Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by March Boedihardjo
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Multiplication operators on $L^p$” by March Boedihardjo<https://arxiv.org/find/math/1/au:+Boedihardjo_M/0/1/0/all/0/1>. Abstract: We show that every operator on $L_p, 1<p<\infty$ defined by multiplication by the identity function on $\mathbb{C}$ is a compact perturbation of an operator that is diagonal with respect to an unconditional basis. We also classify these operators up to similarity modulo compact operators and up to approximate similarity. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.04798

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Abstract of a paper by Arkady Kitover, Mehmet Orhon
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “The dual Radon - Nikodym property for finitely generated Banach $C(K)$-Modules” by Arkady Kitover<https://arxiv.org/find/math/1/au:+Kitover_A/0/1/0/all/0/1>, Mehmet Orhon<https://arxiv.org/find/math/1/au:+Orhon_M/0/1/0/all/0/1>. Abstract: We extend the well-known criterion of Lotz for the dual Radon-Nikodym property (RNP) of Banach lattices to finitely generated Banach $C(K)$-modules and Banach $C(K)$-modules of finite multiplicity. Namely, we prove that if $X$ is a Banach space from one of these classes then its Banach dual $X^*$ has the RNP iff $X$ does not contain a closed subspace isomorphic to $\ell_1$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.04655

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Abstract of a paper by Maciej Ciesielski
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Lower and upper local uniform $K$-monotonicity in symmetric spaces” by Maciej Ciesielski<https://arxiv.org/find/math/1/au:+Ciesielski_M/0/1/0/all/0/1>. Abstract: Using the local approach to the global structure of a symmetric space $E$ we establish a relationship between strict $K$- monotonicity, lower (resp. upper) local uniform $K$-monotonicity, order continuity and the Kadec-Klee property for global convergence in measure. We also answer the question under which condition upper local uniform $K$-monotonicity concludes upper local uniform monotonicity. Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.02632

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Abstract of a paper by Maciej Ciesielski, Grzegorz Lewicki
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Best approximation properties in spaces of measurable functions” by Maciej Ciesielski<https://arxiv.org/find/math/1/au:+Ciesielski_M/0/1/0/all/0/1>, Grzegorz Lewicki<https://arxiv.org/find/math/1/au:+Lewicki_G/0/1/0/all/0/1>. Abstract: We research proximinality of $\mu$-sequentially compact sets and $\mu$-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and $\mu$-compactness of the sets in Banach function spaces. Also the property $S$ is investigated in Fr\'echet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fr\'echet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces $d(w,1)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.02559

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Abstract of a paper by Roman Drnovšek, Marko Kandić
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “Positive operators as commutators of positive operators” by Roman Drnovšek<https://arxiv.org/find/math/1/au:+Drnovsek_R/0/1/0/all/0/1>, Marko Kandić<https://arxiv.org/find/math/1/au:+Kandic_M/0/1/0/all/0/1>. Abstract: It is known that a positive commutator $C=AB-BA$ between positive operators on a Banach lattice is quasinilpotent whenever at least one of $A$ and $B$ is compact. In this paper we study the question under which conditions a positive operator can be written as a commutator between positive operators. As a special case of our main result we obtain that positive compact operators on order continuous Banach lattices which admit order Pelczy\'nski decomposition are commutators between positive operators. Our main result is also applied in the setting of a separable infinite-dimensional Banach lattice $L_p(\mu)$ $(1<p<\infty)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.00882

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Abstract of a paper by Sara Botelho-Andrade, Peter G. Casazza, Desai Cheng, Tin Tran
by Bentuo Zheng (bzheng) 01 Aug '17

01 Aug '17

This is an announcement for the paper “The exact constant for the $\ell_1-\ell_2$ norm inequality” by Sara Botelho-Andrade<https://arxiv.org/find/math/1/au:+Botelho_Andrade_S/0/1/0/all/0/1>, Peter G. Casazza<https://arxiv.org/find/math/1/au:+Casazza_P/0/1/0/all/0/1>, Desai Cheng<https://arxiv.org/find/math/1/au:+Cheng_D/0/1/0/all/0/1>, Tin Tran<https://arxiv.org/find/math/1/au:+Tran_T/0/1/0/all/0/1>. Abstract: A fundamental inequality for Hilbert spaces is the $\ell_1-\ell_2$ -norm inequality which gives that for any $x\in H_n, \|\leq n^{-\sqrt{\|x\|_2}}$. But this is a strict inequality for all but vectors with constant modulus for their coefficients. We will give a trivial method to compute, for each $x$, the constant $c$ for which $\|x\|_1= cn^{-\sqrt{\|x\|_2}}$. Since this inequality is one of the most used results in Hilbert space theory, we believe this will have unlimited applications in the field. We will also show some variations of this result. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1707.00631

