- Banach - mathdept.okstate.edu

Abstract of a paper by Ben Wallis
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Constructing Banach ideals using upper $\ell_p$-estimates" by Ben Wallis. Abstract: Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ with parameters $1\leq p\leq\infty$ and $1\leq\xi\leq\omega_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for all choices $\xi\neq\omega_1$. For the case $\xi=1$, there exists an ideal norm $\|\cdot\|_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal. Archive classification: math.FA Mathematics Subject Classification: 47L20, 46B45, 46A45, 46B25 Submitted from: wallis(a)math.niu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.5948 or http://arXiv.org/abs/1407.5948

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Abstract of a paper by Iian B. Smythe
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Borel equivalence relations in the space of bounded operators" by Iian B. Smythe. Abstract: We consider various notions of equivalence in the space of bounded operators on a Hilbert space, including modulo finite rank operators, modulo Schatten $p$-classes, and modulo compact operators. Using Hjorth's theory of turbulence, the latter two are shown to be not classifiable by countable structures, while the first cannot be reduced to the orbit equivalence relation of any Polish group action. The results for modulo finite rank and modulo compact operators are also shown for the restrictions of these equivalence relations to the space of projection operators. Families of non-classifiable equivalence relations on sequence spaces are described and utilized in these results. Archive classification: math.LO math.OA Mathematics Subject Classification: Primary 03E15, 47B10, Secondary 47C15, 46A45 Remarks: 36 pages Submitted from: ibs24(a)cornell.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.5325 or http://arXiv.org/abs/1407.5325

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Abstract of a paper by Kallol Paul, Debmalya Sain and Lokenath Debnath
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "A conjecture on the characterisation of inner product spaces" by Kallol Paul, Debmalya Sain and Lokenath Debnath. Abstract: We study the properties of strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real normed linear space that are analogous to the properties of orthonormal basis in an inner product space. We relate the notion of strongly orthonormal Hamel basis in the sense of Birkhoff-James with the notions of best approximation and best coapproximation in a finite dimensional real normed linear space. We prove that the existence of best coapproximation to any element of the normed linear space out of any one dimensional subspace and its coincidence with the best approximation to that element out of that subspace characterises a real inner product space of dimension( > 2). Finally we conjecture that a finite dimensional real smooth normed space of dimension ($>2$) is an inner product space iff given any element on the unit sphere there exists a strongly orthonormal Hamel basis in the sense of Birkhoff-James containing that element. Archive classification: math.FA Mathematics Subject Classification: Primary: 46B20, Secondary: 47A30 Submitted from: kalloldada(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.5016 or http://arXiv.org/abs/1407.5016

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Abstract of a paper by Peter G. Casazza
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "Consequences of the Marcus/Spielman/Stivastava solution to the Kadison-Singer Problem" by Peter G. Casazza. Abstract: It is known that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. The recent surprising solution to this problem by Marcus, Spielman and Srivastava was a significant achievement and a significant advance for all these areas of research. We will look at many of the known equivalent forms of the Kadison-Singer Problem and see what are the best new theorems available in each area of research as a consequence of the work of Marcus, Spielman and Srivastave. In the cases where {\it constants} are important for the theorem, we will give the best constants available in terms of a {\it generic constant} taken from \cite{MSS}. Thus, if better constants eventually become available, it will be simple to adapt these new constants to the theorems. Archive classification: math.FA Mathematics Subject Classification: 42A05, 42A10, 42A16, 43A50, 46B03, 46B07, 46L05, Submitted from: casazzap(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.4768 or http://arXiv.org/abs/1407.4768

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Abstract of a paper by Vladimir Kadets, Miguel Martin, Javier Meri, and Dirk Werner
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "Lipschitz slices and the Daugavet equation for Lipschitz operators" by Vladimir Kadets, Miguel Martin, Javier Meri, and Dirk Werner. Abstract: We introduce a substitute for the concept of slice for the case of non-linear Lipschitz functionals and transfer to the non-linear case some results about the Daugavet and the alternative Daugavet equations previously known only for linear operators. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04. Secondary 46B80, 46B22, 47A12 Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.4018 or http://arXiv.org/abs/1407.4018

