November 2014 - Banach - mathdept.okstate.edu

Abstract of a paper by Daher Mohammad
by alspach＠math.okstate.edu 26 Nov '14

26 Nov '14

This is an announcement for the paper "K(X,Y) as subspace complemented of L(X,Y)" by Daher Mohammad. Abstract: Let X,Y be two Banach spaces ; in the first part of this work, we show that K(X,Y) contains a complemented copy of c0 if Y contains a copy of c0 and each bounded sequence in Y has a subsequece which is w* convergente. Afterward we obtain some results of M.Feder and G.Emmanuele: Finally in this part we study the relation between the existence of projection from L(X,Y) on K(X,Y) and the existence of pro- jection from K(X,Y ) on K(X,Y) if Y has the approximation property. In the second part we study the Radon-Nikodym property in L(X,Y): Archive classification: math.FA Mathematics Subject Classification: 46EXX Remarks: 21 pages Submitted from: m.daher(a)orange.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.2217 or http://arXiv.org/abs/1411.2217

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Abstract of a paper by Debmalya Sain, Kallol Paul and Kanhaiya Jha
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Strictly convex space : Strong orthogonality and conjugate diameters" by Debmalya Sain, Kallol Paul and Kanhaiya Jha. Abstract: In a normed linear space X an element x is said to be orthogonal to another element y in the sense of Birkhoff-James, written as $ x \perp_{B}y, $ iff $ \| x \| \leq \| x + \lambda y \| $ for all scalars $ \lambda.$ We prove that a normed linear space X is strictly convex iff for any two elements x, y of the unit sphere $ S_X$, $ x \perp_{B}y $ implies $ \| x + \lambda y \| > 1~ \forall~ \lambda \neq 0. $ We apply this result to find a necessary and sufficient condition for a Hamel basis to be a strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real strictly convex space X. Applying the result we give an estimation for lower bounds of $ \| tx+(1-t)y\|, t \in [0,1] $ and $ \| y + \lambda x \|, ~\forall ~\lambda $ for all elements $ x,y \in S_X $ with $ x \perp_B y. $ We find a necessary and sufficient condition for the existence of conjugate diameters through the points $ e_1,e_2 \in ~S_X $ in a real strictly convex space of dimension 2. The concept of generalized conjuagte diameters is then developed for a real strictly convex smooth space of finite dimension. 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/1411.1464 or http://arXiv.org/abs/1411.1464

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Abstract of a paper by Daniel Pellegrino and Joedson Santos
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Lineability and uniformly dominated sets of summing nonlinear" by Daniel Pellegrino and Joedson Santos. Abstract: In this note we prove an abstract version of a result from 2002 due to Delgado and Pi\~{n}ero on absolutely summing operators. Several applications are presented; some of them in the multilinear framework and some in a completely nonlinear setting. In a final section we investigate the size of the set of non uniformly dominated sets of linear operators under the point of view of lineability. Archive classification: math.FA Submitted from: pellegrino(a)pq.cnpq.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.1100 or http://arXiv.org/abs/1411.1100

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Abstract of a paper by Nigel Kalton and Lutz Weis
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "The $H^{\infty}$--functional calculus and square function estimates" by Nigel Kalton and Lutz Weis. Abstract: Using notions from the geometry of Banach spaces we introduce square functions $\gamma(\Omega,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(\Omega,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $\gamma(\Omega,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e.~g., that a $c_0$--group of operators $T_s$ on a Banach space with finite cotype has an $H^{\infty}$--calculus on a strip if and only if $e^{-a|s|}T_s$ is $R$--bounded for some $a > 0$. Similarly, a sectorial operator $A$ has an $H^{\infty}$--calculus on a sector if and only if $A$ has $R$--bounded imaginary powers. We also consider vector valued Paley--Littlewood $g$--functions on $UMD$--spaces. Archive classification: math.FA Submitted from: Lutz.weis(a)kit.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.0472 or http://arXiv.org/abs/1411.0472

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Abstract of a paper by Daniel Azagra and Carlos Mudarra
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Global approximation of convex functions by differentiable convex functions on Banach spaces" by Daniel Azagra and Carlos Mudarra. Abstract: We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by $C^k$ smooth convex functions. Archive classification: math.FA Mathematics Subject Classification: 46B20, 52A99, 26B25, 41A30 Remarks: 8 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/1411.0471 or http://arXiv.org/abs/1411.0471

