that ${\langle F, G \rangle}$ is a K\"othe dual pair, and that $T$ is $\sigma(L^{\infty}(A),L^{1}(A))$-to-$\sigma(G,F)$ continuous. Then $T_{Y}$ exists provided that $T$ is dominated by a positive operator, in which case $T_{Y}$ is $\sigma(L^{\infty}(A;Y),L^{1}(A;X))$-to-$\sigma(G(Y),F \tilde{\otimes} X)$ continuous; here $F \tilde{\otimes} X$ denotes the closure of $F \otimes X$ in $F(X)$. We also consider situations in which the existence is automatic and we furthermore show that in certain situations it is necessary that $T$ is regular. As an application of this result we consider conditional expectation on Banach space-valued $L^{\infty}$-spaces. Archive classification: math.FA Mathematics Subject Classification: 46E40 (primary), 46E30, 46B10 (secondary) Remarks: 20 pages Submitted from: n.lindemulder(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.02493 or http://arXiv.org/abs/1509.02493

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Abstract of a paper by Olivier Guedon, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "On the interval of fluctuation of the singular values of random" by Olivier Guedon, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann. Abstract: Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio $n/N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $ \exp({-t^{\alpha}})$ with $\alpha \in (0,2]$, extending the known case $\alpha\in[1, 2]$. Archive classification: math.PR cs.IT math.FA math.IT Mathematics Subject Classification: 60B20, 46B06, 15B52, 46B09, 60D05 Remarks: To appear in J. Eur. Math. Soc Submitted from: olivier.guedon(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.02322 or http://arXiv.org/abs/1509.02322

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

15 Sep '15

This is an announcement for the paper "Diametral diameter two properties" by Julio Becerra Guerrero, Gines Lopez Perez and Abraham Rueda Zoca. Abstract: The aim of this note is to define a generalization of the diameter two properties in terms of the abundance of diametral points. We shall also analyze the stability of these properties under $\ell_p$ sums and the problem of inheritance to subspaces. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B22 Remarks: 25 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/1509.02061 or http://arXiv.org/abs/1509.02061

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Abstract of a paper by Gleb Sirotkin and Ben Wallis
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "Sequence-singular operators" by Gleb Sirotkin and Ben Wallis. Abstract: In this paper we study two types of collections of operators on a Banach space on the subject of forming operator ideals. One of the types allows us to construct an uncountable chain of closed ideals in each of the operator algebras $\mathcal{L}(\ell_1\oplus\ell_q)$, $1<q<\infty$, and $\mathcal{L}(\ell_1\oplus c_0)$. This finishes answering a longstanding question of Pietsch. Archive classification: math.FA Mathematics Subject Classification: 46B06, 46B25, 46B45, 47L10, 47L20 Remarks: 13 pages Submitted from: z1019463(a)students.niu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.01485 or http://arXiv.org/abs/1509.01485

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Abstract of a paper by Julio Flores, Francisco L. Hernandez and Pedro Tradacete
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "Disjointly homogeneous Banach lattices and applications" by Julio Flores, Francisco L. Hernandez and Pedro Tradacete. Abstract: This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results. Archive classification: math.FA Mathematics Subject Classification: 47B38, 46E30 Remarks: 20 pages, to appear in Proceedings Positivity VII Conference Submitted from: ptradace(a)math.uc3m.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.01499 or http://arXiv.org/abs/1509.01499

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Abstract of a paper by Stephane Chretien and Sebastien Darses
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "An elementary approach to the problem of column selection in a rectangular matrix" by Stephane Chretien and Sebastien Darses. Abstract: The problem of extracting a well conditioned submatrix from any rectangular matrix (with normalized columns) has been studied for some time in functional and harmonic analysis; see \cite{BourgainTzafriri:IJM87,Tropp:StudiaMath08,Vershynin:IJM01} for methods using random column selection. More constructive approaches have been proposed recently; see the recent contributions of \cite{SpielmanSrivastava:IJM12,Youssef:IMRN14}. The column selection problem we consider in this paper is concerned with extracting a well conditioned submatrix, i.e. a matrix whose singular values all lie in $[1-\epsilon,1+\epsilon]$. We provide individual lower and upper bounds for each singular value of the extracted matrix at the price of conceding only one log factor in the number of columns, when compared to the Restricted Invertibility Theorem of Bourgain and Tzafriri. Our method is fully constructive and the proof is short and elementary. Archive classification: math.FA math.SP Remarks: 5 pages 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/1509.00748 or http://arXiv.org/abs/1509.00748

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Abstract of a paper by Victor Bible and Richard J. Smith
by alspach＠math.okstate.edu 15 Sep '15

15 Sep '15

This is an announcement for the paper "Smooth and polyhedral approximation in Banach spaces" by Victor Bible and Richard J. Smith. Abstract: We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(\Gamma)$, where $\Gamma$ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf. Archive classification: math.FA Mathematics Subject Classification: 46B03 46B20 Remarks: 12 pages Submitted from: victor.bible(a)ucdconnect.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.00369 or http://arXiv.org/abs/1509.00369

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