Banach February 2015

banach@mathdept.okstate.edu

2 participants
28 discussions

Abstract of a paper by Michael Cwikel
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "Some alternative definitions for the ''plus-minus'' interpolation spaces $\left\langle A_{0},A_{1}\right\rangle _{\theta}$ of Jaak Peetre" by Michael Cwikel. Abstract: The Peetre "plus-minus" interpolation spaces $\left\langle A_{0},A_{1}\right\rangle _{\theta}$ are defined variously via conditions about the unconditional convergence of certain Banach space valued series whose terms have coefficients which are powers of 2 or, alternatively, powers of e. It may seem intuitively obvious that using powers of 2, or of e, or powers of some other constant number greater than 1 in such definitions should produce the same space to within equivalence of norms. To allay any doubts, we here offer an explicit proof of this fact, via a "continuous" definition of the same spaces where integrals replace the above mentioned series. This apparently new definition, which is also in some sense a "limiting case" of the above mentioned "discrete" definitions, may be relevant in the study of the connection between the Peetre "plus-minus" interpolation spaces and Calderon complex interpolation spaces when both the spaces of the underlying couple are are Banach lattices on the same measure space. Related results can probably be obtained for the Gustavsson-Peetre variant of the "plus-minus" spaces. Archive classification: math.FA Mathematics Subject Classification: 46B70 Remarks: 12 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/1502.00986 or http://arXiv.org/abs/1502.00986

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Abstract of a paper by Geraldo Botelho and Jamilson R. Campos
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "Type and cotype of multilinear operators" by Geraldo Botelho and Jamilson R. Campos. Abstract: We introduce the notions of type and cotype of multilinear operators between Banach spaces and the resulting classes of such mappings are studied in the setting of the theory of Banach/quasi-Banach ideals of multilinear operators. Distinctions between the linear and the multilinear theories are pointed out, typical multilinear features of the theory are emphasized and many illustrative examples are provided. The classes we introduce are related to the multi-ideals generated by the linear ideals of operators of some type/cotype and are proved to be maximal and Aron-Berner stable. Archive classification: math.FA Submitted from: jamilson(a)dce.ufpb.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1502.00440 or http://arXiv.org/abs/1502.00440

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Abstract of a paper by Jose Bonet
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "Abscissas of weak convergence of vector valued Dirichlet series" by Jose Bonet. Abstract: The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $\sigma(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological properties of the underlying space $X$ is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fr\'echet spaces play a relevant role. Archive classification: math.FA Mathematics Subject Classification: Primary: 46A04, secondary: 30B50, 32A05, 46A03, 46A11, 46B07 Submitted from: jbonet(a)mat.upv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1502.00418 or http://arXiv.org/abs/1502.00418

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Abstract of a paper by Sergio Solimini and Cyril Tintarev
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "Concentration analysis in Banach spaces" by Sergio Solimini and Cyril Tintarev. Abstract: The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach-Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of $\Delta$-convergence by T. C. Lim instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and $\ell^{p}$-spaces, but not in $L^{p}(\mathbb R^{N})$, $p\neq2$. $\Delta$-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies connection of $\Delta$-convergence with Brezis-Lieb Lemma and gives a version of the latter without an assumption of convergence a.e. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B10, 46B50, 46B99 Submitted from: tintarev(a)math.uu.se The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1502.00414 or http://arXiv.org/abs/1502.00414

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Abstract of a paper by H. Ardakani and S.M.S. Modarres Mosadegh
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "Order almost Dunford-Pettis Operators on Banach lattices" by H. Ardakani and S.M.S. Modarres Mosadegh. Abstract: By introducing the concepts of order almost Dunford-Pettis and almost weakly limited operators in Banach lattices, we give some properties of them related to some well known classes of operators, such as, order weakly compact, order Dunford-Pettis, weak and almost Dunford-Pettis and weakly limited operators. Then, we characterize Banach lattices E and F on which each operator from E into F that is order almost Dunford-Pettis and weak almost Dunford-Pettis is an almost weakly limited operator. Archive classification: math.FA Submitted from: h_ardakani(a)stu.yazd.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1502.00369 or http://arXiv.org/abs/1502.00369

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Abstract of a paper by F. Abtahi, H. G. Amini, H. A. Lotfi, and A. Rejali
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "Some intersections of Lorentz spaces" by F. Abtahi, H. G. Amini, H. A. Lotfi, and A. Rejali. Abstract: Let (X,\mu) be a measure space. For p, q\in (0,\infty] and arbitrary subsets P,Q of (0,\infty], we introduce and characterize some intersections of Lorentz spaces, denoted by ILp,Q(X,\mu), ILJ,q(X,\mu) and ILJ,Q(X,\mu). Archive classification: math.FA Mathematics Subject Classification: 43A15, 43A20 Remarks: 10 pages, 0 figures Submitted from: f.abtahi(a)sci.ui.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1502.00159 or http://arXiv.org/abs/1502.00159

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Abstract of a paper by Farhad Jafari and Tyrrell B. McAllister
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "Ellipsoidal cones in normed vector spaces" by Farhad Jafari and Tyrrell B. McAllister. Abstract: We give two characterizations of cones over ellipsoids in real normed vector spaces. Let $C$ be a closed convex cone with nonempty interior such that $C$ has a bounded section of codimension $1$. We show that $C$ is a cone over an ellipsoid if and only if every bounded section of $C$ has a center of symmetry. We also show that $C$ is a cone over an ellipsoid if and only if the affine span of $\partial C \cap \partial(a - C)$ has codimension $1$ for every point $a$ in the interior of $C$. These results generalize the finite-dimensional cases proved in (Jer\'onimo-Castro and McAllister, 2013). Archive classification: math.FA math.MG Mathematics Subject Classification: Primary 46B20, Secondary 52A50, 46B40, 46B10 Remarks: 10 pages, 1 figure Submitted from: tmcallis(a)uwyo.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.07493 or http://arXiv.org/abs/1501.07493

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Abstract of a paper by Eva Pernecka and Richard J. Smith
by alspach＠math.okstate.edu 09 Feb '15

09 Feb '15

This is an announcement for the paper "The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$" by Eva Pernecka and Richard J. Smith. Abstract: We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular, $\mathcal{F}(M)$ has the MAP whenever $M$ is a finite-dimensional compact convex set. This should be compared with a recent result of Godefroy and Ozawa, who showed that there exists a compact convex subset $M$ of a separable Banach space, for which $\mathcal{F}(M)$ fails the approximation property. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B28 Submitted from: richard.smith(a)maths.ucd.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.07036 or http://arXiv.org/abs/1501.07036

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Banach February 2015