Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Amra Rekic-Vukovic, Nermin Okicic, Vedad Pasic, Ivan Arandjelovic
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Continuity of modulus of noncompact convexity for minimalizable measures of noncompactness” by Amra Rekic-Vukovic<http://arxiv.org/find/math/1/au:+Rekic_Vukovic_A/0/1/0/all/0/1>, Nermin Okicic<http://arxiv.org/find/math/1/au:+Okicic_N/0/1/0/all/0/1>, Vedad Pasic<http://arxiv.org/find/math/1/au:+Pasic_V/0/1/0/all/0/1>, Ivan Arandjelovic<http://arxiv.org/find/math/1/au:+Arandjelovic_I/0/1/0/all/0/1>. Abstract: We consider the modulus of noncompact convexity $\Delta_{X, \Phi(\epsilon)}$ associated with the minimalizable measure of noncompactness $\Phi$. We present some properties of this modulus, while the main result of this paper is showing that $\Delta_{X, \Phi(\epsilon)}$ is a subhomogenous and continuous function on $[0, \Phi(\bar{B}_X))$ for an arbitrary minimalizable measure of compactness $\Phi$ in the case of a Banach space $X$ with the Radon-Nikodym property. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.01063

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Abstract of a paper by Anna Kamińska, Hyung-Joon Tag
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Diameter of weak neighborhoods and the Radon-Nikodym property in Orlicz-Lorentz spaces” by Anna Kamińska<http://arxiv.org/find/math/1/au:+Kaminska_A/0/1/0/all/0/1>, Hyung-Joon Tag<http://arxiv.org/find/math/1/au:+Tag_H/0/1/0/all/0/1>. Abstract: Given an Orlicz convex function $\phi$ and a positive weight $w$ we present criteria of diameter two property and of Radon-Nikodym property in the Orlicz-Lorentz function and sequence spaces, $\Lambda_{\phi, w}$ and $\lambda_{\phi, w}$, respectively. We show that in the spaces $\Lambda_{\phi, w}$ or $\lambda_{\phi, w}$ equipped with the Luxemburg norm, the diameter of any relatively weakly subset of the unit ball in these spaces is two if and only if $\phi$ does not satisfy the appropriate growth condition $\Delta_2$, while they do have the Radon-Nikodym property if and only if $\phi$ satisfies the appropriate condition $\Delta_2$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00909

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Abstract of a paper by Malgorzata Czerwinska, Annna Kaminska
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Banach Envelopes in Symmetric Spaces of Measurable Operators” by Malgorzata Czerwinska<http://arxiv.org/find/math/1/au:+Czerwinska_M/0/1/0/all/0/1>, Annna Kaminska<http://arxiv.org/find/math/1/au:+Kaminska_A/0/1/0/all/0/1>. Abstract: We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class $(HC)$ of quasi-normed symmetric sequence or function spaces $E$ for which their Banach envelopes $\hat {E}$ are also symmetric spaces. The class of symmetric spaces satisfying $(HC)$ contains but is not limited to order continuous spaces. Let $\mathcal{M}$ be a non-atomic, semifinite von Neumann algebra with a faithful, normal, $\sigma$-finite trace $\tau$ and $E$ be as symmetric function space on $[0, \tau(1))$ or symmetric sequence space. We compute Banach envelope norms on $E(\mathcal{M}, \tau)$ and $C_E$ for any quasi-normed symmetric space $E$. Then we show under assumption that $E\in (HC)$ that the Banach envelope $\hat{E(\mathcal{M}, \tau)}$ of $E(\mathcal{M}, \tau)$ is equal to $\hat{E(\mathcal{M}, \tau)}$ isometrically. We also prove the analogous result for unitary matrix spaces $C_E$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00319

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Abstract of a paper by Elói M. Galego, Manuel González, Javier Pello
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “On subprojectivity and superprojectivity of Banach spaces” by Elói M. Galego<http://arxiv.org/find/math/1/au:+Galego_E/0/1/0/all/0/1>, Manuel González<http://arxiv.org/find/math/1/au:+Gonzalez_M/0/1/0/all/0/1>, Javier Pello<http://arxiv.org/find/math/1/au:+Pello_J/0/1/0/all/0/1>. Abstract: We obtain some results for and further examples of subprojective and superprojective Banach spaces. We also give several conditions providing examples of non-reflexive superprojective spaces; one of these conditions is stable under $c_0$-sums and projective tensor products. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00308