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Workshop in Geometry of Banach Spaces
by Bentuo Zheng (bzheng) 26 Jul '17

26 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 Casper Benjamin Freksen, Kasper Green Larsen
by Bentuo Zheng (bzheng) 17 Jul '17

17 Jul '17

This is an announcement for the paper “On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss Transforms” by Casper Benjamin Freksen<https://arxiv.org/find/math/1/au:+Freksen_C/0/1/0/all/0/1>, Kasper Green Larsen<https://arxiv.org/find/math/1/au:+Larsen_K/0/1/0/all/0/1>. Abstract: The Johnson-Lindenstrauss lemma is one of the corner stone results in dimensionality reduction. It says that for any set of vectors $X\subset R^n$, there exists a mapping $f: X\rightarrow R^M$ such that $f(X)$ preserves all pairwise distances between vectors in $X$ to within $(1\pm\epsilon)$ if $m=O(\epsilon-2lgN)$. Much effort has gone into developing fast embedding algorithms, with the Fast Johnson-Lindenstrauss transform of Ailon and Chazelle being one of the most well-known techniques. The current fastest algorithm that yields the optimal $m=O(\epsilon-2lgN)$ dimensions has an embedding time of $m=O(nlgN+\epsilon-2lg3N)$. An exciting approach towards improving this, due to Hinrichs and Vyb\'iral, is to use a random $m\times n$ Toeplitz matrix for the embedding. Using Fast Fourier Transform, the embedding of a vector can then be computed in $O(nlgm)$ time. The big question is of course whether $m=O(\epsilon-2lgN)$ dimensions suffice for this technique. If so, this would end a decades long quest to obtain faster and faster Johnson-Lindenstrauss transforms. The current best analysis of the embedding of Hinrichs and Vyb\'iral shows that $m=O(\epsilon-2lg2N)$dimensions suffices. The main result of this paper, is a proof that this analysis unfortunately cannot be tightened any further, i.e., there exists a set of $N$ vectors requiring $m=\Omega(\epsilon-2lg2N)$for the Toeplitz approach to work. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1706.10110

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Abstract of a paper by Francisco L. Hernández, Evgeny M. Semenov, Pedro Tradacete
by Bentuo Zheng (bzheng) 17 Jul '17

17 Jul '17

This is an announcement for the paper “Interpolation and extrapolation of strictly singular operators between $L_p$ spaces” by Francisco L. Hernández<https://arxiv.org/find/math/1/au:+Hernandez_F/0/1/0/all/0/1>, Evgeny M. Semenov<https://arxiv.org/find/math/1/au:+Semenov_E/0/1/0/all/0/1>, Pedro Tradacete<https://arxiv.org/find/math/1/au:+Tradacete_P/0/1/0/all/0/1>. Abstract: We study the interpolation and extrapolation properties of strictly singular operators between different $L_p$ spaces. To this end, the structure of strictly singular non-compact operators between $L_p-L_q$ spaces is analyzed. Among other things, we clarify the relation between strict singularity and the $L$-characteristic set of an operator. In particular, Krasnoselskii's interpolation theorem for compact operators is extended to the class of strictly singular operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1706.08682

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Abstract of a paper by Antonio Avilés, José Rodríguez, Pedro Tradacete
by Bentuo Zheng (bzheng) 17 Jul '17

17 Jul '17

This is an announcement for the paper “The free Banach lattice generated by a Banach space” by Antonio Avilés<https://arxiv.org/find/math/1/au:+Aviles_A/0/1/0/all/0/1>, José Rodríguez<https://arxiv.org/find/math/1/au:+Rodriguez_J/0/1/0/all/0/1>, Pedro Tradacete<https://arxiv.org/find/math/1/au:+Tradacete_P/0/1/0/all/0/1>. Abstract: The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1706.08147

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