Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Ondrej F.K. Kalenda and Jiri Spurny
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Quantification of the reciprocal Dunford-Pettis property" by Ondrej F.K. Kalenda and Jiri Spurny. Abstract: We prove in particular that Banach spaces of the form $C_0(\Omega)$, where $\Omega$ is a locally compact space, enjoy a quantitative version of the reciprocal Dunford-Pettis property. Archive classification: math.FA Remarks: 16 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.4308 or http://arXiv.org/abs/1204.4308

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Abstract of a paper by B. de Pagter and A.W. Wickstead
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Free and projective Banach lattices" by B. de Pagter and A.W. Wickstead. Abstract: We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms and establish some of their fundamental properties. We give much more detailed results about their structure in the case that there are only a finite number of generators and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice $P$ to be projective if whenever $X$ is a Banach lattice, $J$ a closed ideal in $X$, $Q:X\to X/J$ the quotient map, $T:P\to X/J$ a linear lattice homomorphism and $\epsilon>0$ there is a linear lattice homomorphism $\hat{T}:P\to X$ such that (i) $T=Q\circ \hat{T}$ and (ii) $\|\hat{T}\|\le (1+\epsilon)\|T\|$. We establish the connection between projective Banach lattices and free Banach lattices and describe several families of Banach lattices that are projective as well as proving that some are not. Archive classification: math.FA Submitted from: A.Wickstead(a)qub.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.4282 or http://arXiv.org/abs/1204.4282

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Abstract of a paper by Fedor Sukochev and Anna Tomskova
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "(E,F)-multipliers and applications" by Fedor Sukochev and Anna Tomskova. Abstract: For two given symmetric sequence spaces $E$ and $F$ we study the $(E,F)$-multiplier space, that is the space all of matrices $M$ for which the Schur product $M\ast A$ maps $E$ into $F$ boundedly whenever $A$ does. We obtain several results asserting continuous embedding of $(E,F)$-multiplier space into the classical $(p,q)$-multiplier space (that is when $E=l_p$, $F=l_q$). Furthermore, we present many examples of symmetric sequence spaces $E$ and $F$ whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis, extending classical results of S. Kwapie\'{n} and A. Pe{\l}czy\'{n}ski and of G. Bennett for the case when $E=l_p$, $F=l_q$. Archive classification: math.FA Remarks: 16 pages Submitted from: tomskovaanna(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2623 or http://arXiv.org/abs/1204.2623

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Abstract of a paper by Taras Banakh, Bogdan Bokalo, and Nadiya Kolos
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "On \sigma-convex subsets in spaces of scatteredly continuous functions" by Taras Banakh, Bogdan Bokalo, and Nadiya Kolos. Abstract: We prove that for any topological space $X$ of countable tightness, each \sigma-convex subspace $\F$ of the space $SC_p(X)$ of scatteredly continuous real-valued functions on $X$ has network weight $nw(\F)\le nw(X)$. This implies that for a metrizable separable space $X$, each compact convex subset in the function space $SC_p(X)$ is metrizable. Another corollary says that two Tychonoff spaces $X,Y$ with countable tightness and topologically isomorphic linear topological spaces $SC_p(X)$ and $SC_p(Y)$ have the same network weight $nw(X)=nw(Y)$. Also we prove that each zero-dimensional separable Rosenthal compact space is homeomorphic to a compact subset of the function space $SC_p(\omega^\omega)$ over the space $\omega^\omega$ of irrationals. Archive classification: math.GN math.FA Mathematics Subject Classification: 46A55, 46E99, 54C35 Remarks: 6 pages Submitted from: tbanakh(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2438 or http://arXiv.org/abs/1204.2438

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Abstract of a paper by S Dutta and A B Abubaker
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Generalized 3-circular projections in some Banach spaces" by S Dutta and A B Abubaker. Abstract: Recently in a series of papers it is observed that in many Banach spaces, which include classical spaces $C(\Omega)$ and $L_p$-spaces, $1 \leq p < \infty, p \neq 2$, any generalized bi-circular projection $P$ is given by $P = \frac{I+T}{2}$, where $I$ is the identity operator of the space and $T$ is a reflection, that is, $T$ is a surjective isometry with $T^2 = I$. For surjective isometries of order $n \geq 3$, the corresponding notion of projection is generalized $n$-circular projection as defined in \cite{AD}. In this paper we show that in a Banach space $X$, if generalized bi-circular projections are given by $\frac{I+T}{2}$ where $T$ is a reflection, then any generalized $n$-circular projection $P$, $n \geq 3$, is given by $P = \frac{I+T+T^2+\cdots+T^{n-1}}{n}$ where $T$ is a surjective isometry and $T^n = I$. We prove our results for $n=3$ and for $n > 3$, the proof remains same except for routine modifications. Archive classification: math.FA Mathematics Subject Classification: 47L05, 46B20 Remarks: 8 pages Submitted from: sudipta(a)iitk.ac.in The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2360 or http://arXiv.org/abs/1204.2360