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Abstract of a paper by Trond A. Abrahamsen<br>
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "An improvement of a theorem of Heinrich, Mankiewicz, Sims, and Yost" by Trond A. Abrahamsen. Abstract: Heinrich, Mankiewicz, Sims, and Yost proved that every separable subspace of a Banach space Y is contained in a separable ideal in Y. We improve this result by replacing the term "ideal" with the term "almost isometric ideal". As a consequence of this we obtain, in terms of subspaces, characterizations of diameter 2 properties, the Daugavet property along with the properties of being an almost square space and an octahedral space. Archive classification: math.FA Mathematics Subject Classification: 46B20 (Primary) 46B07 (Secondary) Remarks: 13 pages Submitted from: trond.a.abrahamsen(a)uia.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.0425 or http://arXiv.org/abs/1411.0425

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Abstract of a paper by Michael Cwikel and Richard Rochberg
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Lecture notes on complex interpolation of compactness" by Michael Cwikel and Richard Rochberg. Abstract: Suppose that the linear operator $T$ maps $X_0$ compactly to $Y_0$ and also maps $X_1$ boundedly to $Y_1$. We deal once again with the 51 year old question of whether $T$ also always maps the complex interpolation space $[X_0,X_1]_\theta$ compactly to $[Y_0,Y_1]_\theta$. This is a short preliminary version of our promised technical sequel to our earlier paper arXiv:1410.4527 on this topic. It contains the following two small new partial results: (i) The answer to the above question is yes, in the particular case where $Y_0$ is a UMD-space. (ii) The answer to the above question is yes for given spaces $X_0$, $X_1$, $Y_0$ and $Y_1$ if the answer to the "dualized" or "adjoint" version of the question for the duals of these particular spaces is yes. In fact we deduce (i) from (ii) and from an earlier result obtained jointly by one of us with Nigel Kalton. It is remarked that a proof of a natural converse of (ii) would answer the general form of this question completely. Archive classification: math.FA Mathematics Subject Classification: Primary 46B70, 46B50. Secondary 46E15 Remarks: 7 pages Submitted from: mcwikel(a)math.technion.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1411.0171 or http://arXiv.org/abs/1411.0171

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Abstract of a paper by Richard Lechner and Paul F.X. Mueller
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Localization and projections on bi--parameter BMO" by Richard Lechner and Paul F.X. Mueller. Abstract: We prove that for any operator T on bi--parameter BMO the identity factors through T or Id - T. As a consequence, bi--parameter BMO is a primary Banach space. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable Banach space bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by combinatorics of colored dyadic rectangles. Archive classification: math.FA Mathematics Subject Classification: 46B25, 60G46, 46B07, 46B26, 30H35 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/1410.8786 or http://arXiv.org/abs/1410.8786

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Abstract of a paper by E. Casini, E. Miglierina, and L. Piasecki
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "Hyperplanes in the space of convergent sequences and preduals of $\ell_1$" by E. Casini, E. Miglierina, and L. Piasecki. Abstract: The main aim of the present paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of $c$ is isometric to the whole space if and only if it is $1$-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to $\ell_{1}$ and we give a complete description of the preduals of $\ell_{1}$ under the assumption that the standard basis of $\ell_{1}$ is weak$^{*}$-convergent. Archive classification: math.FA Mathematics Subject Classification: 46B45 (Primary), 46B04 (Secondary) Submitted from: enrico.miglierina(a)unicatt.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.7801 or http://arXiv.org/abs/1410.7801

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Abstract of a paper by Afonso S. Bandeira, Dustin G. Mixon, and Joel Moreira
by alspach＠math.okstate.edu 14 Nov '14

14 Nov '14

This is an announcement for the paper "A conditional construction of restricted isometries" by Afonso S. Bandeira, Dustin G. Mixon, and Joel Moreira. Abstract: We study the restricted isometry property of a matrix that is built from the discrete Fourier transform matrix by collecting rows indexed by quadratic residues. We find an $\epsilon>0$ such that, conditioned on a folklore conjecture in number theory, this matrix satisfies the restricted isometry property with sparsity parameter $K=\Omega(M^{1/2+\epsilon})$, where $M$ is the number of rows. Archive classification: math.FA cs.IT math.IT math.NT Remarks: 6 pages Submitted from: moreira(a)math.ohio-state.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.6457 or http://arXiv.org/abs/1410.6457

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