- Banach - mathdept.okstate.edu

Abstract of a paper by Manuel De la Rosa, Leonhard Frerick, Sophie Grivaux, and Alfredo Peris
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "Frequent hypercyclicity, chaos, and unconditional Schauder decompositions" by Manuel De la Rosa, Leonhard Frerick, Sophie Grivaux, and Alfredo Peris. Abstract: We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. This result is extended to complex Frechet spaces with a continuous norm and an unconditional Schauder decomposition, and also to complex Frechet spaces with an unconditional basis, which gives a partial positive answer to a problem posed by Bonet. We also solve a problem of Bes and Chan in the negative by presenting hypercyclic, but non-chaotic operators on \C^\N. We extend the main result to C_0-semigroups of operators. Finally, in contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator. Archive classification: math.FA Submitted from: grivaux(a)math.univ-lille1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1416 or http://arXiv.org/abs/1005.1416

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Abstract of a paper by Daniel A. Klain
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "On the equality conditions of the Brunn-Minkowski theorem" by Daniel A. Klain. Abstract: This article describes a new proof of the equality condition for the Brunn-Minkowski inequality. Archive classification: math.MG math.CA math.FA Mathematics Subject Classification: 52A20, 52A38, 52A39, 52A40 Remarks: 9 pages Submitted from: daniel_klain(a)uml.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1409 or http://arXiv.org/abs/1005.1409

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Abstract of a paper by Veronica Dimant
by alspach＠fourier.math.okstate.edu 20 May '10

20 May '10

This is an announcement for the paper "M-ideals of homogeneous polynomials" by Veronica Dimant. Abstract: We study the problem of whether $\mathcal{P}_w(^nE)$, the space of $n$-homogeneous polynomials which are weakly continuous on bounded sets, is an $M$-ideal in the space of continuous $n$-homogeneous polynomials $\mathcal{P}(^nE)$. We obtain conditions that assure this fact and present some examples. We prove that if $\mathcal{P}_w(^nE)$ is an $M$-ideal in $\mathcal{P}(^nE)$, then $\mathcal{P}_w(^nE)$ coincides with $\mathcal{P}_{w0}(^nE)$ ($n$-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property $(M)$ and derive that if $\mathcal{P}_w(^nE)=\mathcal{P}_{w0}(^nE)$ and $\mathcal{K}(E)$ is an $M$-ideal in $\mathcal{L}(E)$, then $\mathcal{P}_w(^nE)$ is an $M$-ideal in $\mathcal{P}(^nE)$. We also show that if $E^*$ has the approximation property and $\mathcal{P}_w(^nE)$ is an $M$-ideal in $\mathcal{P}(^nE)$, then the set of $n$-homogeneous polynomials whose Aron-Berner extension do not attain the norm is nowhere dense in $\mathcal{P}(^nE)$. Finally, we face an analogous $M$-ideal problem for block diagonal polynomials. Archive classification: math.FA Mathematics Subject Classification: 46G25, 46B04, 47L22, 46B20. Submitted from: vero(a)udesa.edu.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1260 or http://arXiv.org/abs/1005.1260

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Abstract of a paper by Anton Baranov and Yurii Belov
by alspach＠fourier.math.okstate.edu 20 May '10

20 May '10

This is an announcement for the paper "Systems of reproducing kernels and their biorthogonal: completeness or non-completeness?" by Anton Baranov and Yurii Belov. Abstract: Let $\{v_n\}$ be a complete minimal system in a Hilbert space $\mathcal{H}$ and let $\{w_m\}$ be its biorthogonal system. It is well known that $\{w_m\}$ is not necessarily complete. However the situation may change if we consider systems of reproducing kernels in a reproducing kernel Hilbert space $\mathcal{H}$ of analytic functions. We study the completeness problem for a class of spaces with a Riesz basis of reproducing kernels and for model subspaces $K_\Theta$ of the Hardy space. We find a class of spaces where systems biorthogonal to complete systems of reproducing kernels are always complete, and show that in general this is not true. In particular we answer the question posed by N.K. Nikolski and construct a model subspace with a non-complete biorthogonal system. Archive classification: math.CV math.FA Mathematics Subject Classification: 30H05, 46E22, 30D50, 30D55, 47A15 Remarks: 28 pages Submitted from: antonbaranov(a)netscape.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1197 or http://arXiv.org/abs/1005.1197

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Abstract of a paper by Ondrej F.K. Kalenda
by alspach＠fourier.math.okstate.edu 20 May '10

