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banach@mathdept.okstate.edu

February 2017

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Abstract of a paper by A. Aydın, E. Yu. Emelyanov, N. Erkurşun Özcan, M. A. A. Marabeh
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Compact-Like Operators in Lattice-Normed Spaces” by A. Aydın<https://arxiv.org/find/math/1/au:+Aydin_A/0/1/0/all/0/1>, E. Yu. Emelyanov<https://arxiv.org/find/math/1/au:+Emelyanov_E/0/1/0/all/0/1>, N. Erkurşun Özcan<https://arxiv.org/find/math/1/au:+Ozcan_N/0/1/0/all/0/1>, M. A. A. Marabeh<https://arxiv.org/find/math/1/au:+Marabeh_M/0/1/0/all/0/1>. Abstract: A linear operator $T$ between two lattice-normed spaces is said to be $p$-compact if, for any $p$-bounded net $x_{\alpha}$, the net $Tx_{\alpha}$ has a $p$-convergent subnet. $p$-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, AM-compact operators, etc. Similar to M-weakly and L-weakly compact operators, we define $p$-M-weakly and $p$-L-weakly compact operators and study some of their properties. We also study $up$-continuous and $up$-compact operators between lattice-normed vector lattices. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.03073

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Abstract of a paper by Vladimir Kadets, Miguel Martin, Javier Meri, Antonio Perez
by Bentuo Zheng (bzheng) 31 Jan '17

31 Jan '17

This is an announcement for the paper “Spear operators between Banach spaces” by Vladimir Kadets<https://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1>, Miguel Martin<https://arxiv.org/find/math/1/au:+Martin_M/0/1/0/all/0/1>, Javier Meri<https://arxiv.org/find/math/1/au:+Meri_J/0/1/0/all/0/1>, Antonio Perez<https://arxiv.org/find/math/1/au:+Perez_A/0/1/0/all/0/1>. Abstract: The aim of this manuscript is to study $\emph{spear operators}$: bounded linear operators $G$ between Banach spaces $X$ and $Y$ satisfying that for every other bounded linear operator $T: X\rightarrow Y$ there exists a modulus-one scalar $\omega$ such that $$\|G+\omega T\|=1+\|T\|$$. To this end, we introduce two related properties, one weaker called the alternative Daugavet property (if rank-one operators $T$ satisfy the requirements), and one stronger called lushness, and we develop a complete theory about the relations between these three properties. To do this, the concepts of spear vector and spear set play an important role. Further, we provide with many examples among classical spaces, being one of them the lushness of the Fourier transform on $L_1$. We also study the relation of these properties with the Radon-Nikod\'ym property, with Asplund spaces, with the duality, and we provide some stability results. Further, we present some isometric and isomorphic consequences of these properties as, for instance, that $\ell_1$ is contained in the dual of the domain of every real operator with infinite rank and the alternative Daugavet property, and that these three concepts behave badly with smoothness and rotundity. Finally, we study Lipschitz spear operators (that is, those Lipschitz operators satisfying the Lipschitz version of the equation above) and prove that (linear) lush operators are Lipschitz spear operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.02977

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