Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Teresa Bermúdez, Antonio Bonilla, Vladimir Müller, Alfredo Peris
by Bentuo Zheng (bzheng) 17 Jul '17

17 Jul '17

This is an announcement for the paper “Cesàro bounded operators in Banach spaces” by Teresa Bermúdez<https://arxiv.org/find/math/1/au:+Bermudez_T/0/1/0/all/0/1>, Antonio Bonilla<https://arxiv.org/find/math/1/au:+Bonilla_A/0/1/0/all/0/1>, Vladimir Müller<https://arxiv.org/find/math/1/au:+Muller_V/0/1/0/all/0/1>, Alfredo Peris<https://arxiv.org/find/math/1/au:+Peris_A/0/1/0/all/0/1>. Abstract: We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Ces\`aro bounded and strong Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing absolutely Ces\`aro bounded operators on $\ell_p(\mathbb{N}), 1\leq p<\infty$, which are not power bounded, and provide examples of uniformly Kreiss bounded operators which are not absolutely Ces\`aro bounded. These results complement very limited number of known examples (see \cite{Shi} and \cite{AS}). In \cite{AS} Aleman and Suciu ask if every uniformly Kreiss bounded operator $T$ on a Banach spaces satisfies that $\lim_n\|T_n/n\|=0$. We solve this question for Hilbert space operators and, moreover, we prove that, if $T$ is absolutely Ces\`aro bounded on a Banach (Hilbert) space, then $\|T_n\|=o(n)$ ($\|T_n\|=o(n^{1/2})$, respectively). As a consequence, every absolutely Ces\`aro bounded operator on a reflexive Banach space is mean ergodic, and there exist mixing mean ergodic operators on $\ell_p(\mathbb{N}), 1< p<\infty$. Finally, we give new examples of weakly ergodic $3$-isometries and study numerically hypercyclic $m$-isometries on finite or infinite dimensional Hilbert spaces. In particular, all weakly ergodic strict $3$-isometries on a Hilbert space are weakly numerically hypercyclic. Adjoints of unilateral forward weighted shifts which are strict $m$-isometries on $\ell_2(\mathbb{N})$ are shown to be hypercyclic. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1706.03638

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Abstract of a paper by Valentino Magnani, Andrea Pinamonti, Gareth Speight.
by Bentuo Zheng (bzheng) 17 Jul '17

17 Jul '17

This is an announcement for the paper “Porosity and Differentiability of Lipschitz Maps from Stratified Groups to Banach Homogeneous Groups” by Valentino Magnani<https://arxiv.org/find/math/1/au:+Magnani_V/0/1/0/all/0/1>, Andrea Pinamonti<https://arxiv.org/find/math/1/au:+Pinamonti_A/0/1/0/all/0/1>, Gareth Speight<https://arxiv.org/find/math/1/au:+Speight_G/0/1/0/all/0/1>. Abstract: Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $sigma$-porous set. At density points of $A$ we establish a pointwise characterization of differentiability in terms of directional derivatives. We use these new results to obtain an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property. As a consequence we also get an alternative proof of Pansu's Theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1706.01782

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SUMIRFAS at Texas A&M
by Bill Johnson 11 Jul '17

11 Jul '17

The 2017 Summer Informal Regional Functional Analysis Seminar (SUMIRFAS) will take place July 21-23 as part of the annual Workshop in Analysis and Probability at Texas A&M University. SUMIRFAS is an annual three-day conference that aims to cover a broad spectrum of topics in analysis and probability. It is one of the peaks of activity in the Workshop every summer. Although nominally a regional seminar, the conference has grown into an international event. More information can be found at: http://www.math.tamu.edu/~kerr/workshop/sumirfas2017 Preceding SUMIRFAS next week will be the Concentration Week "Probabilistic and Algebraic Methods in Quantum Information Theory". Details can be found via the Workshop homepage at http://www.math.tamu.edu/~kerr/workshop

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1st announcement of BWB 2018
by valentin ferenczi 05 Jul '17

05 Jul '17

1st ANNOUNCEMENT OF BWB 2018 Second Brazilian Workshop in Geometry of Banach Spaces August 13-17, 2018 Maresias, Sao Paulo State, Brazil. (Satellite Conference of the ICM 2018) We are glad to announce that we are organizing the Second Brazilian Workshop in Geometry of Banach Spaces, as a satellite conference of the ICM 2018 (Rio de Janeiro). This international conference will take place at the Beach Hotel Maresias, on the coast of Sao Paulo State, in Maresias, in the week August 13-17, 2018. The scientific program will focus on the theory of geometry of Banach spaces, with emphasis on the following directions: large scale geometry of Banach spaces; nonlinear theory; homological theory and set theory. The webpage of the workshop is under construction and will be available at http://www.ime.usp.br/~banach/bwb2018/ <http://www.ime.usp.br/%7Ebanach/bwb2014/> Registration will start early 2018. Additional scientific, practical and financial information will be given at that time. Plenary speakers: S. A. Argyros (Nat. Tech. U. Athens) G. Godefroy (Paris 6) S. Grivaux* (U. Picardie Jules Verne) R. Haydon* (U. Oxford) W. B. Johnson (Texas A&M) J. Lopez-Abad (U. Paris 7) A. Naor* (U. Princeton) D. Pellegrino (UFPB) G. Pisier* (Paris 6 & Texas A&M) B. Randrianantoanina (Miami U.) C. Rosendal (U. Illinois Chicago) N. Weaver (Washington U.) (* to be confirmed) Scientific committee J. M. F. Castillo (U. Extremadura) R. Deville (U. Bordeaux) V. Ferenczi (U. São Paulo) M. Gonzalez (U. Cantabria) V. Pestov (U. Ottawa & UFSC) G. Pisier (U. Paris 6 & Texas A&M) D. Preiss (U. Warwick) B. Randrianantoanina (Miami U.) We are looking forward to meeting you next year in Brazil, L. Batista, C. Brech, W. Cuellar, V. Ferenczi and P. Kaufmann