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Abstract of a paper by Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Greedy algorithms for reduced bases in Banach spaces" by Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk. Abstract: Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n dimensional space X_n ⊂ X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width d_n(F)_X. However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in A. Buffa, Y. Maday, A.T. Patera, C. Prud’homme, and G. Turinici, ''A Priori convergence of the greedy algorithm for the parameterized reduced basis'', M2AN Math. Model. Numer. Anal., 46(2012), 595–603 in the case X = H is a Hilbert space. The results there were significantly improved on in P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk, ''Convergence rates for greedy algorithms in reduced bases Methods'', SIAM J. Math. Anal., 43 (2011), 1457–1472. The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces. Archive classification: math.FA Mathematics Subject Classification: 41A46, 41A25, 46B20, 15A15 Submitted from: gpetrova(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2290 or http://arXiv.org/abs/1204.2290

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Abstract of a paper by G. Botelho, D. Cariello, V.V. Favaro, D. Pellegrino and J.B. Seoane-Sepulveda
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Subspaces of maximal dimension contained in $L_{p}(\Omega) - \textstyle\bigcup\limits_{q<p}L_{q}(\Omega)$}" by G. Botelho, D. Cariello, V.V. Favaro, D. Pellegrino and J.B. Seoane-Sepulveda. Abstract: Let $(\Omega,\Sigma,\mu)$ be a measure space and $1< p < +\infty$. In this paper we determine when the set $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$ is maximal spaceable, that is, when it contains (except for the null vector) a closed subspace $F$ of $L_{p}(\Omega)$ such that $\dim(F) = \dim\left(L_{p}(\Omega)\right)$. The aim of the results presented here is, among others, to generalize all the previous work (since the 1960's) related to the linear structure of the sets $L_{p}(\Omega) - L_{q}(\Omega)$ with $q < p$ and $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$. We shall also give examples, propose open questions and provide new directions in the study of maximal subspaces of classical measure spaces. Archive classification: math.FA Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2170 or http://arXiv.org/abs/1204.2170

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Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "A remark on condensation of singularities" by Jan-David Hardtke. Abstract: Recently Alan D. Sokal gave a very short and completely elementary proof of the uniform boundedness principle. The aim of this note is to point out that by using a similiar technique one can give a considerably short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is a bit more general than a result of I.\,S. G\'al. Archive classification: math.FA Mathematics Subject Classification: 46A16, 47H99 Remarks: 7 pages Submitted from: hardtke(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2106 or http://arXiv.org/abs/1204.2106

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Abstract of a paper by Jean-Matthieu Auge
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Perturbation of farthest points in weakly compact sets" by Jean-Matthieu Auge. Abstract: If $f$ is a real valued weakly lower semi-continous function on a Banach space $X$ and $C$ a weakly compact subset of $X$, we show that the set of $x \in X$ such that $z \mapsto \|x-z\|-f(z)$ attains its supremum on $C$ is dense in $X$. We also construct a counter example showing that the set of $x \in X$ such that $z \mapsto \|x-z\|+\|z\|$ attains its supremum on $C$ is not always dense in $X$. Archive classification: math.FA Mathematics Subject Classification: Primary 41A65 Remarks: 5 pages Submitted from: jean-matthieu.auge(a)math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2047 or http://arXiv.org/abs/1204.2047

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Abstract of a paper by Jean-Matthieu Auge
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Orbits of linear operators and Banach space geometry" by Jean-Matthieu Auge. Abstract: Let $T$ be a bounded linear operator on a (real or complex) Banach space $X$. If $(a_n)$ is a sequence of non-negative numbers tending to 0. Then, the set of $x \in X$ such that $\|T^nx\| \geqslant a_n \|T^n\|$ for infinitely many $n$'s has a complement which is both $\sigma$-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents $q>0$, such that for every non nilpotent operator $T$, there exists $x \in X$ such that $(\|T^nx\|/\|T^n\|) \notin \ell^{q}(\mathbb{N})$, using techniques which involve the modulus of asymptotic uniform smoothness of $X$. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, 47A16, Secondary 28A05 Remarks: 16 pages Submitted from: jean-matthieu.auge(a)math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2046 or http://arXiv.org/abs/1204.2046

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