Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Marek Cuth
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Noncommutative Valdivia compacta" by Marek Cuth. Abstract: We prove some generalizations of results concerning Valdivia compact spaces (equivalently spaces with a commutative retractional skeleton) to the spaces with a retractional skeleton (not necessarily commutative). Namely, we show that the dual unit ball of a Banach space is Corson provided the dual unit ball of every equivalent norm has a retractional skeleton. Another result to be mentioned is the following. Having a compact space K, we show that K is Corson if and only if every continuous image of K has a retractional skeleton. Archive classification: math.FA Mathematics Subject Classification: 46B26, 54D30 Submitted from: cuthm5am(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.5799 or http://arXiv.org/abs/1301.5799

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Abstract of a paper by G. Garrigos and P. Wojtaszczyk
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Conditional quasi-greedy bases in Hilbert and Banach spaces" by G. Garrigos and P. Wojtaszczyk. Abstract: We show that, for quasi-greedy bases in Hilbert spaces, the associated conditionality constants grow at most as $O(\log N)^{1-\epsilon}$, for some $\epsilon>0$, answering a question by Temlyakov. We show the optimality of this bound with an explicit construction, based on a refinement of the method of Olevskii. This construction leads to other examples of quasi-greedy bases with large $k_N$ in Banach spaces, which are of independent interest. Archive classification: math.FA math.CA Submitted from: gustavo.garrigos(a)uam.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4844 or http://arXiv.org/abs/1301.4844

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Abstract of a paper by Florence Lancien and Christian Le Merdy
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "On functional calculus properties of Ritt operators" by Florence Lancien and Christian Le Merdy. Abstract: We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T : X --> X on some Banach space X with the following property: T has a bounded $\H^\infty$ functional calculus with respect to the unit disc $\D$ (that is, T is polynomially bounded) but T does not have any bounded $\H^\infty$ functional calculus with respect to a Stolz domain of $\D$ with vertex at 1. Also we show that for an R-Ritt operator, the unconditional Ritt condition of Kalton-Portal is equivalent to the existence of a bounded $\H^\infty$ functional calculus with respect to such a Stolz domain. Archive classification: math.FA math.OA Mathematics Subject Classification: 47A60 Submitted from: clemerdy(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4875 or http://arXiv.org/abs/1301.4875

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Abstract of a paper by Yousef Estaremi
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Multiplication and composition operators between two Orlicz spaces" by Yousef Estaremi. Abstract: In this paper we consider composition operator $C_{\varphi} generated by nonsingular measurable transformation $T$ and multiplication operator $M_u$ generated by measurable function $u$ between two different Or- licz spaces, then we investigate boundedness, compactness and essential norm of multiplication and composition operators in term of properties of the mapping $\varphi$, the function $u$ and the measure space $(X, \Sigma, \mu)$. Archive classification: math.FA Submitted from: estaremi(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4830 or http://arXiv.org/abs/1301.4830

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Abstract of a paper by Antonio J. Guirao and Olena Kozhushkina
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "The Bishop-Phelps-Bollob\'as property for numerical radius in $\ell_1(\mathbb{C})$" by Antonio J. Guirao and Olena Kozhushkina. Abstract: We show that the set of bounded linear operators from $X$ to $X$ admits a Bishop-Phelps-Bollob\'as type theorem for numerical radius whenever $X$ is $\ell_1(\mathbb{C})$ or $c_0(\mathbb{C})$. As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollob\'as theorem for $\ell_1(\mathbb{C})$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 47A12 Submitted from: okozhush(a)math.kent.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4574 or http://arXiv.org/abs/1301.4574

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Abstract of a paper by Anna Kaminska and Yves Raynaud
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "New formulas for decreasing rearrangements and a class of spaces" by Anna Kaminska and Yves Raynaud. Abstract: Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{\varphi,w}$ and $m_{\varphi,w}$ respectively, introduced earlier in \cite{HKM} for describing the K\"othe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($\varphi$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{\varphi,w}$ classes in the most general setting, where they may even not be linear, and identify their K\"othe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces $\mathcal{M}_{\varphi,w}$ which proves to be the K\"othe biduals of the $M_{\varphi,w}$ classes. In the case when the class $M_{\varphi,w}$ is a separable quasi-Banach space, $\mathcal{M}_{\varphi,w}$ is its Banach envelope. Archive classification: math.FA Mathematics Subject Classification: 26D07, 39B62, 42B25, 46B10, 46E30 Remarks: 25 pages Submitted from: kaminska(a)memphis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4465 or http://arXiv.org/abs/1301.4465

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Abstract of a paper by William B. Johnson and Gideon Schechtman
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Subspaces of $L_p$ that embed into $L_p(\mu)$ with $\mu$ finite" by William B. Johnson and Gideon Schechtman. Abstract: Enflo and Rosenthal proved that $\ell_p(\aleph_1)$, $1 < p < 2$, does not (isomorphically) embed into $L_p(\mu)$ with $\mu$ a finite measure. We prove that if $X$ is a subspace of an $L_p$ space, $1< p < 2$, and $\ell_p(\aleph_1)$ does not embed into $X$, then $X$ embeds into $L_p(\mu)$ for some finite measure $\mu$. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B26, 46B03 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.4086 or http://arXiv.org/abs/1301.4086

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Abstract of a paper by Manor Mendel and Assaf Naor
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "Spectral calculus and Lipschitz extension for barycentric metric spaces" by Manor Mendel and Assaf Naor. Abstract: The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for $CAT(0)$ targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp. Archive classification: math.MG math.FA Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.3963 or http://arXiv.org/abs/1301.3963

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Abstract of a paper by Lixin Cheng, Duanxu Dai, Yunbai Dong and Yu Zhou
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "On $\eps$-isometry, isometry and linear isometry" by Lixin Cheng, Duanxu Dai, Yunbai Dong and Yu Zhou. Abstract: Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $\gamma>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$ such that $\|Tf(x)-x\|\leq\gamma\eps$ for all $x\in X$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show first that if such an $\eps$-isometry $f$ exists, then there is a linear isometry $U:X^{**}\rightarrow Y^{**}$. Then we prove that universally- right-stable spaces are just Hilbert spaces; every injective space is universally-left-stable; Finally, we verify that a Banach space $X$ which is linear isomorphic to a subspace of $\ell_\infty$ is universally-left-stable if and only if it is linearly isomorphic to $\ell_\infty$; and a separable space $X$ satisfying that $(X,Y)$ is stable for every separable $Y$ if and only if $X$ is linearly isomorphic to $c_0$. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 47A58 (Primary) 26E25, 46A20, 46A24 (Secondary) Remarks: 14 pages, submitted to Israel Journal of Mathematics Submitted from: dduanxu(a)163.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.3374 or http://arXiv.org/abs/1301.3374

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Abstract of a paper by Duanxu Dai
by alspach＠math.okstate.edu 30 Jan '13

30 Jan '13

This is an announcement for the paper "A note on the Cheng-Dong-Zhang Theorem and its applications" by Duanxu Dai. Abstract: In this paper, we first give a short introduction to recent development on the stability of Banach spaces via $\eps$-isometry and then present an application of the Cheng-Dong-Zhang Theorem to the continuous selections of a set valued map via $\eps-$ isometries. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 54C60 (Primary) 26E25, 46A20, 54C65 (Secondary) Remarks: 7 pages Submitted from: dduanxu(a)163.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.3396 or http://arXiv.org/abs/1301.3396

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