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Abstract of a paper by Petr Hajek, Thomas Schlumprecht, Andras Zsak
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Generalization of a theorem of Zippin” by Petr Hajek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>, Andras Zsak<https://arxiv.org/find/math/1/au:+Zsak_A/0/1/0/all/0/1>. Abstract: We generalize and prove a result which was first shown by Zippin, and was explicitly formulated by Benyamini. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.10663

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Abstract of a paper by Anna Kamińska, Han Ju Lee, Hyung-Joon Tag
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “M-ideal properties in Orlicz-Lorentz spaces” by Anna Kamińska<https://arxiv.org/find/math/1/au:+Kaminska_A/0/1/0/all/0/1>, Han Ju Lee<https://arxiv.org/find/math/1/au:+Lee_H/0/1/0/all/0/1>, Hyung-Joon Tag<https://arxiv.org/find/math/1/au:+Tag_H/0/1/0/all/0/1>. Abstract: We provide explicit formulas for the norm of bounded linear functionals on Orlicz-Lorentz function spaces $\Lambda_{\phi, w}$ equipped with two standard Luxemburg and Orlicz norms. Any bounded linear functional is a sum of regular and singular functionals, and we show that the norm of a singular functional is the same regardless of the norm in the space, while the formulas of the norm of general functionals are different for the Luxemburg and Orlicz norm. The relationship between equivalent definitions of the modular $P_{\phi, w}$ generating the dual space to Orlicz-Lorentz space is discussed in order to compute the norm of a bounded linear functional on $\Lambda_{\phi, w}$ equipped with Orlicz norm. As a consequence, we show that the order-continuous subspace of Orlicz-Lorentz space equipped with the Luxemburg norm is an $M$-ideal in $\Lambda_{\phi, w}$, while this is not true for the space with the Orlicz norm when $\phi$ is an Orlicz $N$-function not satisfying the appropriate $\Delta_2$ condition. The analogous results on Orlicz-Lorentz sequence spaces are given. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.10451

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Abstract of a paper by Sheng Zhang
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Asymptotic properties of Banach spaces and coarse quotient maps” by Sheng Zhang<https://arxiv.org/find/math/1/au:+Zhang_S/0/1/0/all/0/1>. Abstract: We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space $Y$ is a coarse quotient of a subset of a Banach space $X$, where the coarse quotient map is coarse Lipschitz, then the $(\beta)$-modulus of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.10207

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Abstract of a paper by R. M. Causey
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Power type ξ-Asymptotically uniformly smooth and ξ-asymptotically uniformly flat norms” by R. M. Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: For each ordinal $\xi$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $\xi$ -$p$ -asymptotically uniformly smooth norm. We also introduce the notion of $\xi$ -asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $\xi$ -asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $\xi$ -asymptotic smoothness of $C(K)$ in terms of the Cantor-Bendixson index of $K$. We also prove that for all ordinals, both the isomorphic properties and isometric properties we study pass from Banach spaces to their injective tensor products. We study the classes of $\xi$ -$p$ -asymptotically uniformly smooth, $\xi$ -$p$ -asymptotically uniformly smoothable, $\xi$ -asymptotically uniformly flat, and $\xi$ -asymptotically uniformly flattenable operators. We show that these classes are either a Banach ideal or a right Banach ideal when assigned an appropriate ideal norm. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.09834

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Abstract of a paper by Florin Catrina, Mikhail I. Ostrovskii
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Images of nowhere differentiable Lipschitz maps of $[0,1]$ into $L_1[0,1]$” by Florin Catrina<https://arxiv.org/find/math/1/au:+Catrina_F/0/1/0/all/0/1>, Mikhail I. Ostrovskii<https://arxiv.org/find/math/1/au:+Ostrovskii_M/0/1/0/all/0/1>. Abstract: The main result: for every $m\in\mathbb{N}$ and $omega>0$ there exists an isometric embedding $F: [0,1]\rightarrow L_1[0,1]$ which is nowhere differentiable, but for each $t\in[0,1]$ the image $F_t$ is an $m$-times continuously differentiable function with absolute values of all of its $m$ derivatives bounded from above by $\omega$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.08916

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Abstract of a paper by Mohammad Sal Moslehian, Ali Zamani, Mahdi Dehghani
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Characterizations of smooth spaces by $\rho_*$-orthogonality” by Mohammad Sal Moslehian<https://arxiv.org/find/math/1/au:+Moslehian_M/0/1/0/all/0/1>, Ali Zamani<https://arxiv.org/find/math/1/au:+Zamani_A/0/1/0/all/0/1>, Mahdi Dehghani<https://arxiv.org/find/math/1/au:+Dehghani_M/0/1/0/all/0/1>. Abstract: The aim of this paper is to present some results concerning the $\rho_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T: X\rightarrow Y$ is a nonzero linear $(I, \rho_*)$-orthogonality preserving mapping between real normed spaces, then $$ 13\|T\|\|x\|\leq \|Tx\|\leq 3\|T\|\|x\|, x\in X $$ where $[T]:=\inf\{\|Tx\|: x\in X, \|x\|=1\}$. We also show that the pair $(X, \perp_{\rho_*})$ is an orthogonality space in the sense of R\"{a}tz. Some characterizations of smooth spaces are given based on the $\rho_*$-orthogonality. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.07032

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