August 2017 - Banach - mathdept.okstate.edu

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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