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Abstract of a paper by Ioannis Gasparis
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "An extension of James's compactness theorem" by Ioannis Gasparis. Abstract: Let X and Y be Banach spaces and F a subset of B_{Y^*}. Endow Y with the topology \tau_F of pointwise convergence on F. Let T: X^* \to Y be a bounded linear operator which is (w^*, \tau_F) continuous. Assume that every vector in the range of T attains its norm at an element of F. Then it is proved that T is (w^*,w) continuous. Archive classification: math.FA Mathematics Subject Classification: 46 Remarks: 15 pages Submitted from: ioagaspa(a)math.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.3655 or http://arXiv.org/abs/1407.3655

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Abstract of a paper by Riku Klen, Antti Rasila, and Jarno Talponen
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "On smoothness of quasihyperbolic balls" by Riku Klen, Antti Rasila, and Jarno Talponen. Abstract: We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. Our main result is that in uniformly smooth Banach spaces a quasihyperbolic ball of a convex domain is $C^1$-smooth. The question about the smoothness of quasihyperbolic balls is old, originating back to the discussions of F.W. Gehring and M. Vuorinen in 1970's. To our belief, the result is new also in the Euclidean setting. We also address some other issues involving the smoothness of quasihyperbolic balls. We introduce an interesting application of quasihyperbolic metrics to renormings of Banach spaces. To provide a useful tool for this approach we turn our attention to the variational stability of quasihyperbolic geodesics. Several examples and illustrations are provided. Archive classification: math.FA math.CV Mathematics Subject Classification: 30C65, 46T05, 46B03 Remarks: 19 pages, 4 figures Submitted from: antti.rasila(a)iki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.2403 or http://arXiv.org/abs/1407.2403

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Abstract of a paper by Miek Messerschmidt
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "Geometric duality theory of cones in dual pairs of vector spaces" by Miek Messerschmidt. Abstract: This paper will generalize what may be termed the ``geometric duality theory'' of real pre-ordered Banach spaces which relates geometric properties of a closed cone in a real Banach space, to geometric properties of the dual cone in the dual Banach space. We show that geometric duality theory is not restricted to real pre-ordered Banach spaces, as is done classically, but can naturally extended to real Banach spaces endowed with arbitrary collections of closed cones. We define geometric notions of normality, conormality, additivity and coadditivity for members of dual pairs of real vector spaces as certain possible interactions between two cones and two convex convex sets containing zero. We show that, thus defined, these notions are dual to each other under certain conditions, i.e., for a dual pair of real vector spaces $(Y,Z)$, the space $Y$ is normal (additive) if and only if its dual $Z$ is conormal (coadditive) and vice versa. These results are set up in a manner so as to provide a framework to prove results in the geometric duality theory of cones in real Banach spaces. As an example of using this framework, we generalize classical duality results for real Banach spaces pre-ordered by a single closed cone, to real Banach spaces endowed with an arbitrary collections of closed cones. As an application, we analyze some of the geometric properties of naturally occurring cones in C*-algebras and their duals. Archive classification: math.FA Mathematics Subject Classification: Primary: 46A20, Secondary: 46B10, 46B20, 46A40, 46B40, 46L05 Submitted from: mmesserschmidt(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.2434 or http://arXiv.org/abs/1407.2434

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Abstract of a paper by Sergo A. Episkoposian
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "$L^1$- convergence of greedy algorithm by generalized Walsh system" by Sergo A. Episkoposian. Abstract: In this paper we consider the generalized Walsh system and a problem $L^1- convergence$ of greedy algorithm of functions after changing the values on small set. Archive classification: math.FA Mathematics Subject Classification: 42A65, 42A20 Submitted from: sergoep(a)ysu.am The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.1496 or http://arXiv.org/abs/1407.1496

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Abstract of a paper by Kallol Paul, Puja Ghosh, and Debmalya Sain
by alspach＠math.okstate.edu 21 Jul '14

21 Jul '14

This is an announcement for the paper "On rectangular constant in normed linear spaces" by Kallol Paul, Puja Ghosh, and Debmalya Sain. Abstract: We study the properties of rectangular constant $ \mu(\mathbb{X}) $ in a normed linear space $\mathbb{X}$. We prove that $ \mu(\mathbb{X}) = 3$ iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound iff the unit sphere contains a straight line segment of length 2. We prove that if the dimension of the space $\mathbb{X}$ is finite then $\mu(\mathbb{X})$ is attained. We also prove that a normed linear space is an inner product space iff we have sup$\{\frac{1+|t|}{\|y+tx\|}$: $x,y \in S_{\mathbb{X}}$ with $x\bot_By\} \leq \sqrt{2}$ $\forall t$ satisfying $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 47A30 Submitted from: kalloldada(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.1353 or http://arXiv.org/abs/1407.1353

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