20 May '10

This is an announcement for the paper "Spaces not containing $\ell_1$ have weak aproximate fixed point property" by Ondrej F.K. Kalenda. Abstract: A nonempty closed convex bounded subset $C$ of a Banach space is said to have the weak approximate fixed point property if for every continuous map $f:C\to C$ there is a sequence $\{x_n\}$ in $C$ such that $x_n-f(x_n)$ converge weakly to $0$. We prove in particular that $C$ has this property whenever it contains no sequence equivalent to the standard basis of $\ell_1$. As a byproduct we obtain a characterization of Banach spaces not containing $\ell_1$ in terms of the weak topology. Archive classification: math.FA Remarks: 5 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/1005.1218 or http://arXiv.org/abs/1005.1218

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Abstract of a paper by D. Azagra, R. Fry, and L. Keener
by alspach＠fourier.math.okstate.edu 20 May '10

20 May '10

This is an announcement for the paper "Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces" by D. Azagra, R. Fry, and L. Keener. Abstract: Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\varepsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \varepsilon$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore we characterize the class of Banach spaces having this approximation property as those Banach spaces $X$ having a Lipschitz, real-analytic separating function (meaning a Lipschitz, real analytic function $Q:X\to [0, +\infty)$ such that $Q(0)=0$ and $Q(x)\geq \|x\|$ for $\|x\|\geq 1$). Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 40 pages Submitted from: dazagra(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1050 or http://arXiv.org/abs/1005.1050

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Abstract of a paper by Rafal Gorak
by alspach＠fourier.math.okstate.edu 20 May '10

20 May '10

This is an announcement for the paper "Coarse version of the Banach Stone theorem" by Rafal Gorak. Abstract: We show that if there exists a Lipschitz homeomorphism $T$ between the nets in the Banach spaces $C(X)$ and $C(Y)$ of continuous real valued functions on compact spaces $X$ and $Y$, then the spaces $X$ and $Y$ are homeomorphic provided $l(T) \times l(T^{-1})<\frac{6}{5}$. By $l(T)$ and $l(T^{-1})$ we denote the Lipschitz constants of the maps $T$ and $T^{-1}$. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is $\frac{17}{16}$. We also estimate the distance of the map $T$ from the isometry of the spaces $C(X)$ and $C(Y)$. Archive classification: math.FA math.GN Mathematics Subject Classification: 46E15, 46B26, 46T99 Submitted from: R.Gorak(a)mini.pw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.0937 or http://arXiv.org/abs/1005.0937

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Abstract of a paper by Pamela Gorkin and Anthony G. OFarrell
by alspach＠fourier.math.okstate.edu 20 May '10

20 May '10

This is an announcement for the paper "Pervasive algebras and maximal subalgebras" by Pamela Gorkin and Anthony G. O'Farrell. Abstract: A uniform algebra $A$ on its Shilov boundary $X$ is {\em maximal} if $A$ is not $C(X)$ and there is no uniform algebra properly contained between $A$ and $C(X)$. It is {\em essentially pervasive} if $A$ is dense in $C(F)$ whenever $F$ is a proper closed subset of the essential set of $A$. If $A$ is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If $A$ is pervasive and proper, and has a nonconstant unimodular element, then $A$ contains an infinite descending chain of pervasive subalgebras on $X$. (2) It is possible to imbed a copy of the lattice of all subsets of $\N$ into the family of pervasive subalgebras of some $C(X)$. (3) In the other direction, if $A$ is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word \lq strongly' is removed. We discuss further examples, involving Dirichlet algebras, $A(U)$ algebras, Douglas algebras, and subalgebras of $H^\infty(\mathbb{D})$. We develop some new results that relate pervasiveness, maximality and relative maximality to support sets of representing measures. Archive classification: math.FA Mathematics Subject Classification: 46J10 Submitted from: AnthonyG.OFarrell(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.0719 or http://arXiv.org/abs/1005.0719

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Abstract of a paper by G. Botelho, D. Diniz, V.V. Favaro and D. Pellegrino
by Dale Alspach 20 May '10

20 May '10

This is an announcement for the paper "Spaceability in Banach and quasi-Banach sequence spaces" by G. Botelho, D. Diniz, V.V. Favaro and D. Pellegrino. Abstract: Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\in\Gamma}\ell_{q}(X)$, where $\Gamma$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain closed infinite-dimensional subspaces of $E$ (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators. Archive classification: math.FA Mathematics Subject Classification: 46A45, 46A16, 46B45 Remarks: 9 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.0596 or http://arXiv.org/abs/1005.0596

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Abstract of a paper by Asghar Rahimi and Peter Balazs
by alspach＠fourier.math.okstate.edu 30 Apr '10

30 Apr '10

This is an announcement for the paper "Multipliers for p-Bessel sequences in Banach spaces" by Asghar Rahimi and Peter Balazs. Abstract: Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters. Archive classification: math.OA math.FA Mathematics Subject Classification: Primary 42C40, Secondary 41A58, 47A58 Remarks: 17 pages Submitted from: Peter.Balazs(a)oeaw.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.5212 or http://arXiv.org/abs/1004.5212

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