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Abstract of a paper by Miek Messerschmidt
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “A strengthening of Wickstead's Theorem: Ordered Banach spaces in which every precompact set is order bounded” by Miek Messerschmidt<http://arxiv.org/find/math/1/au:+Messerschmidt_M/0/1/0/all/0/1>. Abstract: A theorem of Wickstead from 1975 characterizes the ordered Banach spaces with order bounded precompact sets in terms of a geometric property, "coadditivity", relating the space's order with its topology. We strengthen Wickstead's Theorem by showing for an ordered Banach space to have all its precompact sets be order bounded, it is necessary and sufficient for the space to have all its null sequences be order bounded. To establish our strengthening of Wickstead's Theorem, we first prove an Open Mapping Theorem for cone-valued correspondences, which is then employed to prove a Klee-And type theorem for coadditivity (the classical Klee-And Theorem concerns another geometric property, namely "conormality"). By employing this Klee-And type theorem for coadditivity, we establish the equivalence of an ordered Banach space having the coadditivity property from Wickstead's original result with the space having all its null sequences be order bounded. Finally, for the purpose of illustration, we briefly investigate the natural order structures of the James space and the Tsirelson space. The James space is not a Banach lattice, but all its precompact sets are order bounded. The Tsirelson space is a Banach lattice, but not all its precompact sets are order bounded. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00249

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Abstract of a paper by Trond A. Abrahamsen, Vegard Lima, Olav Nygaard, Stanimir Troyanski
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Diameter two properties, convexity and smoothness” by Trond A. Abrahamsen<http://arxiv.org/find/math/1/au:+Abrahamsen_T/0/1/0/all/0/1>, Vegard Lima<http://arxiv.org/find/math/1/au:+Lima_V/0/1/0/all/0/1>, Olav Nygaard<http://arxiv.org/find/math/1/au:+Nygaard_O/0/1/0/all/0/1>, Stanimir Troyanski<http://arxiv.org/find/math/1/au:+Troyanski_S/0/1/0/all/0/1>. Abstract: We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter $2$ properties. We prove that the strong diameter $2$ property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter $2$ property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to $2$ as we want." Spaces with this property are shown to have non-smooth bidual.. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00221

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Abstract of a paper by Jan Rozendaal and Mark Veraar
by Bentuo Zheng (bzheng) 01 Jun '16

01 Jun '16

This is an announcement for the paper “Fourier multiplier theorems involving type and cotype” by Jan Rozendaal and Mark Veraar. Abstract: In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^{d}\to\mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's and is reasonably well-understood. Far less is known for $p<q$. In the scalar case one can deduce results for $p<q$ from the case $p=q$ and additional arguments such as interpolation and Sobolev embedding techniques. However, in the vector-valued case this leads to restrictions both on the smoothness of the multiplier and the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces, and that $m$ satisfies a smoothness condition. For $p<q$ it turns out that the notions of type and cotype and other geometric conditions on $X$ and $Y$ are important to study Fourier multipliers. Moreover, we obtain boundedness results for $T_{m}$ without any smoothness properties of $m$. Under additional smoothness conditions we show that boundedness results can be extrapolated to other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ remains constant. Here we even extend classical results in the scalar case. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.09340

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Abstract of a paper by Jan Rozendaal
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Fourier multiplier theorems involving type and cotype” by Jan Rozendaal. Abstract: In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^{d}\to\mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's and is reasonably well-understood. Far less is known for $p<q$. In the scalar case one can deduce results for $p<q$ from the case $p=q$ and additional arguments such as interpolation and Sobolev embedding techniques. However, in the vector-valued case this leads to restrictions both on the smoothness of the multiplier and the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces, and that $m$ satisfies a smoothness condition. For $p<q$ it turns out that the notions of type and cotype and other geometric conditions on $X$ and $Y$ are important to study Fourier multipliers. Moreover, we obtain boundedness results for $T_{m}$ without any smoothness properties of $m$. Under additional smoothness conditions we show that boundedness results can be extrapolated to other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ remains constant. Here we even extend classical results in the scalar case. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.09340

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Abstract of a paper by Asuman G. Aksoy and Qidi Peng
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “On a Theorem of S. N. Bernstein for Banach Spaces” by Asuman G. Aksoy and Qidi Peng. Abstract: This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We prove that if $X$ is an infinite-dimensional Banach space and $(Y_n)$ is a nested sequence of subspaces of $X$ such that $Y_n\subset Y_{n+1}$ and $\bar{Y_n}\subset Y_{n+1}$ for any $n\in\mathbb{N}$ and if $(d_n)$ be a decreasing sequence of positive numbers tending to 0, then for any $0<c\leq 1$ there exists $x_c\in X$ such that the distance $\rho (x_c, Y_n)$ from $x_c$ to $Y_n$ satisfies $$cd_n\leq\rho(x_c, Y_n)\leq 4c d_n$$. We prove the above by first improving Borodin's result \cite{Borodin} for Banach spaces by weakening the condition on the sequence $(d_n)$. Lastly, we compare subsequences $(d(\phi_n))$ under different choices of $(\phi_n)$ and examine their effects on approximation. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.04592

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Abstract of a paper by S. Ferrari, J. Orihuela and M. Raja
by Bentuo Zheng (bzheng) 31 May '16

31 May '16

This is an announcement for the paper “Generalized metric properties of spheres and renorming of normed spaces” by S. Ferrari, J. Orihuela and M. Raja. Abstract: We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove that there exists a dual norm such that its unit sphere is a Moore space for the weak$^*$topology (has a $G_\delta$-diagonal for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively). The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.08175

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