Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Jesus M. F. Castillo, Valentin Ferenczi and Manuel Gonzalez
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Singular twisted sums generated by complex interpolation" by Jesus M. F. Castillo, Valentin Ferenczi and Manuel Gonzalez. Abstract: We present new methods to obtain singular twisted sums $X\oplus_\Omega X$ (i.e., exact sequences $0\to X\to X\oplus_\Omega X \to X\to 0$ in which the quotient map is strictly singular), in which $X$ is the interpolation space arising from a complex interpolation scheme and $\Omega$ is the induced centralizer. Although our methods are quite general, in our applications we are mainly concerned with the choice of $X$ as either a Hilbert space, or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces, including the only known example so far: the Kalton-Peck space $Z_2$. In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. We then use Rochberg's description of iterated twisted sums to show that there is a sequence $\mathcal F_n$ of H.I. spaces so that $\mathcal F_{m+n}$ is a singular twisted sum of $\mathcal F_m$ and $\mathcal F_n$, while for $l>n$ the direct sum $\mathcal F_n \oplus \mathcal F_{l+m}$ is a nontrivial twisted sum of $\mathcal F_l$ and $\mathcal F_{m+n}$. We also introduce and study the notion of disjoint singular twisted sum of K\"othe function spaces and construct several examples involving reflexive $p$-convex K\"othe function spaces, which include the function version of the Kalton-Peck space $Z_2$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B70, 46M18 Submitted from: castillo(a)unex.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5505 or http://arXiv.org/abs/1410.5505

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Abstract of a paper by Jarno Talponen
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Note on order-isomorphic isometric embeddings of some recent function spaces" by Jarno Talponen. Abstract: We investigate certain recently introduced ODE-determined varying exponent $L^p$ spaces. It turns out that these spaces are finitely representable in a concrete universal varying exponent $\ell^p$ space. Moreover, this can be accomplished in a natural unified fashion. This leads to order-isomorphic isometric embeddings of all of the above $L^p$ spaces to an ultrapower of the above varying exponent $\ell^p$ space. Archive classification: math.FA math.CA Mathematics Subject Classification: 46E30, 46B08, 46B04, 46B42, 46B45, 34-XX Submitted from: talponen(a)iki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.4961 or http://arXiv.org/abs/1410.4961

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Abstract of a paper by Julio Flores, Jordi Lopez-Abad, and Pedro Tradacete
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Banach lattice versions of strict singularity" by Julio Flores, Jordi Lopez-Abad, and Pedro Tradacete. Abstract: We explore the relation between lattice versions of strict singularity for operators from a Banach lattice to a Banach space. In particular, we study when the class of disjointly strictly singular operators, those not invertible on the span of any disjoint sequence, coincides with that of lattice strictly singular operators, i.e. those not invertible on any (infinite dimensional) sublattice. New results are given which help to clarify the existing relation between these two classes. Archive classification: math.FA Mathematics Subject Classification: 46B42, 47B60 Submitted from: ptradace(a)math.uc3m.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.4752 or http://arXiv.org/abs/1410.4752

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Abstract of a paper by Julio Delgado, Michael Ruzhansky and Baoxiang Wang
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Approximation property and nuclearity on mixed-norm $L^p$, modulation and Wiener amalgam spaces" by Julio Delgado, Michael Ruzhansky and Baoxiang Wang. Abstract: In this paper we first prove the metric approximation property for weighted mixed-norm Lebesgue spaces. Then, using Gabor frame representation we show that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and r-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces. Archive classification: math.FA math.OA Mathematics Subject Classification: Primary 46B26, 47B38, Secondary 47G10, 47B06, 42B35 Remarks: 20 pages Submitted from: m.ruzhansky(a)imperial.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.4687 or http://arXiv.org/abs/1410.4687

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Abstract of a paper by Michael Cwikel and Richard Rochberg
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Nigel Kalton and complex interpolation of compact operators" by Michael Cwikel and Richard Rochberg. Abstract: This is the fourth of a series of papers surveying some small part of the remarkable work of our friend and colleague Nigel Kalton. We have written it as part of a tribute to his memory. It does not contain new results. This time we discuss Nigel's partial solutions (obtained jointly with one of us) of the problem of whether the complex method of interpolation preserves the compactness of operators. This problem is now 51 years old and still lacks a complete solution. We also survey some other partial solutions of this problem, obtained before and after the above mentioned joint work. We plan a technical sequel to this paper, which may contain some small new results and will probably conclude this series devoted to Nigel's research. Archive classification: math.FA Mathematics Subject Classification: Primary 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/1410.4527 or http://arXiv.org/abs/1410.4527

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Abstract of a paper by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Diameter two properties in James spaces" by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda. Abstract: We study the diameter two properties in the spaces $JH$, $JT_\infty$ and $JH_\infty$. We show that the topological dual space of the previous Banach spaces fails every diameter two property. However, we prove that $JH$ and $JH_{\infty}$ satisfy the strong diameter two property, and so the dual norm of these spaces is octahedral. Also we find a closed hyperplane $M$ of $JH_\infty$ whose topological dual space enjoys the $w^*$-strong diameter two property and also $M$ and $M^*$ have an octahedral norm. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B22 Remarks: 19 pages Submitted from: glopezp(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.4325 or http://arXiv.org/abs/1410.4325

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Abstract of a paper by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Subspaces of Banach spaces with big slices" by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda. Abstract: We study when diameter two properties pass down to subspaces. We obtain that the slice two property (respectively diameter two property, strong diameter two property) passes down from a Banach space $X$ to a subspace $Y$ whenever $Y$ is complemented by a norm one projection with finite-dimensional kernel (respectively the quotient $X/Y$ is finite dimensional, $X/Y$ is strongly regular). Also we study the same problem for dual properties of the above ones, as having octahedral, weakly octahedral or 2-rough norm. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B22 Remarks: 12 pages Submitted from: glopezp(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.4324 or http://arXiv.org/abs/1410.4324

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Abstract of a paper by Martino Lupini
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Uniqueness, universality, and homogeneity of the noncommutative Gurarij space" by Martino Lupini. Abstract: We realize the noncommutative Gurarij space $\mathbb{NG}$ defined by Oikhberg as the Fra\"{\i}ss\'{e} limit of the class of finite-dimensional $1$-exact operator spaces. As a consequence we deduce that the noncommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable $1$-exact operator spaces. Moreover we show that $\mathbb{NG}$ is isometrically isomorphic to the Gurarij Banach space. Therefore $\mathbb{NG}$ can be thought as a canonical operator space structure on the Gurarij Banach space. Archive classification: math.FA math.LO Mathematics Subject Classification: 46L07 (Primary) 03C30 (Secondary) Remarks: 24 pages Submitted from: mlupini(a)yorku.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.3345 or http://arXiv.org/abs/1410.3345

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Abstract of a paper by David Alonso-Gutierrez, Bernardo Gonzalez, C. Hugo Jimenez, and Rafael Villa
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Rogers-Shephard inequality for log-concave functions" by David Alonso-Gutierrez, Bernardo Gonzalez, C. Hugo Jimenez, and Rafael Villa. Abstract: In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets. Archive classification: math.FA math.MG Mathematics Subject Classification: Primary 52A20, Secondary 39B62, 46N10 Remarks: 24 pages Submitted from: carloshugo(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.2556 or http://arXiv.org/abs/1410.2556

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Abstract of a paper by Jochen Gl\"uck
by alspach＠math.okstate.edu 21 Oct '14

21 Oct '14

This is an announcement for the paper "Spectral and asymptotic properties of contractive semigroups on non-Hilbert spaces" by Jochen Gl\"uck. Abstract: We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of those spaces forces the semigroup's generator to have only trivial (point) spectrum on the imaginary axis. This has interesting consequences for the asymptotic behaviour as $t \to \infty$. For example, we can show that a contractive and eventually norm continuous $C_0$-semigroup on a real-valued $L^p$-space automatically converges strongly if $p \not\in \{1,2,\infty\}$. Archive classification: math.FA Mathematics Subject Classification: 47D06 Remarks: 26 pages Submitted from: jochen.glueck(a)uni-ulm.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.2502 or http://arXiv.org/abs/1410.2502

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